Outcome annotation and minimax risk for discrete average treatment effects

CausalSmith

doi:10.5281/zenodo.22882683

CausalSmith · AI Causal Scientist — CSWP-2026-022 · Stat · pinned commit db544e5 · PDF · Lean code · Slides · AI review · arXiv · GitHub

Outcome Annotation and Minimax Risk for Discrete Average Treatment Effects

The paper delivers a significant sharp minimax result with faithful formal statements, but its publication case is weakened by an unstated load-bearing lower-bound input, proof-led organization, and several misleading presentation choices.

Main criticisms

Common-marginal lower bounds The paragraph titled “One-coordinate marked-Poisson input” identifies a “load-bearing analytic estimate,” but the manuscript supplies only an informal likelihood-expansion intuition rather than a named statement with exact hypotheses and a proof. The proof of the uniform-intensity construction invokes this estimate to obtain the geometric total-variation decay, so the reader-facing lower-bound argument is not self-contained even though the final theorem is verified.
global The manuscript is organized too closely to a generated proof ledger for a journal article. Examples include the full dependency inventory inside the “Binary length” definition, repeated calibration bookkeeping, and many long proofs whose main statistical ideas are obscured by measure-theoretic and constant-tracking details. This makes the contribution difficult to evaluate despite the strong main theorem.

9 findings — 2 major · 6 minor · 1 nit Read the entire review →

The paper may have been revised since this report. One automated referee’s estimate — not peer review.

Submitted: 17 Sep 2026 (v1) · last revised 20 Sep 2026 (v2)
Cite as: CSWP-2026-022 · https://doi.org/10.5281/zenodo.22882683

@techreport{causalsmith_stat_semisupervised_discrete_ate_annotat_0o7smr7,
  title       = {Outcome annotation and minimax risk for discrete average treatment effects},
  author      = {{CausalSmith}},
  year        = {2026},
  month       = sep,
  institution = {CausalSmith Working Papers},
  number      = {CSWP-2026-022},
  doi         = {10.5281/zenodo.22882683},
  url         = {https://causalsmith.org/papers/stat_semisupervised_discrete_ate_annotation_frontier_v1/},
  note        = {Version 2, revised 20 Sep 2026. Machine-generated; formal statements verified in Lean 4 at commit db544e5}
}

CausalSmith (2026). “Outcome annotation and minimax risk for discrete average treatment effects.” CausalSmith Working Paper CSWP-2026-022, version 2 (20 Sep 2026). https://doi.org/10.5281/zenodo.22882683

DOI 10.5281/zenodo.22882683 Cites version 2 of 20 Sep 2026. One DOI per paper; the record it opens lists every version.

Abstract

This paper characterizes the minimax value of outcome annotation in a finite-alphabet causal experiment. A sample of $n$ complete records $(X, A, Y)$ is augmented by $m$ additional records $(X, A)$ from the same population, where $X$ takes $d$ known values, $A$ and $Y$ are binary, and occupied covariate cells satisfy overlap at level $ϵ$ . For the average treatment effect under consistency and conditional exchangeability, the minimax mean-squared risk $R_{ϵ} (n, m, d)$ satisfies, uniformly over the unrestricted cellwise model, $R_{ϵ} (n, m, d) ≍_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} lo g ^{2} ( e n )}} .$ The rate separates outcome information from treatment–covariate marginal information: the $n^{- 1}$ term is the labeled-outcome contribution, while the alphabet-dependent term is governed by the combined $n + m$ treatment–covariate records. The upper bound is attained by a fully specified hybrid estimator that uses polynomial correction in light cells and stabilized inverse counts in heavy cells. The lower bound uses common-marginal priors whose induced treatment–covariate tables have the same distribution across hypotheses, so the auxiliary sample is aligned with the marginal-information component. The characterization yields exact consistency, parametric-rate, strict-improvement, binary-alphabet, inverse-count-baseline, and known-marginal-limit consequences for the annotation experiment.

Introduction

Chart-reviewed outcomes are often the scarce resource in observational causal studies, while baseline covariates and treatment assignments are available for many more units. This paper studies the resulting allocation problem in a finite-alphabet causal model. There are $n$ outcome-labeled records $(X, A, Y)$ and $m$ additional outcome-unlabeled records $(X, A)$ , drawn independently from the same population. The covariate $X$ takes values in a known alphabet of size $d$ , treatment and outcome are binary, and every occupied covariate cell satisfies overlap at level $ϵ$ . Under the usual potential-outcome consistency and conditional-exchangeability conditions, the target is the finite-alphabet average treatment effect $τ (P)$ in Definition 8.

The main result gives the finite-sample minimax frontier for this annotation experiment. For every $0 < ϵ < 1/2$ , Theorem 4 proves that the minimax mean-squared risk in Definition 11 obeys $R_{ϵ} (n, m, d) ≍_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} lo g ^{2} ( e n )}}$ for all $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ . The rate has two resources. Outcome marks enter through the $n^{- 1}$ term. Treatment–covariate frequencies enter through the second term, where the relevant sample size is the pooled marginal-information size $n + m$ . Thus the theorem gives a quantitative rule for how additional treatment–covariate records change the attainable accuracy of average-treatment-effect estimation in the unrestricted finite-cell model.

The upper bound is constructive. The estimator in Algorithm 1 splits the labeled sample into outcome, pilot, and factorial pools, splits the auxiliary sample into pilot and factorial pools, and then combines the treatment–covariate records for marginal estimation. Cells with small pilot counts use a Chebyshev polynomial correction and a factorial-moment lift, following the polynomial-approximation ideas used in sharp discrete functional estimation (Jiao et al., 2015; Wu et al., 2016) and the rare-cell perspective developed in sample amplification and unseen-species prediction (Valiant et al., 2017; Orlitsky et al., 2016). Cells with larger pilot counts use stabilized inverse-count weights. Theorem 2 shows that this total, finitely calibrated estimator attains the frontier rate uniformly over the model class.

The matching lower bound uses a common-marginal construction. Theorem 3 supplies priors whose induced random treatment–covariate marginal $P_{X A}$ , the full $2 d$ -cell table, has the same distribution across the two hypotheses, while the outcome regressions shift the target. This aligns the converse with the two-sample annotation design: auxiliary observations reveal the same random marginal object under both branches, and the remaining testing difficulty produces the label floor and the rare-cell approximation term. A separate parametric label floor in Lemma 22 supplies the $n^{- 1}$ component.

The frontier connects several familiar regimes. At $m = 0$ , the rate becomes $min {1, n^{- 1} + d^{2} / (n^{2} lo g^{2} (e n))}$ . This refines the bracket of Zeng et al. (2026): their plug-in upper bound $d^{2} / n^{2} + 1/ n$ holds over the unrestricted binary-outcome, discrete-covariate overlap class, while their minimax lower bound $d^{2} / {n^{2} lo g^{2} n} + 1/ n$ is stated for $d ≲ n lo g n$ ; they leave closing the logarithmic gap by a polynomial estimator as future work. The present result supplies that construction and, with auxiliary records, replaces the marginal-information sample size $n$ by $n + m$ . With fixed $n$ and $d$ , Theorem 5 shows that $R_{ϵ} (n, m, d)$ converges as $m \to \infty$ to the known-marginal risk in Definition 15, whose order is $n^{- 1}$ by Theorem 1. For sequences, Theorem 4 states that consistency is equivalent to $d_{n} / {(n + m_{n}) lo g (e n)} \to 0$ , and that parametric risk is equivalent to $d_{n} = O ((n + m_{n}) lo g (e n) / n)$ . The same theorem gives the exact criterion under which auxiliary treatment–covariate records improve the asymptotic order relative to the labeled-only endpoint.

The paper contributes to causal estimation under potential outcomes (Rubin, 1974; Rosenbaum et al., 1983; Imbens et al., 2015), semiparametric and doubly robust average-treatment-effect estimation (Robins et al., 1994; Hahn, 1998; Hirano et al., 2003; Bang et al., 2005; Farrell, 2015; Athey et al., 2018; Chernozhukov et al., 2018; Kennedy, 2016), and auxiliary-data methods for causal and distributional estimation, including semi-supervised treatment-effect, surrogate-outcome, missing-outcome, regression, and quantile settings (Cheng et al., 2021; Chakrabortty et al., 2019; Chakrabortty et al., 2024; Chakrabortty et al., 2022; Kallus et al., 2025; Anniwaer et al., 2025; Xu et al., 2025). Those literatures characterize identification, regular asymptotics, efficiency, and gains from structure or auxiliary measurements. The present analysis gives a finite-alphabet minimax benchmark for the binary-treatment, binary-outcome chart-review design under independent random outcome labeling, occupied-cell overlap, and unrestricted cellwise propensities and outcome regressions.

The appendix records supporting details for the displayed mathematical layer. The paper proceeds as follows. Section 2 positions the result within causal inference, semi-supervised estimation, and non-smooth discrete functional estimation. Section 3 defines the finite-alphabet annotation experiment, the causal target, and minimax risk. Section 4 states the frontier theorem, the known-marginal benchmark, and the common-marginal converse. Section 5 gives the algorithmic construction behind the upper bound. Section 6 discusses the resulting regimes, the inverse-count baseline, the binary-alphabet specialization, and future directions. Sections A, B, and C collect the upper-bound ingredients, common-marginal lower bounds, known-marginal limit, verification note, and proofs.

This paper studies a two-sample semi-supervised discrete causal experiment: a complete outcome-labeled sample of size $n$ is augmented by an outcome-unlabeled treatment-covariate sample of size $m$ on a finite covariate alphabet of size $d$ . In that experiment, the target is the average treatment effect under the usual potential-outcomes interpretation, while the statistical difficulty is governed by how outcome labels and treatment-covariate cell information are split across the two samples. The contribution is therefore most closely connected to five literatures: identification in potential-outcomes models, semiparametric efficiency for average treatment effects with missing or observed outcomes, high-dimensional causal inference under structure, semi-supervised treatment-effect estimation, and minimax estimation of non-smooth discrete functionals.

The identification framework follows the potential-outcomes tradition of Rubin (1974) and the propensity-score formulation of Rosenbaum et al. (1983); see also Imbens et al. (2015) for a systematic account. In that tradition, consistency and conditional exchangeability connect observed outcome regressions to the causal estimand. The present analysis specializes this framework to a finite covariate space and asks how accurately the resulting discrete average treatment effect can be estimated when additional observations reveal treatment and covariates but omit the outcome.

A large semiparametric literature characterizes regular estimation of average treatment effects under missingness, inverse weighting, augmentation, and doubly robust structure. Classical efficiency calculations include Robins et al. (1994), Hahn (1998), and Hirano et al. (2003), with augmented and robust estimation developed further by Bang et al. (2005). Modern high-dimensional versions combine orthogonal scores with flexible first-stage learning, as in Farrell (2015), Athey et al. (2018), and Chernozhukov et al. (2018); Kennedy (2016) gives a broad review of semiparametric causal estimation. Those results identify regular-asymptotic regimes in which nuisance estimation error can be controlled through smoothness, sparsity, sample splitting, or orthogonality. Here the object is the finite-alphabet minimax risk in an unrestricted discrete model, where rare treatment-covariate cells create a non-smooth estimation problem.

Work on high-dimensional and structured semi-supervised inference provides a complementary perspective. Zhang et al. (2019), Chakrabortty et al. (2018), and Deng et al. (2024) develop semi-supervised methods for means and regression targets in high-dimensional settings. Related robustness and prediction-assisted ideas appear in Angelopoulos et al. (2023). These papers show how auxiliary samples and predictive structure can improve estimation in rich covariate spaces. The present setting uses the finite alphabet itself as the primitive statistical object and characterizes the risk uniformly over the unrestricted cell-level causal model.

A related auxiliary-data literature studies how unlabeled samples, missing-outcome structure, transfer information, regression targets, quantile targets, and surrogate outcomes improve estimation. Cheng et al. (2021), Hou et al. (2025), Kato (2025), Chakrabortty et al. (2024), Anniwaer et al. (2025), and Xu et al. (2025) develop semi-supervised, transfer-style, or regular-asymptotic causal estimators, while Chakrabortty et al. (2019), Chakrabortty et al. (2022), and Kallus et al. (2025) analyze missing-outcome, quantile, and surrogate-outcome settings. Hou, Mukherjee, and Cai establish root- $n$ inference and efficiency for partially annotated treatment and response under structured nuisance estimation; Kato studies a two-sample unlabeled-covariate design through generalized Riesz regression and characterizes its efficiency bound. The present paper instead takes an unrestricted finite-cell model and asks for the global finite-sample minimax risk, including rare-cell regimes where regular asymptotics need not apply.

The discrete nature of the model also connects the analysis to minimax estimation of non-smooth functionals of distributions. For entropy, support-size, and related functionals, Jiao et al. (2015) and Wu et al. (2016) show that polynomial approximation helps determine sharp rates, while Valiant et al. (2017) and Orlitsky et al. (2016) develop sample-amplification and unseen-species perspectives on rare cells. The average treatment effect in the annotation experiment has an analogous cell-wise difficulty: estimating outcome means requires labeled observations, while estimating the treatment-covariate weights benefits from the larger treatment-covariate sample. The resulting minimax rate reflects that interaction between labeled outcome information and auxiliary marginal information.

The most direct benchmark is Zeng et al. (2026), who study the same unrestricted binary-outcome, discrete-covariate overlap class with only complete records. Their plug-in upper bound holds for all $d$ , whereas their minimax lower bound is stated for $d ≲ n lo g n$ ; the two differ by a logarithmic factor in the alphabet-dependent term. The present paper closes that gap at the supervised endpoint and extends the problem to separate outcome-labeled and outcome-unlabeled samples. Table 1 summarizes the sampling design, structural regime, target, and guarantee for the nearest comparisons.

Closest comparisons by auxiliary information, structural regime, target, and inferential guarantee.
Work	Auxiliary information	Regime and target	Guarantee
Zeng et al. (2026)	none	discrete covariates; ATE	supervised minimax bracket
Hou et al. (2025)	partially annotated treatment and response	structured nuisance models; ATE	root- $n$ inference and efficiency
Kato (2025)	unlabeled covariates	two-sample Riesz model; treatment effects	efficiency characterization
This paper	unlabeled $(X, A)$ records	unrestricted finite cells; ATE	exact finite-sample minimax order in $n, m, d$

Model and experiment

Throughout the paper, $x$ indexes a finite covariate cell, $a \in {0, 1}$ indexes a treatment arm, and sums over cells run over the known alphabet $[d] = {1, \dots, d}$ . Generic constants may depend on the overlap level $ϵ$ when this dependence is indicated, while sample-size dependence is displayed explicitly. Norms and asymptotic notation are used in their standard finite-dimensional sense, and all probability laws below live on finite measurable spaces unless stated otherwise.

The statistical object is an observed record with a finite covariate, a binary treatment, and a binary outcome. Writing $X$ for the covariate, $A$ for treatment, and $Y$ for the observed outcome, the complete record takes values in the following finite space.

Definition 1 [synth_11] (Complete observed-record space).

For each natural number $d$ , define the complete observed-record space by $Obs (d) : = [d] \times {0, 1} \times {0, 1} .$ An element of $Obs (d)$ is a triple $(X, A, Y)$ , where $X \in [d]$ is the finite-alphabet covariate, $A \in {0, 1}$ is the binary treatment indicator, and $Y \in {0, 1}$ is the binary observed outcome.

⊢ Lean

Let $P$ denote a probability law on this observed record space. For a cell $x$ , the covariate mass $p_{x}$ records how much population probability lies in that cell.

Definition 2 [synth_16] (Covariate cell mass).

For an observed-data law $P$ on $[d] \times {0, 1}^{2}$ and a covariate value $x \in [d]$ , define the covariate cell mass $p_{x} = a \in {0, 1} \sum y \in {0, 1} \sum P (X = x, A = a, Y = y) .$

⊢ Lean

We also write $s_{a x}$ for the treatment-covariate mass in arm $a$ . The treatment propensity $e_{x}$ is the conditional treatment probability in the cell, with the finite-space convention made explicit next.

Definition 3 [synth_10] (Cell treatment propensity).

For an observed-data law $P$ on $[d] \times {0, 1}^{2}$ and a covariate cell $x \in [d]$ , define the treatment propensity $e_{x}$ by $e_{x} := \frac{\sum _{y \in {0, 1}} P ( X = x , A = 1 , Y = y )}{\sum _{a \in {0, 1}} \sum _{y \in {0, 1}} P ( X = x , A = a , Y = y )} .$ The ratio is interpreted with the usual totalized real division convention, so it is $0$ when the denominator is $0$ .

⊢ Lean

Assumption 1 [ass:overlap] (Occupied-cell overlap).

For every covariate cell $x \in [d]$ with cell mass $p_{x} > 0$ , the propensity score satisfies $ϵ \leq e_{x} \leq 1 - ϵ .$

⊢ Lean

The occupied-cell overlap condition in Assumption 1 is the standard strong-overlap restriction on cells that occur with positive probability (Zeng et al., 2026). It keeps both treatment arms represented wherever the covariate distribution places mass and therefore gives the outcome regressions in the ATE functional a common observed-data interpretation.

The experiment combines complete outcome-labeled records with additional treatment-covariate records. We first record the sampling law for the labeled observations and then define the marginal law governing the auxiliary observations.

Assumption 2 [ass:labeled-iid] (Labeled sampling).

The labeled sample is distributed as $L_{n} \sim P^{\otimes n} .$

⊢ Lean

The labeled sampling condition in Assumption 2 is the standard independent and identically distributed sampling condition for complete records (Zeng et al., 2026). These observations supply the outcome information used to learn the arm-specific regressions.

Definition 4 [synth_14] (Treatment-covariate marginal).

For any $d$ and any observed-data law $P$ on $[d] \times {0, 1}^{2}$ , the treatment-covariate marginal $P_{X A}$ is the probability mass function on $[d] \times {0, 1}$ obtained by pushing $P$ forward under the projection $(x, a, y) \mapsto (x, a)$ : $P_{X A} = P \circ ((x, a, y) \mapsto (x, a))^{- 1} .$

⊢ Lean

The marginal $P_{X A}$ records the $2 d$ treatment-covariate cell probabilities while integrating out the outcome. The auxiliary sample $U_{m}$ is drawn from this same population marginal.

Assumption 3 [ass:auxiliary-iid] (Auxiliary sampling).

The auxiliary sample is distributed as $U_{m} \sim P_{X A}^{\otimes m} .$

⊢ Lean

Assumption 3 is the standard same-population outcome-unlabeled sampling condition (Chakrabortty et al., 2024). It lets the auxiliary records refine information about the treatment-covariate distribution relevant for weighting cell-level outcome contrasts.

Assumption 4 [ass:sample-independence] (Sample independence).

The labeled and auxiliary samples are independent: $L_{n} ⊥ ⊥ U_{m} .$

⊢ Lean

The independent two-sample design in Assumption 4 is standard in semisupervised treatment-effect settings with separate labeled and outcome-unlabeled samples (Chakrabortty et al., 2024). Under this design, the product experiment separates outcome information from additional marginal information.

The causal interpretation uses a finite potential-outcome overlay on the same observed-data law. Let $Y (a)$ denote the binary potential outcome under arm $a$ ; the following assumptions connect the observed-data functional to the causal average treatment effect in the usual potential-outcome framework (Rubin, 1974; Imbens et al., 2015).

Assumption 5 [ass:consistency] (Potential-outcome consistency).

The observed binary outcome equals the potential outcome under the realized treatment: $Y = Y (A)$ almost surely.

⊢ Lean

Assumption 5 is the standard consistency condition (Zeng et al., 2026). It identifies the observed outcome within each treatment arm with the corresponding potential outcome.

Assumption 6 [ass:conditional-exchangeability] (Conditional exchangeability).

The potential outcomes are conditionally independent of treatment given the finite covariate: $(Y (0), Y (1)) ⊥ ⊥ A ∣ X .$

⊢ Lean

Assumption 6 is the standard conditional exchangeability condition (Zeng et al., 2026). Together with Assumptions 5 and 1, it gives the observed conditional means their usual causal meaning within each finite covariate cell (Rosenbaum et al., 1983; Imbens et al., 2015).

More explicitly, let $Q$ be a full-data law for $(X, A, Y (0), Y (1))$ , and let $P$ be its observed-data pushforward under $(x, a, y_{0}, y_{1}) ⟼ (x, a, y_{a}) .$ Consistency and conditional exchangeability imply, for every occupied cell $x$ and arm $a$ , $E_{Q} {Y (a) ∣ X = x} = E_{Q} {Y (a) ∣ X = x, A = a} = E_{P} {Y ∣ X = x, A = a} .$ Overlap makes the last conditional mean identified in every occupied cell. Therefore $E_{Q} {Y (1) - Y (0)} = x = 1 \sum d p_{x} [E_{P} {Y ∣ X = x, A = 1} - E_{P} {Y ∣ X = x, A = 0}],$ which is exactly the observed-data functional defined below. Thus the minimax analysis is over observed laws $P$ , while the causal interpretation is supplied by any compatible full-data law $Q$ satisfying these assumptions.

We now collect the finite-space probability notation, the model class, and the target functional used in the risk analysis.

Definition 5 [synth_13] (Probability laws on a measurable space).

For a measurable space $Z$ , let $Prob (Z)$ denote the set of all probability measures on $Z$ . For the finite spaces used below, including $[d] \times {0, 1}^{2}$ and $[d] \times {0, 1}$ , the measurable structure is the full power set.

Definition 6 [def:model-class] (Model class

M_{d, ϵ}

For an overlap constant $ϵ$ and alphabet size $d$ , define $M_{d, ϵ} := {P \in Prob ([d] \times {0, 1}^{2}) : d \geq 2, 0 < ϵ < 1/2, P satisfies Assumption 1} .$

⊢ Lean

Thus $M_{d, ϵ}$ is the class of observed-data laws on the finite alphabet satisfying occupied-cell overlap at level $ϵ$ . All suprema in the main risk statements range over this unrestricted cell-level model.

For each arm and cell, the outcome regression $μ_{a x}$ is the conditional success probability computed from the observed law.

Definition 7 [synth_17] (Outcome regression).

For a law $P$ on $[d] \times {0, 1}^{2}$ , a treatment arm $a \in {0, 1}$ , and a covariate cell $x \in [d]$ , define $μ_{a x} (P) := \frac{P ( X = x , A = a , Y = 1 )}{\sum _{y \in {0, 1}} P ( X = x , A = a , Y = y )} .$ The quotient is interpreted as total real division, so $μ_{a x} (P) = 0$ when the treatment–covariate cell mass $\sum_{y \in {0, 1}} P (X = x, A = a, Y = y)$ is zero. In particular, $μ_{1 x} (P)$ and $μ_{0 x} (P)$ are obtained by taking $a = 1$ and $a = 0$ , respectively.

⊢ Lean

Definition 8 [def:ate-functional] (Average treatment effect functional).

For a finite alphabet size $d$ and an observed-data law $P$ on $[d] \times {0, 1}^{2}$ , define the average treatment effect functional by $τ (P) := x = 1 \sum d p_{x} (P) (μ_{1 x} (P) - μ_{0 x} (P)) .$ Here $p_{x} (P)$ is the covariate cell mass at $x$ , and $μ_{a x} (P)$ is the binary outcome regression in treatment arm $a$ and covariate cell $x$ , totalized by division in the arm cell.

⊢ Lean

The functional $τ (P)$ averages the within-cell treated-control contrast using the population covariate distribution. Under the potential-outcome conditions above, it is the finite-alphabet causal average treatment effect.

Finally, the annotation experiment and minimax risk formalize the statistical problem studied in the rest of the paper.

Definition 9 [def:annotation-experiment] (Annotation experiment

E_{n, m, d, ϵ}

Define the annotation experiment by $E_{n, m, d, ϵ} := {P^{\otimes n} \otimes P_{X A}^{\otimes m} : P \in M_{d, ϵ}, n \geq 1, m \geq 0, d \geq 2} .$ The product law corresponds to $n$ independent complete outcome-labeled records and $m$ independent additional outcome-unlabeled treatment-covariate records, with independence between the two samples.

⊢ Lean

Definition 9 defines $E_{n, m, d, ϵ}$ , the experiment in which outcome labels and treatment-covariate information enter through separate independent samples. This formulation makes the amount of outcome information and the amount of marginal information explicit.

The realized data live on the corresponding product sample space.

Definition 10 [synth_18] (Joint sample space).

For natural numbers $n, m, d$ , the joint sample space is $Sample (n, m, d) = ([n] \to [d] \times {0, 1} \times {0, 1}) \times ([m] \to [d] \times {0, 1}) .$ Equivalently, an element consists of $n$ complete records $(X, A, Y)$ and $m$ outcome-unlabeled records $(X, A)$ .

⊢ Lean

Definition 11 [def:minimax-risk] (Minimax risk

R_{ϵ} (n, m, d)

Define the minimax mean-squared risk by $R_{ϵ} (n, m, d) := T \in T_{n, m, d} in f P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [(T - τ (P))^{2}],$ where $T_{n, m, d}$ is the class of estimators measurable with respect to the labeled sample and auxiliary sample.

⊢ Lean

The risk $R_{ϵ} (n, m, d)$ measures the best possible uniform mean-squared accuracy for estimating the ATE functional in Definition 8 under the experiment in Definition 9. The estimator class $T_{n, m, d}$ contains all measurable procedures based on the two samples, so the minimax risk isolates the statistical difficulty created by finite cells, overlap, and the split between outcome-labeled and outcome-unlabeled information.

Main minimax results

The minimax analysis separates two sources of information in the annotation experiment. Outcome marks contribute through the $n$ complete records, while treatment–covariate frequencies contribute through the combined table information carried by the $n$ labeled records and the $m$ auxiliary records. The resulting rate uses the total marginal-information size $N$ and the labeled-sample logarithmic scale.

Definition 12 [def:frontier-rate-handle] (Frontier rate

r_{ϵ} (n, m, d)

Put $N = n + m, ℓ_{n} = lo g (e n) .$ The unequal-information annotation-frontier rate is $r_{ϵ} (n, m, d) := min {1, \frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}}} .$

⊢ Lean

The rate $r_{ϵ} (n, m, d)$ has two components. The term $1/ n$ is the labeled-outcome contribution. The term $d^{2} / (N^{2} ℓ_{n}^{2})$ is the discrete marginal-estimation contribution, where the alphabet dimension is compared with the amount of treatment–covariate information available from both samples.

The upper construction and the converse both use finite calibration devices. For the estimator, the calibration fixes dyadic overlap, polynomial degree, light-cell scale, and the data splits used below.

Definition 13 [synth_7] (Finite calibration scales for the mixed estimator).

Set $A_{⋆} = 7056, c^{\circ} = \frac{1}{256} .$ For $0 < ϵ < 1/2$ , let $j_{ϵ}$ be the least integer $j \geq 1$ such that $2^{- j} \leq ϵ$ , and define the dyadic overlap certificate $\overset{ϵ}{ˉ} (ϵ) = 2^{- j_{ϵ}} .$ Then $0 < \overset{ϵ}{ˉ} (ϵ) \leq ϵ < 2 \overset{ϵ}{ˉ} (ϵ)$ . Let $H_{ϵ}$ be the least positive integer $H$ satisfying $H \geq 4, \frac{H}{4} \geq 15 + \frac{24}{c ^{\circ}}, H c^{\circ} \overset{ϵ}{ˉ} (ϵ) \geq 672, H c^{\circ} \overset{ϵ}{ˉ} (ϵ) \geq 24.$ For $n \geq 1$ , put $h_{n} = 1 + ⌈ lo g_{2} n ⌉, L (n) = max {2, ⌊ c^{\circ} h_{n} ⌋} .$ For $n \geq 24$ and $m \geq 0$ , define $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ \frac{n - M _{0}}{2} ⌋, n_{f} = n - M_{0} - n_{p},$ $m_{p} = ⌊ m /2 ⌋, m_{f} = m - m_{p}, M_{p} = n_{p} + m_{p}, M_{f} = n_{f} + m_{f},$ and $t_{p} = \frac{M _{p}}{8}, t = \frac{M _{f}}{8} .$ The light-cell scale and pilot-count threshold are $B (n, m, ϵ) = \frac{H _{ϵ} L ( n )}{min ( t _{p} , t )}, k_{0} (n, m, ϵ) = ⌊ \frac{t _{p} B ( n , m , ϵ )}{4} ⌋ .$ For $n < 24$ , the calibration predicate using these quantities is declared false.

The lower-bound argument compares two randomized laws through their induced distribution over the treatment–covariate table. The following notation records that marginal distribution for a branch prior.

Definition 14 [synth_12] (Induced law of the treatment--covariate marginal).

Let $b \in {0, 1}$ and let $Π_{b}$ be a probability measure on $M_{d, ϵ}$ . If $P \sim Π_{b}$ , define $L_{Π_{b}} (P_{X A}) := (P \mapsto P_{X A})_{#} Π_{b} .$ Thus, for every measurable set $B \subseteq Prob ([d] \times {0, 1})$ , $L_{Π_{b}} (P_{X A}) (B) = Π_{b} {P \in M_{d, ϵ} : P_{X A} \in B} .$ Equivalently, $L_{Π_{b}} (P_{X A})$ is the distribution of the full $2 d$ -cell treatment–covariate marginal table induced when the complete-data law $P$ is drawn from $Π_{b}$ .

A useful benchmark gives the target risk when the treatment–covariate marginal is observed exactly. In that experiment, procedures may use the labeled sample together with the exact $P_{X A}$ table.

Definition 15 [def:known-marginal-risk] (Known-marginal minimax risk).

For nonnegative integers $n$ and $d$ and a real overlap parameter $ϵ$ , the known-marginal minimax risk is $R_{ϵ}^{known} (n, d) := T \in T_{n, d}^{known} in f P \in M_{d, ϵ} sup \int (T (L_{n}, P_{X A}) - τ (P))^{2} d P^{\otimes n} .$ Here $M_{d, ϵ}$ is the model class in Definition 6, $τ (P)$ is the average treatment effect functional in Definition 8, and $T_{n, d}^{known}$ is the class of measurable estimators $T$ based on the labeled sample together with the exact treatment-covariate table $P_{X A}$ , with $T (L_{n}, P_{X A}) \in [- 1, 1]$ for every input.

⊢ Lean

Here $T_{n, d}^{known}$ is the estimator class for the known-marginal problem, and $R_{ϵ}^{known} (n, d)$ records the corresponding minimax mean-squared error.

Theorem 1 [thm:known-marginal-boundary] (Known marginal boundary).

Suppose the overlap constant $ϵ$ satisfies $0 < ϵ < 1/2$ . For the known-marginal minimax risk $R_{ϵ}^{known} (n, d)$ defined in Definition 15, there exist real constants $c_{ϵ}$ and $C_{ϵ}$ with $0 < c_{ϵ} \leq C_{ϵ}$ such that, for every $n, d \in N$ with $n \geq 1$ and $d \geq 2$ , $\frac{c _{ϵ}}{n} \leq R_{ϵ}^{known} (n, d) \leq \frac{C _{ϵ}}{n} .$

⊢ Lean

The benchmark establishes the labeled-sample boundary when the full marginal table is available. Once $P_{X A}$ is known, the remaining uniform risk is of order $1/ n$ , so additional treatment–covariate observations move the original annotation problem toward this labeled-outcome scale.

The upper bound uses an explicit hybrid estimator. Its calibration first fixes a binary-length scale and a clipping map for the final statistic. The long Context item in the next generated definition records formal dependencies and may be skipped on a first reading. Also, the type $T_{n, m, d}$ in the estimator display denotes precisely the map from the two sample spaces to $R$ displayed immediately below it.

Definition 16 [synth_8] (Binary length

b_{n}

(Context.) Work under Definition 22, Definition 21, Definition 17, Definition 24, Definition 20, Definition 26, Definition 13, Definition 25, Definition 3, Definition 1, Definition 14, Definition 5, Definition 4, Definition 2, Definition 7, Definition 10, Definition 23, Definition 28, Definition 29, Definition 18, Assumption 1, Assumption 2, Assumption 3, Assumption 4, Assumption 5, Assumption 6, Definition 6, Definition 8, Definition 9, Definition 11, Definition 12, Algorithm 1, Definition 19, Definition 27, Definition 30, Definition 15, Theorem 1, Theorem 2, Theorem 7, Lemma 5, Lemma 14, Lemma 17, Lemma 22, Lemma 1, Lemma 3, Lemma 4, Lemma 15, Lemma 29, Lemma 18, Lemma 19, Lemma 25, Lemma 6, Lemma 28, Lemma 7, Lemma 8, Lemma 9, Lemma 10, Lemma 11, Lemma 12, Lemma 23, Lemma 26, Theorem 3, Theorem 4, Lemma 2, Theorem 6, Lemma 13, Lemma 16, Lemma 24, Lemma 27, Lemma 20, Lemma 21, and Theorem 5.
(Integer.) Let $n \geq 1$ be an integer.

Define $b_{n} := 1 + ⌈ lo g_{2} n ⌉ .$ Equivalently, in the notation of Definition 13, $b_{n} = h_{n}$ .

Definition 17 [synth_3] (Clipping to the unit interval).

For $z \in R$ , define $clip_{[- 1, 1]} (z) = max {- 1, min {1, z}} .$ Equivalently, $clip_{[- 1, 1]} (z)$ is the Euclidean projection of $z$ onto the interval $[- 1, 1]$ .

The estimator itself combines a light-cell polynomial correction with a heavy-cell inverse-count correction. The construction is total: outside its finite calibration region it returns zero, and inside that region it averages a clipped Poisson-prefix statistic. For orientation before the full display, if $W_{a, L} (s_{0}, s_{1}) = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} s_{0}^{r_{0}} s_{1}^{r_{1}},$ then its factorial lift at intensity $t$ is $W_{a, L} (K_{0}, K_{1}) = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} \frac{( K _{0} ) _{r_{0}} ( K _{1} ) _{r_{1}}}{t ^{r_{0} + r_{1}}} .$ The complete definition and its moment identity appear in Definition 24.

Algorithm 1 [def:mixed-estimator-handle] (Mixed estimator definition).

For $n, m, d \in N$ and a real overlap constant $ϵ$ , define $τ_{n, m, d}^{mix} : T_{n, m, d}$ as the following map from $([d] \times {0, 1} \times {0, 1})^{n} \times ([d] \times {0, 1})^{m}$ to $R$ . Put $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ \frac{n - M _{0}}{2} ⌋, n_{f} = n - M_{0} - n_{p},$ $m_{p} = ⌊ m /2 ⌋, m_{f} = m - m_{p}, M_{p} = n_{p} + m_{p}, M_{f} = n_{f} + m_{f},$ and $u = \frac{M _{0}}{8}, t_{p} = \frac{M _{p}}{8}, t = \frac{M _{f}}{8} .$ Let $b_{n}$ denote the binary-length calibration quantity used here, and set $L = L (n), B = B (n, m, ϵ), k_{0} = k_{0} (n, m, ϵ),$ with $\overset{ϵ}{ˉ} = \overset{ϵ}{ˉ} (ϵ)$ , $A_{⋆} = 7056$ , and $c^{\circ} = 1/256$ . The calibration predicate $C_{ϵ} (n, m)$ holds exactly when all of the following finite inequalities hold:

(Sample size.) $24 \leq n$ .
(Block proportions.) $\frac{1}{32} \leq \frac{u}{n} \leq \frac{1}{24}, \frac{1}{32} \leq \frac{t _{p}}{n + m} \leq \frac{3}{32}, \frac{1}{32} \leq \frac{t}{n + m} \leq \frac{3}{32}, \frac{1}{3} \leq \frac{t}{t _{p}} \leq 3.$
(Degree and scale.) $\frac{c ^{\circ} b _{n}}{2} \leq L, 8 \leq t_{p} B, L \leq tB .$
(Polynomial size.) $L^{4} A_{⋆}^{2 L} \leq n, L^{6} A_{⋆}^{2 L} \leq n .$
(Tail calibration.) $15 L + 12 b_{n} \leq \frac{t _{p} B}{4}, n^{12} \leq (1 + \overset{ϵ}{ˉ} \frac{t}{t _{p}})^{k_{0}}, 12 b_{n} \leq \overset{ϵ}{ˉ} tB .$

If $C_{ϵ} (n, m)$ fails, set $τ_{n, m, d}^{mix} (L_{n}, U_{m}) = 0.$ If $C_{ϵ} (n, m)$ holds, set $τ_{n, m, d}^{mix} (L_{n}, U_{m}) = r_{0} = 0 \sum M_{0} r_{p} = 0 \sum M_{p} r_{f} = 0 \sum M_{f} Pr {Poi (u) = r_{0}} Pr {Poi (t_{p}) = r_{p}} Pr {Poi (t) = r_{f}} Φ_{ϵ} (L_{n}, U_{m}; r_{0}, r_{p}, r_{f}),$ where $Φ_{ϵ} (L_{n}, U_{m}; r_{0}, r_{p}, r_{f}) = clip_{[- 1, 1]} x \in [d] \sum Z_{x} .$ For each $x \in [d]$ , let $J_{x}$ be the pilot-prefix count in cell $x$ , let $S_{a x}$ be the outcome-block prefix count with covariate $x$ , treatment $a$ , and outcome $1$ , and let $K_{a x}$ be the factorial-pool prefix count with covariate $x$ and treatment $a$ . Then $Z_{x} = ⎩ ⎨ ⎧ \frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}, \frac{S _{1 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{1 x} + 1} - \frac{S _{0 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{0 x} + 1}, J_{x} \leq k_{0}, k_{0} < J_{x} .$ Here $W_{a, L, x}$ is the factorial lift, at degree $L$ , scale $B$ , and intensity $t$ , of the light-cell polynomial weight $W_{a, L} (s_{0}, s_{1}) = \frac{s _{0} + s _{1}}{B} G_{L} (s_{a} / B),$ where $H_{L} (z) = \frac{1 - T _{L} ( 1 - 2 z )}{2 L ^{2}}, E_{L} (z) = \frac{H _{L} ( z )}{z}, G_{L} (z) = \frac{1 - E _{L} ( z )}{z},$ with polynomial continuation at zero.

⊢ Lean

The statistic $τ_{n, m, d}^{mix}$ uses labeled outcomes through the $S_{a x}$ counts and uses the larger treatment–covariate pool through the pilot and factorial counts. Light cells receive the polynomial correction familiar from discrete functional estimation (Jiao et al., 2015; Wu et al., 2016), while heavy cells use direct inverse-count weighting.

Theorem 2 [thm:uniform-mixed-upper] (Uniform mixed upper bound).

Fix an overlap constant $ϵ$ with $0 < ϵ < 1/2$ . There exists a constant $C_{ϵ} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , the total computable hybrid estimator $τ_{n, m, d}^{mix}$ of Algorithm 1 satisfies $P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq C_{ϵ} r_{ϵ} (n, m, d),$ where $τ (P)$ is the average treatment effect functional in Definition 8, and $r_{ϵ} (n, m, d)$ is the annotation-frontier rate in Definition 12.

⊢ Lean

The upper bound establishes that the displayed rate is attainable uniformly over the model class. Its form reflects the experiment: increasing $m$ improves the marginal-frequency term through $N = n + m$ , while the labeled-outcome term remains governed by $n$ . Here total computable means that the displayed finite sums define an everywhere-defined statistic, including at calibration failure; implementation-oriented scope is discussed in Section 6.

The matching converse is organized through priors with a common induced marginal. In the rare-cell regime, the construction below generates a shared random treatment–covariate table and perturbs the treated outcome regression across two branches.

Definition 18 [synth_22] (Common-marginal finite-atom construction).

Fix $0 < ϵ < 1/2$ , an integer $L \geq 2$ , scales $B > 0$ and $a > 0$ , and write $κ = \frac{1 - 2 ϵ}{ϵ}, I = [a / κ, B] .$ Let $σ$ be a finite signed measure on $I$ with total variation one, let $h = d σ / d ∣ σ ∣$ , and define a probability law $ν$ on ${0} \cup I$ by $ν (d p) = \frac{a}{p + a} ∣ σ ∣ (d p) (p \in I), ν ({0}) = 1 - \int_{I} \frac{a}{p + a} ∣ σ ∣ (d p) .$ For an integer $k \leq d - 1$ , the finite-atom construction has atom-generator law $Γ = ν^{\otimes k}$ on $w = (p_{1}, \dots, p_{k})$ . Given $w$ , form $k$ rare cells and one reservoir cell. For each rare cell with $p_{i} > 0$ , set $s_{i} = ϵ (p_{i} + a), e_{i} = \frac{s _{i}}{p _{i}},$ put $μ_{0 i} = 0$ , and set the treated outcome regression in branch $b \in {0, 1}$ to $μ_{1 i}^{(b)} = \frac{1 + ( 2 b - 1 ) h ( p _{i} )}{2} .$ At atoms with $p_{i} = 0$ , assign the same legal propensity and outcome values in both branches. The reservoir has raw mass $p_{*} (w) = 1 - k E_{ν} [p],$ propensity $1/2$ , and outcome regressions equal to zero. Alphabet cells not used by the $k$ rare cells or the reservoir have mass zero.

The raw total mass is $S (w) = p_{*} (w) + i = 1 \sum k p_{i} .$ The branch law $P_{b} (w)$ is obtained by dividing every raw cell mass by $S (w)$ and keeping the propensities and outcome regressions just specified. The corresponding raw target is $r_{b} (w) = i = 1 \sum k p_{i} μ_{1 i}^{(b)},$ so the normalized target value of the generated law is $r_{b} (w) / S (w)$ . The branch priors are the pushforwards $Π_{b} = Γ \circ P_{b}^{- 1}, b \in {0, 1} .$ Thus the two branches use the same generated rare-cell masses, reservoir mass, propensities, normalization factor, and treatment–covariate table, while differing only in the branch sign of the treated rare-cell outcome regressions.

The finite-atom construction makes the shared marginal table explicit: the branches draw the same rare-cell masses and propensities, then separate the targets through outcome regressions. The common-marginal handle chooses between this rare-cell recipe and a parametric point-mass regime according to the comparison in the following definition.

Definition 19 [def:common-marginal-prior-handle] (Common-marginal prior handle).

Fix $n, m, d \in N$ , $ϵ \in R$ , and a calibration consisting of constants $C_{ϵ}, C_{ϵ}^{\circ}, b_{0}, b_{ϵ}, γ_{ϵ}, c_{ϵ}^{\circ}, c_{ϵ}, C_{d}$ with all eight constants positive and $c_{ϵ} \leq 1/4$ . A common-marginal prior handle at $(n, m, d)$ consists of integers $L, k$ , real numbers $B, a$ , and a calibrated recipe supplying two induced priors, denoted $Π_{0} (H)$ and $Π_{1} (H)$ , on laws on $[d] \times {0, 1}^{2}$ , subject to $L = ⌈ C_{ϵ}^{\circ} lo g (e n) ⌉, B = \frac{b _{0} L}{n + m}, a = \frac{γ _{ϵ} B}{L ^{2}},$ and $k = min {d - 1, ⌊ c_{ϵ}^{\circ} (n + m) L ⌋} .$ The recipe uses the finite-atom construction exactly when $min {1, \frac{d ^{2}}{( n + m ) ^{2} lo g ( e n ) ^{2}}} > \frac{C _{ϵ}}{n} .$ For a handle $H$ and a branch $b \in {0, 1}$ , the common-marginal prior is the branch-indexed measure $Π_{b} (H) = {Π_{1} (H), Π_{0} (H), b = 1, b = 0.$

⊢ Lean

The two branches $Π_{0}, Π_{1}$ are calibrated so that the auxiliary sample sees the same random marginal table under both hypotheses. Hence the distinguishing information in the lower-bound experiment must pass through the labeled outcomes.

Theorem 3 [thm:common-marginal-converse] (Common marginal converse).

Let $ϵ$ , the overlap parameter, satisfy $0 < ϵ < 1/2$ . Then there exist a constant $c_{ϵ} > 0$ and a collection of $ϵ$ -dependent calibration constants as in Definition 19 such that, for every $n, m, d \in N$ with $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , the regime-dependent handle in Definition 19 supplies priors $Π_{0}$ and $Π_{1}$ satisfying $L_{Π_{0}} (P_{X A}) = L_{Π_{1}} (P_{X A})$ and $c_{ϵ} r_{ϵ} (n, m, d) \leq R_{ϵ} (n, m, d),$ where $r_{ϵ} (n, m, d)$ is the frontier rate in Definition 12 and $R_{ϵ} (n, m, d)$ is the minimax risk in Definition 11.

⊢ Lean

The converse gives the complementary half of the sample-size separation. Matching the induced $P_{X A}$ law across the two priors makes the auxiliary records contribute through the marginal-frequency side of the construction; outcome-marked observations remain the resource that controls the $1/ n$ component.

Combining the attainable estimator with the common-marginal converse yields the sharp risk statement and its sequence consequences.

Theorem 4 [thm:sharp-annotation-frontier] (Sharp annotation frontier).

For every overlap parameter $ϵ$ satisfying $0 < ϵ < 1/2$ , there exist constants $c_{ϵ}$ and $C_{ϵ}$ , depending only on $ϵ$ , with $0 < c_{ϵ} \leq C_{ϵ},$ such that the following assertions hold.

(Finite-sample bracket.) For every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , $c_{ϵ} r_{ϵ} (n, m, d) \leq R_{ϵ} (n, m, d) \leq C_{ϵ} r_{ϵ} (n, m, d),$ where, with $N = n + m$ and $ℓ_{n} = lo g (e n)$ , $r_{ϵ} (n, m, d) = min {1, \frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}}} .$

(Unannotated endpoint.) For every $n \geq 1$ and every $d \geq 0$ , $r_{ϵ} (n, 0, d) = min {1, \frac{1}{n} + \frac{d ^{2}}{n ^{2} lo g ^{2} ( e n )}} .$

(Known-marginal limit.) For every fixed $n \geq 1$ and $d \geq 2$ , $m \to \infty lim R_{ϵ} (n, m, d) = R_{ϵ}^{known} (n, d) .$

(Sequence regimes.) For any sequences $(m_{n})_{n \geq 1}$ and $(d_{n})_{n \geq 1}$ of nonnegative integers satisfying $d_{n} \geq 2$ for every $n \geq 1$ , set $N_{n} = n + m_{n}$ and $ℓ_{n} = lo g (e n)$ . Then $R_{ϵ} (n, m_{n}, d_{n}) \to 0 ⟺ \frac{d _{n}}{N _{n} ℓ _{n}} \to 0,$ $R_{ϵ} (n, m_{n}, d_{n}) = O (n^{- 1}) ⟺ d_{n} = O (\frac{N _{n} ℓ _{n}}{n}),$ and $R_{ϵ} (n, m_{n}, d_{n}) = o (R_{ϵ} (n, 0, d_{n}))$ if and only if both $\frac{d _{n}}{n ℓ _{n}} \to \infty$ and $\frac{d _{n} / ( N _{n} ℓ _{n} )}{min { 1 , d _{n} / ( n ℓ _{n} )}} \to 0.$

⊢ Lean

Theorem 4 characterizes the minimax risk up to constants depending only on the overlap parameter $ϵ$ . In its generated statement, the phrase unannotated endpoint means $m = 0$ , namely the labeled-only endpoint with no auxiliary outcome-unlabeled records. The consistency threshold compares the alphabet size with $N_{n} ℓ_{n}$ ; the parametric-rate threshold allows $d_{n}$ to grow as $N_{n} ℓ_{n} / n$ ; and the strict-improvement criterion identifies when auxiliary treatment–covariate records improve on that labeled-only endpoint. The theorem therefore gives a theoretical sample-size frontier for the two sources of information; its conservative calibrated witness is not an operational allocation rule.

The known-marginal bullet in Theorem 4 is a summary pointer. The following formally primary statement records that limiting link with the known-treatment--covariate-marginal benchmark experiment and gives the exact large- $m$ endpoint for fixed labeled sample size and alphabet size.

Theorem 5 [thm:known-marginal-limit] (Known marginal limit).

Fix an overlap parameter $ϵ$ , a labeled sample size $n$ , and a finite covariate alphabet size $d$ . Suppose

(Overlap.) $0 < ϵ < 1/2$ .

(Labeled size.) $n \geq 1$ .

(Alphabet size.) $d \geq 2$ .

Then the minimax mean-squared risk $R_{ϵ} (n, m, d)$ from Definition 11 converges, as the auxiliary sample size $m \to \infty$ , to the known-marginal minimax risk $R_{ϵ}^{known} (n, d)$ from Definition 15: $m \to \infty lim R_{ϵ} (n, m, d) = R_{ϵ}^{known} (n, d) .$

⊢ Lean

The limit is the formal version of the marginal-information interpretation. As the auxiliary sample grows with $n$ and $d$ fixed, the treatment–covariate table is learned to the known-marginal benchmark, and the risk converges to the $n^{- 1}$ boundary in Theorem 1.

The hybrid estimator

This section gives an algorithmic reading of the estimator used in Algorithm 1. The construction separates three statistical roles that are intertwined in the target functional: the labeled records supply outcome marks, the combined treatment–covariate records estimate marginal cell frequencies, and the calibration chooses when a cell is treated by polynomial extrapolation rather than by direct inverse-count weighting.

The first step is deterministic sample splitting. The labeled sample is divided into an outcome block of size $M_{0}$ , a pilot contribution of size $n_{p}$ , and a factorial contribution of size $n_{f}$ . The auxiliary sample is divided into pilot and factorial contributions of sizes $m_{p}$ and $m_{f}$ . Pooling the corresponding labeled and auxiliary treatment–covariate records gives pilot and factorial pools of sizes $M_{p}$ and $M_{f}$ . This split makes the outcome counts independent of the treatment–covariate counts used to choose and evaluate the weights, matching the independence structure required by the risk analysis.

The finite calibration in Algorithm 1 then fixes the polynomial degree $L$ , the light-cell scale $B$ , and the pilot threshold $k_{0}$ . The predicate $C_{ϵ} (n, m)$ is a computable list of inequalities involving these quantities, the dyadic overlap certificate $\overset{ϵ}{ˉ}$ , and the prefix intensities $u, t_{p}, t$ . When the predicate holds, the selected scales keep the polynomial coefficients and Poisson tails within the finite-sample ranges used by the upper-bound argument. When it fails, the estimator is defined by the zero rule, which keeps the procedure total for every admissible $n, m, d$ .

The estimator uses Poisson-prefix derandomization to express an independent-Poisson statistic as a deterministic function of the fixed samples. For each triple of prefix lengths $(r_{0}, r_{p}, r_{f})$ , the rule keeps the first $r_{0}$ observations in the outcome block, the first $r_{p}$ observations in the pilot pool, and the first $r_{f}$ observations in the factorial pool, then averages the resulting clipped statistic against the three Poisson probabilities. This finite sum is the conditional expectation of the randomized prefix statistic, so the implemented estimator remains an ordinary measurable statistic of the observed data.

For each covariate cell $x$ , the statistic forms three counts. The pilot count $J_{x}$ decides the regime. The outcome counts $S_{1 x}$ and $S_{0 x}$ count labeled successes in the two treatment arms within the outcome prefix. The factorial counts $K_{1 x}$ and $K_{0 x}$ count treatment–covariate observations in the factorial prefix. Thus outcome information enters only through the success counts, while the larger annotated and unannotated treatment–covariate pool enters through $J_{x}$ and $K_{a x}$ .

Cells with $J_{x} \leq k_{0}$ use the light-cell polynomial weight. The Chebyshev component $T_{L}$ defines $H_{L}$ , its continuation $E_{L}$ , and the correction polynomial $G_{L}$ , giving the weight polynomial $W_{a, L}$ in Algorithm 1. Its factorial lift $W_{a, L, x}$ replaces monomials in the unknown cell probabilities by falling-factorial count statistics. This applies the polynomial-approximation strategy used in sharp discrete functional estimation (Jiao et al., 2015; Wu et al., 2016) to the inverse propensity-type weights needed for the average treatment effect, with the rare-cell organization motivated by sample-amplification and unseen-species perspectives (Valiant et al., 2017; Orlitsky et al., 2016).

Cells with $J_{x} > k_{0}$ use the heavy-cell inverse-count rule. In those cells, the factorial prefix supplies enough treatment–covariate mass for the ratios $(K_{0 x} + K_{1 x} + 1) / (K_{a x} + 1)$ to act as stabilized inverse-count weights. The added ones keep the weights finite at every realized count configuration, while the pilot threshold routes cells with limited marginal information to the polynomial branch.

Finally, Algorithm 1 sums the cell contributions and applies $clip_{[- 1, 1]}$ . The clipping matches the natural range of the average treatment effect functional and controls exceptional prefix realizations without changing the structure of the light- and heavy-cell rules. The resulting statistic, $τ_{n, m, d}^{mix}$ , is therefore a fully specified finite-sample estimator built from deterministic splitting, calibrated polynomial extrapolation, inverse-count stabilization, and bounded aggregation.

Theorem 2 is the performance guarantee attached to this construction. Under the overlap model and product-sample experiment stated earlier, the calibrated estimator attains the rate in Definition 12 uniformly over the model class. The two pieces of the rate correspond to the two resources used by the algorithm: labeled outcomes determine the $1/ n$ component, while treatment–covariate counts from both samples determine the marginal-frequency component through $n + m$ .

Proof roadmap.

Poissonization first makes the outcome, pilot, and factorial counts independent. The light-cell argument uses Chebyshev approximation and unbiased factorial moments; the heavy-cell argument uses the exact stabilized inverse-count mean and moment bounds. Pilot tails control routing errors, and summing the cellwise bias and variance bounds yields the Poissonized risk. Prefix conditioning then transfers that bound to the fixed samples. For the converse, moment-matched common-marginal priors keep the auxiliary table indistinguishable while separating the ATE, and a separate binary submodel supplies the $1/ n$ label floor. The dependency map at the start of Section A links these ideas to the detailed certificates.

Interpretation and extensions

The main minimax theorem yields three interpretable sequence consequences. In the notation of Definition 12, the risk is consistent precisely along sequences for which the labeled sample grows and the treatment–covariate information is sufficiently rich relative to the alphabet size. The first component, $1/ n$ , is the irreducible cost of observing binary outcomes; the second component, $d^{2} / {(n + m)^{2} ℓ_{n}^{2}}$ , is the cost of learning enough about the treatment–covariate distribution to stabilize the inverse-probability structure of Definition 8. Thus the auxiliary sample improves risk through the marginal table rather than through direct outcome information.

The parametric regime is the portion of the experiment in which the marginal-frequency component is of order $1/ n$ or smaller. There the sharp rate in Theorem 4 matches the labeled-outcome benchmark up to constants depending only on the overlap constant. In that range, the hybrid estimator of Algorithm 1 uses the auxiliary treatment–covariate records to make the nuisance marginal problem small enough that the remaining mean-squared error is governed by outcome sampling.

A complementary regime is where annotation yields strict improvement over the supervised endpoint. With $m = 0$ , the marginal table is learned only from the labeled sample, and the second term in Definition 12 is $d^{2} / (n^{2} ℓ_{n}^{2})$ . Adding outcome-unlabeled treatment–covariate records replaces $n$ by $n + m$ in that component. Consequently, when the alphabet is large enough for the marginal-frequency term to matter and the auxiliary sample makes $n + m$ substantially larger than $n$ , Theorem 4 gives a quantitative risk reduction inside the same overlap model and target functional. This is the sense in which outcome annotation has value here: it converts unlabeled treatment–covariate information into a sharper stabilized estimate of the average treatment effect.

The hybrid rule in Algorithm 1 is calibrated to attain the logarithmic rate in Definition 12. A simpler comparison is obtained by using inverse-count stabilization everywhere, without the light-cell polynomial correction. The next definition records this baseline in the same sample-splitting and Poisson-prefix format, so that the comparison isolates the role of the polynomial branch. Every quotient in this definition uses totalized real division: when $u = 0$ , division by $u$ evaluates to zero. Consequently the clipped prefix statistic remains defined for $n = 1, 2$ , where $M_{0} = u = 0$ .

Definition 20 [synth_5] (Poisson-prefix inverse-count estimator).

For $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , put $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ \frac{n - M _{0}}{2} ⌋, n_{f} = n - M_{0} - n_{p}, m_{p} = ⌊ m /2 ⌋, m_{f} = m - m_{p},$ and $u = M_{0} /8$ . Given a labeled sample $L_{n}$ and an auxiliary sample $U_{m}$ , define $τ_{n, m, d}^{IC, prefix} (r_{0}, r_{f}) = 0$ when $r_{0} > M_{0}$ or $r_{f} > n_{f} + m_{f}$ . For $0 \leq r_{0} \leq M_{0}$ and $0 \leq r_{f} \leq n_{f} + m_{f}$ , let $S_{a x} (r_{0})$ be the number of records among the first $r_{0}$ labeled observations with $(X, A, Y) = (x, a, 1)$ . Let $K_{a x} (r_{f})$ be the number of records with $(X, A) = (x, a)$ in the factorial prefix formed from the last $n_{f}$ labeled records and the last $m_{f}$ auxiliary records, taking the first $r_{f}$ records of that pooled factorial block in its fixed order. Define $τ_{n, m, d}^{IC, prefix} (r_{0}, r_{f}) = clip_{[- 1, 1]} [x = 1 \sum d {\frac{S _{1 x} ( r _{0} )}{u} \frac{K _{0 x} ( r _{f} ) + K _{1 x} ( r _{f} ) + 1}{K _{1 x} ( r _{f} ) + 1} - \frac{S _{0 x} ( r _{0} )}{u} \frac{K _{0 x} ( r _{f} ) + K _{1 x} ( r _{f} ) + 1}{K _{0 x} ( r _{f} ) + 1}}] .$

The statistic in Definition 20 uses the outcome prefix to estimate treated and control success masses and the factorial prefix to form stabilized inverse-count weights. The added ones make the expression finite in every cell and for every realized prefix length. Averaging over the Poisson prefix lengths derandomizes the construction while preserving the same count structure.

Theorem 6 [thm:inverse-count-baseline] (Inverse-count baseline bound).

Fix an overlap constant $ϵ$ with $0 < ϵ < 1/2$ . Then there is a constant $C_{ϵ} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , the clipped derandomized inverse-count estimator defined for each sample $Z = (L_{n}, U_{m})$ by $τ_{n, m, d}^{IC} (Z) = r_{0} = 0 \sum M_{0} r_{f} = 0 \sum n_{f} + m_{f} P {Poi (M_{0} /8) = r_{0}} P {Poi ((n_{f} + m_{f}) /8) = r_{f}} τ_{n, m, d}^{IC, prefix} (Z; r_{0}, r_{f})$ is measurable. Here $M_{0} = ⌊ n /3 ⌋$ , $n_{p} = ⌊(n - M_{0}) /2 ⌋$ , $n_{f} = n - M_{0} - n_{p}$ , $m_{p} = ⌊ m /2 ⌋$ , and $m_{f} = m - m_{p}$ are the block sizes from Algorithm 1, and $τ_{n, m, d}^{IC, prefix} (Z; r_{0}, r_{f})$ is the sample-and-prefix-count statistic of Definition 20.

Moreover, $P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq C_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$ Here $M_{d, ϵ}$ is the overlap class in Definition 6, $τ (P)$ is the average treatment effect functional in Definition 8, and the expectation is under the annotation experiment of Definition 9. Consequently, for the minimax risk in Definition 11, $R_{ϵ} (n, m, d) \leq C_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$

⊢ Lean

Theorem 6 shows that direct inverse-count stabilization already converts auxiliary treatment–covariate observations into a risk bound with the same two-resource form as the sharp theorem: a labeled-outcome term and a marginal-frequency term. Its marginal term is $d^{2} / (n + m)^{2}$ , whereas Theorem 4 attains the logarithmically improved $d^{2} / {(n + m)^{2} ℓ_{n}^{2}}$ rate through the calibrated light-cell polynomial component. The comparison identifies the gain supplied by the hybrid construction in sparse cells while keeping the same model, target, and product-sample experiment.

The general rate also has a simple specialization at the smallest nontrivial alphabet size. When the covariate alphabet is binary, the marginal-frequency component is uniformly absorbed into the labeled-outcome scale.

Theorem 7 [prop:binary-alphabet-parametric-rate] (Binary parametric rate).

Let $0 < ϵ < 1/2$ . For the minimax risk $R_{ϵ} (n, m, d)$ defined in Definition 11, at binary alphabet size $d = 2$ there exist constants $c_{ϵ}, C_{ϵ} \in R$ with $0 < c_{ϵ} \leq C_{ϵ}$ such that, for every integer $n \geq 1$ and every integer $m \geq 0$ , $\frac{c _{ϵ}}{n} \leq R_{ϵ} (n, m, 2) \leq \frac{C _{ϵ}}{n} .$

⊢ Lean

Theorem 7 gives the exact order of the minimax risk for $d = 2$ : the experiment is parametric for every auxiliary sample size. The upper bound follows the same scale as the labeled-outcome term in the main theorem, while the lower bound records that binary outcomes still require order $1/ n$ mean-squared error uniformly over the overlap class. Thus, in the binary-alphabet case, auxiliary treatment–covariate records fit naturally into the constant factor rather than changing the rate.

The result complements semisupervised causal and surrogate-outcome analyses in which auxiliary observations improve estimation through nuisance-function, design-distribution, or surrogate-response learning (Cheng et al., 2021; Chakrabortty et al., 2024; Kallus et al., 2025). Here the finite-alphabet structure makes that role explicit in minimax form: unlabeled treatment–covariate records sharpen the marginal component of the risk, and the sharp theorem characterizes exactly when that sharpening changes the rate.

Limitations and future work.

The analysis treats independent random outcome labeling from a shared population, binary treatment and outcome, a known finite covariate alphabet, occupied-cell overlap, and unrestricted cellwise nuisance functions. These conditions define a transparent discrete benchmark in which the value of outcome annotation can be characterized exactly. The constants in the explicit calibration are deliberately proof-oriented rather than implementation-oriented: for example, already at degree $L = 2$ , the stronger displayed polynomial-size condition $L^{6} 705 6^{2 L} \leq n$ requires $n$ on the order of $1.6 \times 1 0^{17}$ . Thus the theorem establishes a finite-sample minimax rate and an explicit witness, but it does not recommend this conservative calibration as a practical estimator at ordinary sample sizes. Natural extensions include sharper implementable tuning, continuous or growing covariate representations, multi-valued treatments or outcomes, adaptive annotation designs, and structural restrictions on the cellwise outcome regressions or propensity scores.

Appendices

Upper-bound ingredients

This appendix collects the analytic ingredients used by the upper bounds in Theorem 2 and by the inverse-count comparison in Theorem 6. The statements are organized around the two treatment–covariate masses that enter the estimators, the all-heavy inverse-count calculation, the polynomial certificate for light cells, and the fixed-sample transfer that connects the Poissonized construction to the annotation experiment.

The proof dependency map is as follows. The inverse-count moment lemmas feed the heavy-cell analysis; Definition 24, Lemma 5, and Lemma 16 feed the light-cell analysis; the calibrated mean and second-moment bounds combine in Lemma 13; and Lemma 14 converts that Poissonized bound into Theorem 2. Section B independently supplies the common-marginal converse, after which the upper and lower bounds meet in Theorem 4. This map is the shortest reading path through the detailed certificates below.

The inverse-count analysis uses one scalar inequality to insert the marginal treatment–covariate intensity into the variance calculation. It is stated first because the later cellwise bounds apply it separately to each covariate cell and treatment arm.

Lemma 1 [lem:poisson-inverse-marginal-energy-scalar] (Marginal energy insertion).

Let $u, p, w, ϵ, λ_{0}, λ_{1}, q_{0}, q_{1}, r_{2, 0}, r_{2, 1}, r_{4, 0}, r_{4, 1} \in R$ . Suppose that

(Basic scales.) $u > 0$ , $p \geq 0$ , $0 < ϵ \leq 1$ , and $w \geq 0$ .

(Outcome masses.) $0 \leq q_{0} \leq p$ and $0 \leq q_{1} \leq p$ .

(Marginal intensities.) For each $a \in {0, 1}$ , $ϵ w \leq λ_{a} \leq w$ .

(Second-moment rates.) $r_{2, 0}, r_{2, 1} \geq 0$ , $(1 + λ_{0})^{2} r_{2, 0} \leq 8$ , and $(1 + λ_{1})^{2} r_{2, 1} \leq 8$ .

(Fourth-moment rates.) $r_{4, 0}, r_{4, 1} \geq 0$ , $(1 + λ_{0})^{4} r_{4, 0} \leq 96$ , and $(1 + λ_{1})^{4} r_{4, 1} \leq 96$ .

Put $c_{mar} = 8 ϵ^{- 2} + 96 (ϵ^{- 2} + ϵ^{- 3}) .$ Then $λ_{1} {(q_{1} u^{- 1} + q_{1}^{2}) (λ_{0} + λ_{0}^{2}) r_{4, 1} + (q_{0} u^{- 1} + q_{0}^{2}) r_{2, 0}} + λ_{0} {(q_{0} u^{- 1} + q_{0}^{2}) (λ_{1} + λ_{1}^{2}) r_{4, 0} + (q_{1} u^{- 1} + q_{1}^{2}) r_{2, 1}} \leq (p u^{- 1} + p^{2}) \frac{2 c _{mar}}{1 + w} .$

⊢ Lean

Proof of Lemma 1.

First derive the scalar bidirectional energy bound. The lower and upper intensity bounds, together with $0 < ϵ \leq 1$ and $w \geq 0$ , imply $0 \leq λ_{0}, 0 \leq λ_{1},$ and, for the denominator attached to $λ_{1}$ , $ϵ (1 + w) = ϵ + ϵ w \leq 1 + λ_{1}, 1 + w \leq ϵ^{- 1} (1 + λ_{1}), w \leq ϵ^{- 1} (1 + λ_{1}) .$ Using also $λ_{0} \leq w$ , we have $(1 + w) λ_{0} \leq (1 + w) w \leq ϵ^{- 2} (1 + λ_{1})^{2},$ and hence $λ_{0} r_{2, 1} \leq 8 \frac{λ _{0}}{( 1 + λ _{1} ) ^{2}} \leq \frac{8 ϵ ^{- 2}}{1 + w} .$ Similarly, $(1 + w) λ_{1} λ_{0} \leq ϵ^{- 2} (1 + λ_{1})^{4}, (1 + w) λ_{1} λ_{0}^{2} \leq ϵ^{- 3} (1 + λ_{1})^{4},$ so the fourth-moment rate gives $λ_{1} (λ_{0} + λ_{0}^{2}) r_{4, 1} \leq \frac{96 ( ϵ ^{- 2} + ϵ ^{- 3} )}{1 + w} .$ Combining these two estimates yields $λ_{1} (λ_{0} + λ_{0}^{2}) r_{4, 1} + λ_{0} r_{2, 1} \leq \frac{c _{mar}}{1 + w} .$ The same calculation with the roles of $0$ and $1$ interchanged gives $λ_{0} (λ_{1} + λ_{1}^{2}) r_{4, 0} + λ_{1} r_{2, 0} \leq \frac{c _{mar}}{1 + w} .$ Therefore ${λ_{1} (λ_{0} + λ_{0}^{2}) r_{4, 1} + λ_{0} r_{2, 1}} + {λ_{0} (λ_{1} + λ_{1}^{2}) r_{4, 0} + λ_{1} r_{2, 0}} \leq \frac{2 c _{mar}}{1 + w} .$
Define the three real quantities $η_{0} = q_{0} u^{- 1} + q_{0}^{2}, η_{1} = q_{1} u^{- 1} + q_{1}^{2}, η_{p} = p u^{- 1} + p^{2} .$ Since $u > 0$ , we have $u^{- 1} \geq 0$ . For each displayed outcome mass, $η_{p} - η_{0} = (p - q_{0}) (u^{- 1} + p + q_{0}) \geq 0, η_{p} - η_{1} = (p - q_{1}) (u^{- 1} + p + q_{1}) \geq 0,$ because $0 \leq q_{0}, q_{1} \leq p$ . Thus $η_{0} \leq η_{p}, η_{1} \leq η_{p}, η_{p} \geq 0.$
All coefficients multiplying $η_{0}$ and $η_{1}$ below are nonnegative, by the nonnegativity of the intensities, the moment rates, and $λ_{a} + λ_{a}^{2}$ . Hence the comparison from the previous step gives $λ_{1} {η_{1} (λ_{0} + λ_{0}^{2}) r_{4, 1} + η_{0} r_{2, 0}} + λ_{0} {η_{0} (λ_{1} + λ_{1}^{2}) r_{4, 0} + η_{1} r_{2, 1}} \leq η_{p} [{λ_{1} (λ_{0} + λ_{0}^{2}) r_{4, 1} + λ_{0} r_{2, 1}} + {λ_{0} (λ_{1} + λ_{1}^{2}) r_{4, 0} + λ_{1} r_{2, 0}}] .$ Applying the scalar bidirectional bound from Step 1 and using $η_{p} \geq 0$ , we obtain $λ_{1} {(q_{1} u^{- 1} + q_{1}^{2}) (λ_{0} + λ_{0}^{2}) r_{4, 1} + (q_{0} u^{- 1} + q_{0}^{2}) r_{2, 0}} + λ_{0} {(q_{0} u^{- 1} + q_{0}^{2}) (λ_{1} + λ_{1}^{2}) r_{4, 0} + (q_{1} u^{- 1} + q_{1}^{2}) r_{2, 1}} \leq (p u^{- 1} + p^{2}) \frac{2 c _{mar}}{1 + w} .$

∎

Lemma 1 converts arm-level inverse-count moment bounds into a covariate-cell contribution controlled by the total marginal scale $w$ . The overlap inequalities in the statement match the occupied-cell overlap structure of $M_{d, ϵ}$ , while the final display has the form needed to sum over cells.

We next isolate the cell-level notation used throughout the appendix. The treatment–covariate mass records how often an arm appears in a covariate cell, while the outcome-marked mass records the same cell together with a successful binary outcome.

Definition 21 [synth_2] (Cell arm mass).

For a law $P$ on $[d] \times {0, 1}^{2}$ , a covariate cell $x \in [d]$ , and a treatment arm $a \in {0, 1}$ , define the treatment-covariate cell mass $s_{a x} = y \in {0, 1} \sum P (X = x, A = a, Y = y) .$

⊢ Lean

The mass $s_{a x}$ is the denominator scale for inverse treatment-cell weights. Under the overlap model in Definition 6, occupied covariate cells have both arm masses controlled by the same covariate mass, so inverse-count weights can be calibrated uniformly over the model class.

Definition 22 [synth_1] (Outcome-marked cell mass).

For an observed-data law $P$ on the finite record space $[d] \times {0, 1} \times {0, 1}$ , define the outcome-marked cell mass $q_{a x}$ for $x \in [d]$ and $a \in {0, 1}$ by $q_{a x} = P (X = x, A = a, Y = 1) .$ Equivalently, $q_{a x}$ is the probability mass assigned by $P$ to the complete atom $(x, a, 1)$ .

⊢ Lean

The outcome-marked mass $q_{a x}$ supplies the numerator scale. Combining $q_{a x}$ with an inverse estimate of $s_{a x}$ gives a cellwise contribution to the average treatment effect functional in Definition 8.

The simplest upper-bound component treats every cell as sufficiently observed and uses inverse counts throughout. This statistic is useful both as a baseline and as the heavy-cell part of the hybrid estimator.

Lemma 2 [lem:poisson-inverse-count-risk] (Poisson inverse-count risk).

Fix an overlap constant $ϵ$ with $0 < ϵ < 1/2$ . There is a constant $C = C_{ϵ} > 0$ such that, for every $d$ , every $u, t > 0$ , and every observed-data law $P \in M_{d, ϵ}$ as in Definition 6, the following holds.

In the independent Poisson count experiment with independent counts $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}), a \in {0, 1}, x \in [d],$ define $Z^{H} = x = 1 \sum d {\frac{S _{1 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{1 x} + 1} - \frac{S _{0 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{0 x} + 1}} .$ Writing $τ (P)$ for the average treatment effect functional in Definition 8, $E_{P} Z^{H} - τ (P) \leq C min {1, d / t}, Var_{P} (Z^{H}) \leq C (u^{- 1} + t^{- 1}) .$

⊢ Lean

Proof of Lemma 2.

Fix $0 < ϵ < 1/2$ . Put $c_{out} := 1 + 10 ϵ^{- 1} + 8 ϵ^{- 2}, c_{mar} := 8 ϵ^{- 2} + 96 (ϵ^{- 2} + ϵ^{- 3}),$ $c_{bias} := 2 max {1, (ϵe)^{- 1}}, C_{ϵ} := c_{bias} + c_{out} + 4 c_{mar} + 1.$ These constants are positive. We prove the two asserted inequalities with $C = C_{ϵ}$ . Fix $d$ , $u, t > 0$ , and $P \in M_{d, ϵ}$ . For each cell write $p_{x} := a \in {0, 1} \sum y \in {0, 1} \sum P (x, a, y), s_{a x} := y \in {0, 1} \sum P (x, a, y), q_{a x} := P (x, a, 1) .$ Thus $\sum_{x} p_{x} = 1$ , $q_{a x} \leq s_{a x} \leq p_{x}$ , and $\sum_{x, a} q_{a x} \leq 1$ . By Definition 6, the overlap condition gives $ϵ p_{x} \leq s_{a x} \leq p_{x}, a \in {0, 1} .$

For $a \in {0, 1}$ , let $\overset{a}{ˉ}$ denote the opposite arm and define the scalar arm mean $m_{a x} := E [\frac{S _{a x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1}] .$ Independence of the Poisson coordinates and linearity of expectation give $E_{P} Z^{H} = x \sum (m_{1 x} - m_{0 x}) .$ Indeed, after expanding the summand defining $Z^{H}$ , each term is integrable by the polynomial growth bound $\frac{S _{a x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1} \leq ∣ u^{- 1} ∣ (1 + T)^{2},$ where $T$ is the total count over all displayed Poisson coordinates, and all polynomial moments of the finite product Poisson law are finite. The product structure then reduces the expectation to the one-cell arm mean above.
For a Poisson variable $J$ with mean $λ \geq 0$ , $E \frac{1}{J + 1} = {(1 - e^{- λ}) / λ, 1, λ > 0, λ = 0.$ Define the arm outcome regression by $μ_{a x} := {q_{a x} / s_{a x}, 0, s_{a x} > 0, s_{a x} = 0.$ Then $q_{a x} = s_{a x} μ_{a x}$ for all cells and arms, including zero-mass arms, and $0 \leq μ_{a x} \leq 1$ . Using $E (S_{a x} / u) = q_{a x}$ and the preceding identity with $λ = t s_{a x}$ , one obtains the continuous formula $m_{a x} = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} e^{- t s_{a x}} .$ Since $0 \leq μ_{a x} \leq 1$ , $p_{x} - s_{a x} \leq p_{x}$ , and $s_{a x} \geq ϵ p_{x}$ , $∣ m_{a x} - p_{x} μ_{a x} ∣ \leq p_{x} e^{- ϵ t p_{x}} .$ Consequently, using Definition 8, $E_{P} Z^{H} - τ (P) \leq 2 x \sum p_{x} e^{- ϵ t p_{x}} .$ For every $z \geq 0$ , $z e^{- ϵ t z} \leq (ϵe t)^{- 1}$ , while $\sum_{x} p_{x} = 1$ . Hence $E_{P} Z^{H} - τ (P) \leq 2 min {1, \frac{d}{ϵe t}} .$ Writing $ξ = d / t$ , the elementary inequality $2 min {1, ξ / (ϵe)} \leq 2 max {1, (ϵe)^{- 1}} min {1, ξ} = c_{bias} min {1, d / t}$ gives $E_{P} Z^{H} - τ (P) \leq C_{ϵ} min {1, d / t} .$
We next bound the variance. For a full count array $ω$ , let $Δ_{a x}^{S} F (ω)$ be the change in $F$ after adding one to $S_{a x}$ , and let $Δ_{a x}^{K} F (ω)$ be the change after adding one to $K_{a x}$ . The finite-product Poisson add-one Poincare inequality gives, for $F = Z^{H}$ , $Var_{P} (Z^{H}) \leq x, a \sum u q_{a x} E {(Δ_{a x}^{S} Z^{H})^{2}} + x, a \sum t s_{a x} E {(Δ_{a x}^{K} Z^{H})^{2}} .$ The square-integrability hypotheses required for this inequality follow from the same polynomial growth estimate used above, applied also to the add-one increments.
For the outcome-count increments, $Δ_{a x}^{S} Z^{H} = \pm u^{- 1} (1 + \frac{K _{\overset{a}{ˉ} x}}{K _{a x} + 1}),$ where the sign is irrelevant after squaring. If $λ = t s_{a x}$ and $ν = t s_{\overset{a}{ˉ} x}$ , then $ϵ ν \leq λ$ . For independent $J \sim Poi (λ)$ and $V \sim Poi (ν)$ , set $r_{1} := E \frac{1}{J + 1}, r_{2} := E \frac{1}{( J + 1 ) ^{2}} .$ The Poisson reciprocal bounds give $λ r_{1} \leq 1, (1 + λ)^{2} r_{2} \leq 8, r_{1} \geq 0, r_{2} \geq 0.$ Moreover $E V = ν$ and $E V^{2} \leq ν + ν^{2}$ . Thus product-measure factorization gives $E (1 + \frac{V}{J + 1})^{2} \leq 1 + 2 ν r_{1} + (ν + ν^{2}) r_{2} .$ The overlap comparison $ϵ ν \leq λ$ , together with $ϵ > 0$ , implies $ν \leq ϵ^{- 1} λ$ . Hence $2 ν r_{1} \leq 2 ϵ^{- 1} λ r_{1} \leq 2 ϵ^{- 1},$ and, using $r_{2} \leq 8 (1 + λ)^{- 2}$ , $ν r_{2} \leq 8 ϵ^{- 1} \frac{λ}{( 1 + λ ) ^{2}} \leq 8 ϵ^{- 1}, ν^{2} r_{2} \leq 8 ϵ^{- 2} \frac{λ ^{2}}{( 1 + λ ) ^{2}} \leq 8 ϵ^{- 2} .$ Consequently $E (1 + \frac{V}{J + 1})^{2} \leq 1 + 10 ϵ^{- 1} + 8 ϵ^{- 2} = c_{out} .$ Multiplying by the rate $u q_{a x}$ and summing yields $x, a \sum u q_{a x} E {(Δ_{a x}^{S} Z^{H})^{2}} \leq c_{out} u^{- 1} x, a \sum q_{a x} \leq c_{out} u^{- 1} .$
It remains to control the marginal-count increments. In one fixed cell, write $θ_{a x} := t s_{a x}, Θ_{x} := t p_{x},$ and, for $K_{a x} \sim Poi (θ_{a x})$ , write $η_{2, a} := E \frac{1}{( K _{a x} + 1 ) ^{2}}, η_{4, a} := E \frac{1}{(( K _{a x} + 1 ) ( K _{a x} + 2 ) ) ^{2}} .$ The reciprocal estimates are $η_{2, a} \geq 0, (1 + θ_{a x})^{2} η_{2, a} \leq 8, η_{4, a} \geq 0, (1 + θ_{a x})^{4} η_{4, a} \leq 96.$ For a full count array $ω$ , define $cr_{a x} (ω) := \frac{S _{a x} ( ω )}{u} \frac{K _{\overset{a}{ˉ} x} ( ω )}{( K _{a x} ( ω ) + 1 ) ( K _{a x} ( ω ) + 2 )}, rc_{a x} (ω) := \frac{S _{a x} ( ω )}{u} \frac{1}{K _{a x} ( ω ) + 1} .$ The add-one formulas for $K_{a x}$ are $Δ_{1 x}^{K} Z^{H} = - {cr_{1 x} + rc_{0 x}}, Δ_{0 x}^{K} Z^{H} = cr_{0 x} + rc_{1 x} .$ The two displayed components are nonnegative. Hence $cr_{a x} \leq ∣ Δ_{a x}^{K} Z^{H} ∣, rc_{\overset{a}{ˉ} x} \leq ∣ Δ_{a x}^{K} Z^{H} ∣,$ so the component squares are integrable because the add-one increment is square-integrable. Applying $(r + s)^{2} \leq 2 (r^{2} + s^{2})$ to the preceding decomposition gives, for each $a$ , $E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 2 (E cr_{a x}^{2} + E rc_{\overset{a}{ˉ} x}^{2}) .$ Product-measure factorization gives $E cr_{a x}^{2} = E (\frac{S _{a x}}{u})^{2} η_{4, a} E K_{\overset{a}{ˉ} x}^{2}, E rc_{a x}^{2} = E (\frac{S _{a x}}{u})^{2} η_{2, a} .$ The Poisson second-moment bounds are $E (\frac{S _{a x}}{u})^{2} \leq q_{a x} u^{- 1} + q_{a x}^{2}, E K_{a x}^{2} \leq θ_{a x} + θ_{a x}^{2} .$ Combining the last three displays yields $E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 2 [(q_{a x} u^{- 1} + q_{a x}^{2}) (θ_{\overset{a}{ˉ} x} + θ_{\overset{a}{ˉ} x}^{2}) η_{4, a} + (q_{\overset{a}{ˉ} x} u^{- 1} + q_{\overset{a}{ˉ} x}^{2}) η_{2, \overset{a}{ˉ}}] .$ Since $q_{a x} \leq p_{x}$ and $u > 0$ , $q_{a x} u^{- 1} + q_{a x}^{2} \leq p_{x} u^{- 1} + p_{x}^{2} .$ The hypotheses of Lemma 1 hold after identifying its generic cell mass and total intensity with $p = p_{x}, w = Θ_{x},$ with its arm rates equal to $θ_{0 x}, θ_{1 x}$ , its marked masses equal to $q_{0 x}, q_{1 x}$ , and its four reciprocal moments equal to $(η_{2, 0}, η_{2, 1}, η_{4, 0}, η_{4, 1})$ . These arm rates are simply $t s_{a x}$ , which are nonnegative because $t > 0$ and $s_{a x} \geq 0$ . Also $p_{x} \geq 0$ , $u > 0$ , $0 < ϵ \leq 1$ , and $Θ_{x} \geq 0$ ; the outcome masses satisfy $0 \leq q_{a x} \leq p_{x}$ ; overlap gives $ϵ Θ_{x} \leq θ_{a x}$ , while $s_{a x} \leq p_{x}$ gives $θ_{a x} \leq Θ_{x}$ ; and the displayed reciprocal estimates give the required second- and fourth-moment rate conditions. Therefore Lemma 1 yields $θ_{1 x} {(q_{1 x} u^{- 1} + q_{1 x}^{2}) (θ_{0 x} + θ_{0 x}^{2}) η_{4, 1} + (q_{0 x} u^{- 1} + q_{0 x}^{2}) η_{2, 0}} + θ_{0 x} {(q_{0 x} u^{- 1} + q_{0 x}^{2}) (θ_{1 x} + θ_{1 x}^{2}) η_{4, 0} + (q_{1 x} u^{- 1} + q_{1 x}^{2}) η_{2, 1}} \leq (p_{x} u^{- 1} + p_{x}^{2}) \frac{2 c _{mar}}{1 + Θ _{x}} .$ Multiplying the add-one-square bounds by their Poisson rates $θ_{a x} = t s_{a x}$ , summing over $a$ , and using the preceding display gives the cell estimate $a \sum t s_{a x} E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 4 c_{mar} \frac{p _{x} u ^{- 1} + p _{x}^{2}}{1 + t p _{x}} .$
Summing the preceding display over $x$ , $x, a \sum t s_{a x} E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 4 c_{mar} x \sum \frac{p _{x} u ^{- 1} + p _{x}^{2}}{1 + t p _{x}} .$ Since $1 + t p_{x} \geq 1$ and $p_{x} u^{- 1} \geq 0$ , $\frac{p _{x} u ^{- 1} + p _{x}^{2}}{1 + t p _{x}} \leq p_{x} u^{- 1} + \frac{p _{x}^{2}}{1 + t p _{x}} .$ Thus $x, a \sum t s_{a x} E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 4 c_{mar} x \sum (p_{x} u^{- 1} + \frac{p _{x}^{2}}{1 + t p _{x}}) .$ The first sum is $u^{- 1}$ . For the second, since $t > 0$ and $p_{x} \geq 0$ , $\frac{p _{x}^{2}}{1 + t p _{x}} \leq \frac{p _{x}}{t},$ and hence $x \sum \frac{p _{x}^{2}}{1 + t p _{x}} \leq t^{- 1} x \sum p_{x} = t^{- 1} .$ Thus $x, a \sum t s_{a x} E {(Δ_{a x}^{K} Z^{H})^{2}} \leq 4 c_{mar} (u^{- 1} + t^{- 1}) .$
Combining the Poisson Poincare bound with the outcome and marginal energy estimates, $Var_{P} (Z^{H}) \leq c_{out} u^{- 1} + 4 c_{mar} (u^{- 1} + t^{- 1}) .$ Since $u^{- 1} \leq u^{- 1} + t^{- 1}$ and $C_{ϵ} = c_{bias} + c_{out} + 4 c_{mar} + 1$ , $Var_{P} (Z^{H}) \leq C_{ϵ} (u^{- 1} + t^{- 1}) .$ Together with the bias bound from Step 2, this proves the claim.

∎

Lemma 2 separates the two costs of the all-heavy construction: the bias is governed by the number of cells relative to the treatment–covariate intensity, and the variance is governed by the outcome and count intensities. This is the parametric inverse-count behavior that becomes effective once the pilot stage classifies a cell as heavy.

To use this calculation with sampled prefixes, the Poisson record experiment is identified with the independent count experiment. The next two statements record the count law for two independent prefixes and then name the prefix distribution used by the inverse-count transfer.

Lemma 3 [lem:poisson-prefix-count-law] (Poisson prefix count law).

Let $P$ be a probability law on $[d] \times {0, 1}^{2}$ , and let $u, t \geq 0$ . Write $q_{a x} = P (x, a, 1), s_{a x} = P (x, a, 0) + P (x, a, 1) .$ Under the prefix law that independently draws an outcome-marked prefix with $Poi (u)$ records from $P$ and a treatment–covariate prefix with $Poi (t)$ records from the marginal $P_{X A}$ , define $S_{a x} = # {outcome-marked prefix records equal to (x, a, 1)}, K_{a x} = # {treatment-covariate prefix records equal to (x, a)} .$ Then the pushforward of the prefix law under the map sending the two prefixes to the array $(S_{a x}, K_{a x})_{x \in [d], a \in {0, 1}}$ is the product measure $x \in [d] ⨂ a \in {0, 1} ⨂ {Poi (u q_{a x}) \otimes Poi (t s_{a x})} .$ Equivalently, the coordinates are mutually independent, with $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x})$ for every $x \in [d]$ and $a \in {0, 1}$ .

⊢ Lean

Proof of Lemma 3.

Let $Ω_{obs} = [d] \times {0, 1}^{2}$ and $Ω_{aux} = [d] \times {0, 1}$ . For a finite prefix $ξ = (z_{1}, \dots, z_{r})$ in a finite space $Ω$ , write $H_{ξ} (z) = # {i : z_{i} = z}, z \in Ω.$ The elementary Poisson splitting identity says that if $R \sim Poi (λ)$ and, conditionally on $R$ , $z_{1}, \dots, z_{R}$ are i.i.d. with law $ν$ on $Ω$ , then $(H_{ξ} (z))_{z \in Ω} \sim z \in Ω ⨂ Poi (λ ν {z}) .$ Indeed, for any nonnegative integers $(r_{z})_{z \in Ω}$ , with $r_{∙} = \sum_{z} r_{z}$ , $Pr {H_{ξ} (z) = r_{z} \forall z} = e^{- λ} \frac{λ ^{r_{∙}}}{r _{∙} !} \frac{r _{∙} !}{\prod _{z} r _{z} !} z \prod ν {z}^{r_{z}} = z \prod e^{- λ ν {z}} \frac{( λ ν { z } ) ^{r_{z}}}{r _{z} !} .$ Applied to the outcome-marked prefix with intensity $u$ and the treatment–covariate prefix with intensity $t$ , and using the independence of the two prefixes, this gives $(H_{z}^{obs})_{z \in Ω_{obs}} \sim z \in Ω_{obs} ⨂ Poi (u P {z}), (H_{w}^{aux})_{w \in Ω_{aux}} \sim w \in Ω_{aux} ⨂ Poi (t P_{X A} {w}),$ with the two histogram arrays independent. Mapping the first histogram to $E (H^{obs}) (x, a) = H^{obs} (x, a, 1)$ selects one coordinate for each $(x, a)$ . Since the selected observed coordinates are distinct, and the auxiliary histogram is independent of the observed histogram, the array $(E (H^{obs}) (x, a), H^{aux} (x, a))_{x \in [d], a \in {0, 1}}$ has law $x \in [d] ⨂ a \in {0, 1} ⨂ {Poi (u P {(x, a, 1)}) \otimes Poi (t P_{X A} {(x, a)})} .$ This array is exactly $(S_{a x}, K_{a x})_{x, a}$ . In the degenerate case $[d] = \emptyset$ , the same display is the empty product measure on the one-point array space.

It remains to identify the two singleton masses in the displayed product. By the definition of $q_{a x}$ , $P {(x, a, 1)} = P (x, a, 1) = q_{a x} .$ Hence the first Poisson intensity is $u q_{a x}$ .

For the auxiliary marginal, $P_{X A}$ is the pushforward of $P$ under $(x, a, y) \mapsto (x, a)$ . Therefore $P_{X A} {(x, a)} = y \in {0, 1} \sum P (x, a, y) = P (x, a, 0) + P (x, a, 1) = s_{a x} .$ Thus the second Poisson intensity is $t s_{a x}$ . Since $u, t \geq 0$ , these products are the corresponding nonnegative Poisson parameters, and substituting the two identities into the preceding product law yields $x \in [d] ⨂ a \in {0, 1} ⨂ {Poi (u q_{a x}) \otimes Poi (t s_{a x})} .$ The product representation is equivalent to mutual independence of all coordinates and to the stated marginal laws for every $x$ and $a$ .

∎

Definition 23 [synth_19] (Poisson-prefix law).

For $P \in M_{d, ϵ}$ and intensities $u, t \geq 0$ , let $P_{P, u, t}^{pref}$ denote the following probability law. Draw two prefix lengths independently, $R_{0} \sim Poi (u), R_{f} \sim Poi (t) .$ Conditional on $(R_{0}, R_{f}) = (r_{0}, r_{f})$ , draw an outcome prefix $((X_{i}, A_{i}, Y_{i}))_{i = 1}^{r_{0}} \sim P^{\otimes r_{0}}$ and, independently, a factorial treatment–covariate prefix $((X_{j}, A_{j}))_{j = 1}^{r_{f}} \sim P_{X A}^{\otimes r_{f}} .$ Thus the two prefixes are independent, the outcome prefix has complete-record sampling law $P$ , and the factorial prefix has treatment–covariate sampling law $P_{X A}$ . Under $P_{P, u, t}^{pref}$ , the associated counts $S_{a x} = i = 1 \sum R_{0} 1 {X_{i} = x, A_{i} = a, Y_{i} = 1}, K_{a x} = j = 1 \sum R_{f} 1 {X_{j} = x, A_{j} = a}$ are independent across the two prefixes, with $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}) .$

Lemma 3 and Definition 23 justify applying the count-risk calculation directly to Poisson prefixes of the original records. The prefix representation is convenient for derandomization because every admissible prefix length is a finite statistic of the fixed sample pools.

Lemma 4 [lem:inverse-count-poisson-prefix-transfer] (Poisson prefix transfer).

Let $n, m, d$ be nonnegative integers, let $ϵ \in R$ , and let $P \in M_{d, ϵ}$ as in Definition 6. Define $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ (n - M_{0}) /2 ⌋, n_{f} = n - M_{0} - n_{p}, m_{f} = m - ⌊ m /2 ⌋,$ and set $u = M_{0} /8$ and $t = (n_{f} + m_{f}) /8$ . Let $τ_{n, m, d}^{IC}$ be the clipped derandomized inverse-count estimator obtained by averaging, over $0 \leq r_{0} \leq M_{0}$ and $0 \leq r_{f} \leq n_{f} + m_{f}$ , the finite-prefix inverse-count output with weights $Pr {Poi (u) = r_{0}} Pr {Poi (t) = r_{f}} .$ Let $τ_{u, t}^{IC, pref}$ be the clipped inverse-count statistic computed from two independent Poisson prefixes: an outcome-marked prefix with law induced by $P$ and length $Poi (u)$ , and a treatment–covariate prefix with law $P_{X A}$ and length $Poi (t)$ . Then, for the annotation experiment in Definition 9 and the ATE functional $τ (P)$ in Definition 8, $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [(τ_{n, m, d}^{IC} - τ (P))^{2}] \leq E_{P_{P, u, t}^{pref}} [(τ_{u, t}^{IC, pref} - τ (P))^{2}] + 4 [Pr {Poi (u) > M_{0}} + Pr {Poi (t) > n_{f} + m_{f}}] .$

⊢ Lean

Proof of Lemma 4.

Put $M_{f} := n_{f} + m_{f}, m_{p} := ⌊ m /2 ⌋ .$ All lists below use the empty-list convention when one of the displayed lengths is zero.

For finite prefixes $ω_{0}$ of outcome-marked observations and $ω_{f}$ of treatment–covariate observations, write $Φ_{u} (ω_{0}, ω_{f}) := the clipped finite-prefix inverse-count output at scale u$ so that, by clipping, $- 1 \leq Φ_{u} (ω_{0}, ω_{f}) \leq 1 for every (ω_{0}, ω_{f}) .$ Also, $∣ τ (P) ∣ = x = 1 \sum d p_{x} (μ_{1 x} - μ_{0 x}) \leq x = 1 \sum d p_{x} ∣ μ_{1 x} - μ_{0 x} ∣ \leq x = 1 \sum d p_{x} = 1,$ using Definition 8, the fact that $P$ is a probability law in Definition 6, and $μ_{a x} \in [0, 1]$ , with the standard zero-mass-cell convention.

For fixed pools $o = (o_{1}, \dots, o_{M_{0}})$ and $v = (v_{1}, \dots, v_{M_{f}})$ , define the capped prefix value $C (o, v, r_{0}, r_{f}) := {Φ_{u} ((o_{1}, \dots, o_{r_{0}}), (v_{1}, \dots, v_{r_{f}})), 0, r_{0} \leq M_{0}, r_{f} \leq M_{f}, otherwise .$ Let $R_{0} \sim Poi (u)$ and $R_{f} \sim Poi (t)$ be independent, and set $R_{u, t} (o, v) := E [C (o, v, R_{0}, R_{f})] .$ If $(O, V) \sim P^{\otimes M_{0}} \otimes P_{X A}^{\otimes M_{f}}$ , Jensen’s inequality for the conditional expectation over $(R_{0}, R_{f})$ gives $E [(R_{u, t} (O, V) - τ (P))^{2}] \leq E [(C (O, V, R_{0}, R_{f}) - τ (P))^{2}] .$ On the event $Ω_{fit} = {R_{0} \leq M_{0}, R_{f} \leq M_{f}}$ , the selected prefixes have the same law as the independent Poisson prefixes restricted to their lengths fitting in the two pools. Hence $E [(C - τ (P))^{2} 1_{Ω_{fit}}] \leq E_{P_{P, u, t}^{pref}} [(Φ_{u} - τ (P))^{2}] .$ On $Ω_{fit}^{c}$ , the capped value is $0$ , and the interval bound above gives $(0 - τ (P))^{2} \leq (1 - (- 1))^{2} = 4.$ The union bound gives $Pr (Ω_{fit}^{c}) \leq Pr {Poi (u) > M_{0}} + Pr {Poi (t) > M_{f}} .$ Combining these displays, $E [(R_{u, t} (O, V) - τ (P))^{2}] \leq E_{P_{P, u, t}^{pref}} [(Φ_{u} - τ (P))^{2}] + 4 [Pr {Poi (u) > M_{0}} + Pr {Poi (t) > M_{f}}] .$
For a full sample $z = ((O_{1}, \dots, O_{n}), (V_{1}, \dots, V_{m}))$ , with $O_{i} = (X_{i}, A_{i}, Y_{i})$ and $V_{j} \in [d] \times {0, 1}$ , define the fixed-pool projection $Ψ (z) := ((O_{1}, \dots, O_{M_{0}}), ((X_{M_{0} + n_{p} + 1}, A_{M_{0} + n_{p} + 1}), \dots, (X_{n}, A_{n}), V_{m_{p} + 1}, \dots, V_{m})) .$ For every $z$ , $τ_{n, m, d}^{IC} (z) = R_{u, t} (Ψ (z)) .$ Indeed, expanding $R_{u, t}$ gives $r_{0}, r_{f} \geq 0 \sum Pr {Poi (u) = r_{0}} Pr {Poi (t) = r_{f}} C (Ψ (z), r_{0}, r_{f}),$ and the terms with $r_{0} > M_{0}$ or $r_{f} > M_{f}$ are zero, leaving exactly the finite average over $0 \leq r_{0} \leq M_{0}$ and $0 \leq r_{f} \leq M_{f}$ in the derandomized inverse-count estimator.
Under the annotation experiment in Definition 9, $Ψ_{#} (P^{\otimes n} \otimes P_{X A}^{\otimes m}) = P^{\otimes M_{0}} \otimes P_{X A}^{\otimes M_{f}} .$ The first component selects the first $M_{0}$ labeled records. The second component concatenates the treatment–covariate projections of the final $n_{f}$ labeled records with the final $m_{f}$ auxiliary records; independence of the original product samples and the definition of $P_{X A}$ give the displayed product law. Therefore the change of variables formula yields $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [(τ_{n, m, d}^{IC} - τ (P))^{2}] = E_{P^{\otimes M_{0}} \otimes P_{X A}^{\otimes M_{f}}} [(R_{u, t} - τ (P))^{2}] .$ Measurability is immediate because all sample spaces involved are finite products and the statistic is clipped.
Applying the transfer bound from the first step to the fixed-pool risk in the last display gives $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [(τ_{n, m, d}^{IC} - τ (P))^{2}] \leq E_{P_{P, u, t}^{pref}} [(Φ_{u} - τ (P))^{2}] + 4 [Pr {Poi (u) > M_{0}} + Pr {Poi (t) > M_{f}}] .$ Under $P_{P, u, t}^{pref}$ , the random value $Φ_{u}$ is precisely $τ_{u, t}^{IC, pref}$ , and $M_{f} = n_{f} + m_{f}$ . This is the claimed inequality.

∎

Lemma 4 converts the Poisson-prefix risk into a fixed-sample inverse-count estimator, with the excess risk controlled by the two prefix-overflow probabilities. This is the transfer step behind the inverse-count comparison in Theorem 6.

For light cells, the hybrid estimator replaces reciprocal weights by polynomial weights. The next definition records the factorial-moment lift used to turn the polynomial into an unbiased Poisson statistic, following the same polynomial-estimation principle used in discrete functional estimation (Jiao et al., 2015; Wu et al., 2016; Valiant et al., 2017; Orlitsky et al., 2016).

Definition 24 [synth_4] (Factorial lift of the light-cell polynomial).

Fix $a \in {0, 1}$ , an integer $L \geq 2$ , and constants $B, t > 0$ . Let $W_{a, L} (s_{0}, s_{1}) = \frac{s _{0} + s _{1}}{B} G_{L} (s_{a} / B) = r_{0}, r_{1} \geq 0 \sum c_{a, r_{0}, r_{1}} s_{0}^{r_{0}} s_{1}^{r_{1}},$ where $G_{L}$ is the Chebyshev correction polynomial and only finitely many coefficients $c_{a, r_{0}, r_{1}}$ are nonzero. For integer counts $K_{0}, K_{1} \geq 0$ , define the factorial lift $W_{a, L} (K_{0}, K_{1}) = r_{0}, r_{1} \geq 0 \sum c_{a, r_{0}, r_{1}} \frac{( K _{0} ) _{r_{0}} ( K _{1} ) _{r_{1}}}{t ^{r_{0} + r_{1}}},$ with $(k)_{r} = k (k - 1) \dots (k - r + 1)$ and $(k)_{0} = 1$ . Thus each monomial of $W_{a, L}$ is replaced by the corresponding unbiased Poisson factorial-moment statistic.

The lift in Definition 24 is tailored to independent Poisson counts: each falling factorial has expectation equal to the corresponding power of the intensity. The remaining task is to control the polynomial and the second moment of its lift uniformly over the light-cell range.

Lemma 5 [lem:chebyshev-factorial-certificate] (Chebyshev factorial certificate).

There is a universal constant $A > 1$ such that, for every integer $L \geq 2$ , the polynomial continuations $E_{L} (z) = H_{L} (z) / z, G_{L} (z) = {1 - E_{L} (z)} / z$ satisfy the following properties:

(Degree.) The polynomial $G_{L}$ has degree at most $L - 2$ .

(Chebyshev certificate.) For every $z \in [0, 1]$ , $0 \leq E_{L} (z) \leq 1, z G_{L} (z) + E_{L} (z) = 1,$ and, whenever $z > 0$ , $E_{L} (z) \leq (L^{2} z)^{- 1} .$

(Factorial-lift moment.) For every $a \in {0, 1}$ and every $t, B, s_{0}, s_{1} \in R$ with $t > 0, B > 0, s_{0} \geq 0, s_{1} \geq 0, L \leq tB,$ if $K_{0} \sim Poi (t s_{0})$ and $K_{1} \sim Poi (t s_{1})$ are independent, then the factorial lift $W_{a, L}$ satisfies $E [W_{a, L} (K_{0}, K_{1})^{2}] \leq A^{L} (1 + \frac{s _{0} + s _{1}}{B})^{2 L} .$ Moreover, if $s_{0} + s_{1} \leq B$ , then $E [W_{a, L} (K_{0}, K_{1})^{2}] \leq A^{L} .$

⊢ Lean

Proof of Lemma 5.

Choose $A = 7056$ , so $A > 1$ .

1. Fix an integer $L \geq 2$ . Write $Q_{L} (z) = T_{L} (1 - 2 z), H_{L} (z) = \frac{1 - Q _{L} ( z )}{2 L ^{2}}, E_{L} (z) = H_{L} (z) / z, G_{L} (z) = {1 - E_{L} (z)} / z .$ Since $T_{L}$ has degree $L$ and $1 - 2 z$ has degree at most $1$ , $de g Q_{L} \leq L .$ Multiplication by the scalar $(2 L^{2})^{- 1}$ does not increase degree, and subtraction from $1$ gives $de g H_{L} \leq de g (1 - Q_{L}) \leq max {0, de g Q_{L}} \leq L .$ Because $E_{L}$ is the quotient of $H_{L}$ by $z$ , its degree is at most $L - 1$ . Hence $de g (1 - E_{L}) \leq max {0, de g E_{L}} \leq L - 1,$ using $L \geq 2$ . Dividing once more by $z$ gives $de g G_{L} \leq L - 2.$

2. By Lemma 16, for every $z \in [0, 1]$ , $0 \leq E_{L} (z) \leq 1, z G_{L} (z) + E_{L} (z) = 1,$ and, for $z > 0$ , $E_{L} (z) \leq \frac{1}{L ^{2} z} .$ These are exactly the Chebyshev-certificate assertions.

3. It remains to check the moment assertions with the same constant. Let $a \in {0, 1}$ , $t > 0$ , $B > 0$ , $s_{0}, s_{1} \geq 0$ , and $L \leq tB$ , and let $K_{0} \sim Poi (t s_{0})$ and $K_{1} \sim Poi (t s_{1})$ be independent. Applying Lemma 16 with $p = s_{0} + s_{1}$ gives $E [W_{a, L} (K_{0}, K_{1})^{2}] \leq 705 6^{L} (1 + \frac{s _{0} + s _{1}}{B})^{2 L} .$ If in addition $s_{0} + s_{1} \leq B$ , the same calibration yields $E [W_{a, L} (K_{0}, K_{1})^{2}] \leq 705 6^{L} .$ Since $A = 7056$ , these are the two required factorial-lift moment bounds.

∎

Lemma 5 provides the approximation identity and moment growth bound needed for the light-cell part of the hybrid statistic. The identity $z G_{L} (z) + E_{L} (z) = 1$ expresses the reciprocal-weight correction, and the displayed decay of $E_{L}$ quantifies the approximation error at positive mass. The factorial-lift moment bound then keeps the variance compatible with the polynomial degree used in Algorithm 1.

The hybrid mean analysis separates the deterministic light-cell approximation error from the pilot-classification error on heavy cells. The next statements give the corresponding cell-summed bounds and the envelope calculation that puts them on the target rate scale.

Lemma 6 [lem:calibrated-envelope-absorption] (Calibrated envelope absorption).

Let $ϵ > 0$ and $C_{0} > 0$ . There is a constant $C > 0$ , depending on $ϵ$ and $C_{0}$ , such that the following holds.

(Sample sizes.) $n, m, d$ are integers with $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ .

(Calibration.) The finite calibration predicate $C_{ϵ} (n, m)$ from Algorithm 1 holds.

(Estimator scales.) With the block sizes and scales from Algorithm 1, $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊(n - M_{0}) /2 ⌋, n_{f} = n - M_{0} - n_{p}, m_{f} = m - ⌊ m /2 ⌋,$ and $u = M_{0} /8, t = (n_{f} + m_{f}) /8, L = L (n), B = B (n, m, ϵ), A_{⋆} = 7056.$ Let $N = n + m$ and $ℓ_{n} = lo g (e n)$ , as in Definition 12.

Then $4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{0} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 414 n^{- 12} + (\frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12})^{2} \leq C (\frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}}) .$

⊢ Lean

Proof of Lemma 6.

Let $H_{ϵ}$ be the calibration constant appearing in Algorithm 1, viewed as a real number, and set $K_{ϵ} (C_{0}) = 4096 H_{ϵ} 51 2^{2} + 4096 H_{ϵ}^{2} 512 + 64 C_{0} + \frac{8192 H _{ϵ}^{2} 51 2 ^{2}}{ϵ ^{2}} + \frac{128}{ϵ} + 414 + 2 (\frac{64 H _{ϵ} 512}{ϵ})^{2} + 72.$ We take $C = 1 + K_{ϵ} (C_{0}) .$ Since $ϵ > 0$ , $C_{0} > 0$ , and $H_{ϵ} \geq 0$ , every displayed summand in $K_{ϵ} (C_{0})$ is nonnegative except for none, and hence $C > 0$ .
Fix $n, m, d \in N$ with $n \geq 1$ , $d \geq 2$ , and $C_{ϵ} (n, m)$ . Write $N = n + m, ℓ_{n} = lo g (e n), A = A_{⋆}^{L}, η = n^{- 12},$ where $L = L (n)$ , and define the comparison scale $Ξ = \frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}} .$ All quantities in the displayed inequalities below are understood as real numbers. The sample-size assumptions give $1 \leq n, n \leq N, 1 \leq d .$ Moreover, $ℓ_{n} = lo g (e n) = 1 + lo g n \geq 1,$ because $n \geq 1$ . From $L \geq 2$ in Algorithm 1, $L > 0$ , and $A = A_{⋆}^{L} \geq 0$ . The calibration inequalities also give $u > 0$ , $t_{p} > 0$ , $t > 0$ , and $B > 0$ : indeed $u / n \geq 1/32$ , $t_{p} / N \geq 1/32$ , $t / N \geq 1/32$ , and $8 \leq t_{p} B$ .
The block-proportion part of $C_{ϵ} (n, m)$ supplies $\frac{n}{32} \leq u, \frac{N}{32} \leq t .$ The scale definition $B = H_{ϵ} L / min {t_{p}, t}$ , together with $\frac{N}{32} \leq t_{p}, \frac{N}{32} \leq t,$ gives $B = \frac{H _{ϵ} L}{min { t _{p} , t }} \leq \frac{32 H _{ϵ} L}{N} .$ The same calculation with the upper block bounds $t_{p} \leq 3 N /32$ and $t \leq 3 N /32$ yields the full auxiliary estimate $\frac{32 H _{ϵ} L}{3 N} \leq B \leq \frac{32 H _{ϵ} L}{N} .$ Only the upper bound is used below.
Let $b_{n}$ be the binary-length calibration quantity from Algorithm 1. The logarithmic estimate gives $ℓ_{n} \leq b_{n} .$ The degree calibration inequality is $\frac{c ^{\circ} b _{n}}{2} \leq L, c^{\circ} = \frac{1}{256},$ so $b_{n} \leq 512 L$ . Therefore $ℓ_{n} \leq 512 L .$ The polynomial-size condition $L^{6} A_{⋆}^{2 L} \leq n$ gives $(L^{3} A_{⋆}^{L})^{2} = L^{6} A_{⋆}^{2 L} \leq n .$ Finally, since $n \geq 1$ , $0 \leq η = n^{- 12} \leq \frac{1}{n} .$
Put $Θ = L^{3} A = L^{3} A_{⋆}^{L} .$ From $Θ^{2} \leq n$ , the elementary inequality $(x - y)^{2} \geq 0$ , applied with $x = \frac{Θ}{n}, y = \frac{d}{N ℓ _{n}},$ gives $\frac{Θ d}{N n ℓ _{n}} \leq \frac{1}{2} {(\frac{Θ}{n})^{2} + (\frac{d}{N ℓ _{n}})^{2}} \leq \frac{1}{2} (\frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}}) = \frac{Ξ}{2} .$ Also, from $ℓ_{n} \leq 512 L$ and positivity of $ℓ_{n}, L$ , $\frac{1}{L} \leq \frac{512}{ℓ _{n}}, \frac{1}{L ^{2}} \leq \frac{51 2 ^{2}}{ℓ _{n}^{2}} .$
The reciprocal block bounds are $u^{- 1} \leq \frac{32}{n}, t^{- 1} \leq \frac{32}{n},$ because $u \geq n /32$ and $t \geq N /32 \geq n /32$ . Hence $C_{0} (u^{- 1} + t^{- 1}) \leq 64 C_{0} \frac{1}{n},$ $\frac{4}{ϵ t} = \frac{4}{ϵ} t^{- 1} \leq \frac{128}{ϵ} \frac{1}{n},$ and $414 η \leq 414 \frac{1}{n} .$ Since $1/ n \leq Ξ$ , these estimates combine to $C_{0} (u^{- 1} + t^{- 1}) + \frac{4}{ϵ t} + 414 η \leq (64 C_{0} + \frac{128}{ϵ} + 414) Ξ.$
The two core normalized products obey $\frac{Θ d}{N n L ^{2}} \leq \frac{Θ d}{N n} \frac{51 2 ^{2}}{ℓ _{n}^{2}} \leq 51 2^{2} \frac{Θ d}{N n ℓ _{n}} \leq 51 2^{2} \frac{Ξ}{2},$ where the middle step uses $ℓ_{n} \geq 1$ , and $\frac{Θ d}{N ^{2} L} \leq \frac{Θ d}{N ^{2}} \frac{512}{ℓ _{n}} \leq 512 \frac{Θ d}{N n ℓ _{n}} \leq 512 \frac{Ξ}{2},$ where $N \geq n$ is used in the second inequality.
The first light-envelope term satisfies $4 A \frac{min { 1 , d B }}{u} \leq 4 A \frac{d B}{u} \leq 4 A \frac{d ( 32 H _{ϵ} L / N )}{n /32} = 4096 H_{ϵ} \frac{Θ d}{N n L ^{2}} \leq (4096 H_{ϵ} 51 2^{2}) \frac{Ξ}{2} .$ The second light-envelope term satisfies $4 A d B^{2} \leq 4 A d (\frac{32 H _{ϵ} L}{N})^{2} = 4096 H_{ϵ}^{2} \frac{Θ d}{N ^{2} L} \leq (4096 H_{ϵ}^{2} 512) \frac{Ξ}{2} .$ Adding the two bounds and using $Ξ \geq 0$ gives $4 A {\frac{min { 1 , d B }}{u} + d B^{2}} \leq (4096 H_{ϵ} 51 2^{2} + 4096 H_{ϵ}^{2} 512) Ξ.$
Since $d \geq 1$ , $d \leq d^{2} .$ Consequently, $\frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} \leq \frac{8 d}{ϵ ^{2} L ^{4}} (\frac{32 H _{ϵ} L}{N})^{2} = \frac{8192 H _{ϵ}^{2}}{ϵ ^{2}} \frac{d}{N ^{2} L ^{2}} .$ Using $1/ L^{2} \leq 51 2^{2} / ℓ_{n}^{2}$ and $d \leq d^{2}$ , $\frac{d}{N ^{2} L ^{2}} \leq \frac{51 2 ^{2} d ^{2}}{N ^{2} ℓ _{n}^{2}} .$ Because $d^{2} / (N^{2} ℓ_{n}^{2}) \leq Ξ$ , we obtain $\frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} \leq \frac{8192 H _{ϵ}^{2} 51 2 ^{2}}{ϵ ^{2}} Ξ.$
For the squared bias term, $\frac{2 d B}{ϵ L ^{2}} \leq \frac{2 d}{ϵ L ^{2}} \frac{32 H _{ϵ} L}{N} = \frac{64 H _{ϵ}}{ϵ} \frac{d}{N L} \leq \frac{64 H _{ϵ} 512}{ϵ} \frac{d}{N ℓ _{n}} .$ Both sides are nonnegative, so squaring gives $(\frac{2 d B}{ϵ L ^{2}})^{2} \leq (\frac{64 H _{ϵ} 512}{ϵ})^{2} \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}} \leq (\frac{64 H _{ϵ} 512}{ϵ})^{2} Ξ.$ Also $0 \leq η \leq 1/ n \leq 1$ , hence $(6 η)^{2} = 36 η^{2} \leq 36 η \leq \frac{36}{n} .$ The inequality $(x + y)^{2} \leq 2 x^{2} + 2 y^{2}$ , applied to $x = \frac{2 d B}{ϵ L ^{2}}, y = 6 η,$ therefore yields $(\frac{2 d B}{ϵ L ^{2}} + 6 η)^{2} \leq {2 (\frac{64 H _{ϵ} 512}{ϵ})^{2} + 72} Ξ.$
Combining the light-envelope, reciprocal, variance-polynomial, and squared-bias estimates gives $4 A {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{0} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 414 η + (\frac{2 d B}{ϵ L ^{2}} + 6 η)^{2} \leq K_{ϵ} (C_{0}) Ξ.$ Since $0 \leq Ξ$ and $K_{ϵ} (C_{0}) \leq C$ , the right-hand side is at most $C Ξ$ . Substituting back $A = A_{⋆}^{L}, η = n^{- 12}, Ξ = \frac{1}{n} + \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}},$ with $N = n + m$ and $ℓ_{n} = lo g (e n)$ from Definition 12, is precisely the asserted inequality.

∎

Lemma 7 compares the light polynomial contribution with the inverse-count contribution on cells whose covariate mass is at most $B$ . The bound has a Chebyshev approximation term and a factorial-pool intensity term, matching the light-branch mean terms that later enter the calibrated envelope.

Lemma 7 [lem:hybrid-light-branch-mean-bound] (Light branch mean bound).

Let $P \in M_{d, ϵ}$ be a law in the finite model class of Definition 6. Let $u, t, B > 0$ and let $L \geq 2$ be an integer. For each $x \in [d]$ , write $p_{x} = P (X = x), q_{a x} = P (X = x, A = a, Y = 1), s_{a x} = P (X = x, A = a) .$ For each $x \in [d]$ , let $S_{0 x}, S_{1 x}, K_{0 x}, K_{1 x}$ be independent Poisson counts with $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}), a \in {0, 1} .$ Using the light factorial-lift weights and heavy inverse-count weights from Algorithm 1, set $P_{x} = \frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}, H_{x} = \frac{S _{1 x}}{u} W_{1, x}^{H} - \frac{S _{0 x}}{u} W_{0, x}^{H},$ where the light weights use degree $L$ , scale $B$ , and intensity $t$ . Then $x : p_{x} \leq B \sum (E P_{x} - E H_{x})^{2} \leq \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} .$

⊢ Lean

Proof of Lemma 7.

Let $θ_{x} := p_{x} (μ_{1 x} - μ_{0 x}), ℓ_{x} := E P_{x}, h_{x} := E H_{x},$ where $μ_{a x} = P (Y = 1 ∣ X = x, A = a)$ , with the usual harmless value when $s_{a x} = 0$ . Let $S_{B} = {x \in [d] : p_{x} \leq B}$ . The cell masses are nonnegative and $\sum_{x \in [d]} p_{x} = 1$ , since $P$ is a probability law. We prove the bound by first controlling each summand and then summing over $S_{B}$ .

Step 1. Fix $x \in S_{B}$ . The light-branch expectation equals $ℓ_{x} = q_{1 x} \frac{p _{x}}{B} G_{L} (s_{1 x} / B) - q_{0 x} \frac{p _{x}}{B} G_{L} (s_{0 x} / B) .$ To verify the first equality, write the light polynomial as $W_{a, L} (z_{0}, z_{1}) = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} z_{0}^{r_{0}} z_{1}^{r_{1}} .$ The factorial lift in Definition 24 is $W_{a, L, x} = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} \frac{( K _{0 x} ) _{r_{0}} ( K _{1 x} ) _{r_{1}}}{t ^{r_{0} + r_{1}}} .$ Independence and the Poisson factorial-moment identity $E [(K_{a x})_{r}] = (t s_{a x})^{r}$ therefore give $E W_{a, L, x} = W_{a, L} (s_{0 x}, s_{1 x})$ . Independently, $E (S_{a x} / u) = q_{a x}$ , which proves the displayed formula for $ℓ_{x}$ . The light Chebyshev estimate then gives $∣ ℓ_{x} - θ_{x} ∣ \leq \frac{2 B}{ϵ L ^{2}} .$ Indeed $p_{x} \leq B$ , $s_{0 x} + s_{1 x} = p_{x}$ , and Assumption 1 gives $s_{a x} \geq ϵ p_{x}$ for each arm $a$ whenever $p_{x} > 0$ . If $p_{x} = 0$ , then $s_{0 x} = s_{1 x} = q_{0 x} = q_{1 x} = 0$ , so the displayed bound is immediate. If $p_{x} > 0$ , the shifted Chebyshev certificate in Lemma 5, together with the definitions of $E_{L}$ and $G_{L}$ in Algorithm 1, gives, for $z_{a} = s_{a x} / B \in [0, 1]$ , $0 \leq E_{L} (z_{a}) \leq \frac{1}{L ^{2} z _{a}}, 1 - z_{a} G_{L} (z_{a}) = E_{L} (z_{a}) .$ Writing $q_{a x} = s_{a x} μ_{a x}$ , the light-branch error becomes a linear combination of the two terms $p_{x} μ_{a x} E_{L} (s_{a x} / B)$ . Since $0 \leq μ_{a x} \leq 1$ and $s_{a x} \geq ϵ p_{x}$ , each arm contributes at most $B / (ϵ L^{2})$ , giving the stated two-arm bound.

Step 2. The heavy inverse-count branch satisfies $∣ h_{x} - θ_{x} ∣ \leq 2 p_{x} exp (- ϵ t p_{x}) .$ Write $u_{out} := u$ for the outcome-block intensity. For each arm $a \in {0, 1}$ , set $η_{a x} := E [\frac{S _{a x}}{u _{out}} \frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1}] .$ Since the outcome pool is independent of the factorial pool and $E S_{a x} = u_{out} q_{a x}$ , $η_{a x} = q_{a x} E [\frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1}] .$ The armwise inverse-count expectation formula in Lemma 8, applied to the independent factorial counts, gives $η_{a x} = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} exp (- t s_{a x}) .$ Consequently $∣ η_{a x} - p_{x} μ_{a x} ∣ = (p_{x} - s_{a x}) μ_{a x} exp (- t s_{a x}) \leq p_{x} exp (- ϵ t p_{x}),$ because $0 \leq p_{x} - s_{a x} \leq p_{x}$ , $0 \leq μ_{a x} \leq 1$ , and occupied-cell overlap gives $s_{a x} \geq ϵ p_{x}$ ; when $p_{x} = 0$ , both sides vanish. Since $h_{x} - θ_{x} = (η_{1 x} - p_{x} μ_{1 x}) - (η_{0 x} - p_{x} μ_{0 x}),$ the triangle inequality gives the displayed heavy-branch bound.

Step 3. Combining the two preceding bounds, $(ℓ_{x} - h_{x})^{2} \leq \frac{8 B ^{2}}{ϵ ^{2} L ^{4}} + 8 p_{x}^{2} exp (- 2 ϵ t p_{x}) .$ To see this, write $ℓ_{x} - h_{x} = (ℓ_{x} - θ_{x}) - (h_{x} - θ_{x})$ and use $(r - s)^{2} \leq 2 r^{2} + 2 s^{2}$ with $∣ r ∣ \leq 2 B / (ϵ L^{2})$ and $∣ s ∣ \leq 2 p_{x} exp (- ϵ t p_{x})$ .

Step 4. For every $x$ , $p_{x}^{2} exp (- 2 ϵ t p_{x}) \leq \frac{p _{x}}{2 ϵe t} .$ Indeed, $p_{x}^{2} exp (- 2 ϵ t p_{x}) = p_{x} {p_{x} exp (- (2 ϵ) t p_{x})} .$ The elementary envelope $z exp (- α t z) \leq (α e t)^{- 1}$ , applied with $z = p_{x}$ and $α = 2 ϵ$ , gives $p_{x} exp (- (2 ϵ) t p_{x}) \leq \frac{1}{2 ϵe t} .$ Multiplying by the nonnegative factor $p_{x}$ gives the displayed bound. The envelope itself follows from $e (α t z) \leq exp (α t z)$ , multiplied by $exp (- α t z)$ , and then divided by the positive number $α e t$ .

Step 5. Summing the pointwise bound over $S_{B}$ gives $x \in S_{B} \sum (ℓ_{x} - h_{x})^{2} \leq x \in S_{B} \sum {\frac{8 B ^{2}}{ϵ ^{2} L ^{4}} + 8 \frac{p _{x}}{2 ϵe t}} .$ Since $∣ S_{B} ∣ \leq d$ and $\sum_{x \in S_{B}} p_{x} \leq \sum_{x \in [d]} p_{x} = 1$ , $x \in S_{B} \sum (ℓ_{x} - h_{x})^{2} \leq \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{8}{2 ϵe t} .$ Finally $1 \leq e$ , so $8/ (2 ϵe t) \leq 4/ (ϵ t)$ . Substituting $ℓ_{x} = E P_{x}$ and $h_{x} = E H_{x}$ yields $x : p_{x} \leq B \sum (E P_{x} - E H_{x})^{2} \leq \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t},$ as claimed.

∎

The comparison in Lemma 7 is anchored by an exact arm-level identity for the heavy inverse-count mean. The identity expresses the inverse-count arm contribution as the desired cell term plus an exponentially damped residual.

Lemma 8 [lem:poisson-inverse-arm-mean-formula] (Inverse-count arm mean).

Let $P$ be an observed-data law on $[d] \times {0, 1}^{2}$ , let $x \in [d]$ , and let $a \in {0, 1}$ . Write $p_{x}$ for the covariate-cell mass, $s_{a x}$ for the treatment-covariate cell mass, $q_{a x}$ for the outcome-marked mass, and $μ_{a x} := q_{a x} / s_{a x}$ for the totalized outcome regression. If $u > 0$ and $t > 0$ , and if $K_{a x} \sim Poi (t s_{a x})$ and $K_{1 - a, x} \sim Poi (t s_{1 - a, x})$ are independent, then $\frac{q _{a x}}{u / u} E [1 + \frac{K _{1 - a, x}}{K _{a x} + 1}] = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} exp (- t s_{a x}) .$

⊢ Lean

Proof of Lemma 8.

1. Put $\overset{a}{ˉ} := 1 - a$ and write $Λ_{a} := u q_{a x}, Θ_{a} := t s_{a x}, Θ_{\overset{a}{ˉ}} := t s_{\overset{a}{ˉ} x} .$ The cell probabilities are nonnegative, so $s_{a x} \geq 0$ , $s_{\overset{a}{ˉ} x} \geq 0$ , and $q_{a x} \geq 0$ . Since $u, t > 0$ , the nonnegative rates above are the real values of the corresponding Poisson intensities. Also, $s_{a x} + s_{\overset{a}{ˉ} x} = p_{x},$ because the two treatment arms partition the covariate cell $x$ .

2. Define $Ω_{a} := E [1 + \frac{K _{\overset{a}{ˉ} x}}{K _{a x} + 1}], K_{a x} \sim Poi (Θ_{a}), K_{\overset{a}{ˉ} x} \sim Poi (Θ_{\overset{a}{ˉ}}),$ with the two counts independent. The product-Poisson identity used for this expectation is $Θ_{a} Ω_{a} = Θ_{a} + Θ_{\overset{a}{ˉ}} (1 - e^{- Θ_{a}}) .$ Indeed, for $r (k) := (k + 1)^{- 1}$ , $Ω_{a} = 1 + \int v r (k) d (Poi (Θ_{a}) \otimes Poi (Θ_{\overset{a}{ˉ}})) (k, v) .$ Fubini factorizes the cross term as $\int v r (k) d (Poi (Θ_{a}) \otimes Poi (Θ_{\overset{a}{ˉ}})) = (\int r (k) d Poi (Θ_{a}) (k)) (\int v d Poi (Θ_{\overset{a}{ˉ}}) (v)) .$ The Poisson first moment gives $\int v d Poi (Θ_{\overset{a}{ˉ}}) (v) = Θ_{\overset{a}{ˉ}},$ and the shifted reciprocal identity gives $Θ_{a} \int \frac{1}{k + 1} d Poi (Θ_{a}) (k) = 1 - e^{- Θ_{a}} .$ Multiplying the displayed formula for $Ω_{a}$ by $Θ_{a}$ yields the claimed identity.

3. Substituting $Θ_{a} = t s_{a x}$ , $Θ_{\overset{a}{ˉ}} = t s_{\overset{a}{ˉ} x}$ , and $s_{\overset{a}{ˉ} x} = p_{x} - s_{a x}$ , the preceding display becomes $(t s_{a x}) Ω_{a} = t s_{a x} + t (p_{x} - s_{a x}) (1 - e^{- t s_{a x}}) .$ Because $t > 0$ , cancellation of $t$ gives $s_{a x} Ω_{a} = s_{a x} + (p_{x} - s_{a x}) (1 - e^{- t s_{a x}}) .$ Multiplying by $μ_{a x}$ and rearranging, $(s_{a x} μ_{a x}) Ω_{a} = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} e^{- t s_{a x}} .$

4. Finally, $q_{a x} = s_{a x} μ_{a x} .$ This is immediate from $μ_{a x} = q_{a x} / s_{a x}$ when $s_{a x} \neq = 0$ . When $s_{a x} = 0$ , the bound $0 \leq q_{a x} \leq s_{a x}$ , obtained by summing the nonnegative outcome masses in the arm cell, gives $q_{a x} = 0$ , and the same identity holds under the totalized division convention. Hence, since the normalized outcome count has mean $q_{a x}$ , $q_{a x} E [1 + \frac{K _{\overset{a}{ˉ} x}}{K _{a x} + 1}] = (s_{a x} μ_{a x}) Ω_{a} = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} e^{- t s_{a x}},$ as claimed.

∎

Lemma 8 supplies the cellwise bias formula used when the heavy branch is compared with the target ATE contribution. The exponential factor is controlled by the treatment–covariate intensity, so the identity connects the heavy branch to the same marginal scale that appears in Lemma 2.

A second auxiliary bound controls the event that a truly heavy cell is nevertheless routed to the light polynomial branch. It combines the pilot probability of such a routing event with the polynomial second moment under the factorial-pool law.

Lemma 9 [lem:heavy-light-second-moment-sum] (Heavy light second moment).

Let $d$ be a finite alphabet size, let $P$ be an observed-data law on $[d] \times {0, 1}^{2}$ , and let $n, m$ be integers. Suppose that:

(Positive overlap parameter.) The overlap parameter satisfies $ϵ > 0$ .

(Sample size.) The labeled sample size satisfies $n \geq 1$ .

(Calibration.) The finite calibration predicate $C_{ϵ} (n, m)$ from Algorithm 1 holds.

Set $u = \frac{M _{0}}{8}, t_{p} = \frac{M _{p}}{8}, t = \frac{M _{f}}{8}, L = L (n), B = B (n, m, ϵ), k_{0} = k_{0} (n, m, ϵ),$ with these quantities defined as in Algorithm 1. For each $x \in [d]$ , let $π_{x} = P {J_{x} \leq k_{0}},$ where, conditionally on $P$ , the pilot counts have independent components $J_{a x} \sim Poi (t_{p} s_{a x})$ , $a \in {0, 1}$ . Let $μ_{x}$ be the product law under which the outcome and factorial counts are independent with $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}), a \in {0, 1},$ and define the light polynomial cell contribution $P_{x} = \frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x},$ where the factorial lift is taken at degree $L$ , scale $B$ , and intensity $t$ . Then $x : p_{x} > B \sum π_{x} E_{μ_{x}} [P_{x}^{2}] \leq 132 n^{- 12} .$

⊢ Lean

Proof of Lemma 9.

Let $Heavy_{B} = {x \in [d] : p_{x} > B}, p_{x} = P_{P} {X = x} .$ The quantities $u, t_{p}, t, L, B, k_{0}$ are those fixed in Algorithm 1. For each $x$ , put $Ξ_{x} = \int [\frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}]^{2} d μ_{x} .$ Thus the left-hand side is $\sum_{x \in Heavy_{B}} π_{x} Ξ_{x}$ .

The cell masses satisfy $0 \leq p_{x} \leq 1 for every x \in [d], x \in [d] \sum p_{x} = 1.$ Hence, for the finite set $Heavy_{B}$ , $x \in Heavy_{B} \sum p_{x} \leq 1, x \in Heavy_{B} \sum p_{x}^{2} \leq x \in Heavy_{B} \sum p_{x} \leq 1.$ The calibration predicate gives $u, t_{p}, t, B > 0$ , $L \geq 2$ , $L \leq tB$ , and $1/32 \leq u / n$ . Since $n \geq 1$ , $u \geq \frac{n}{32} \geq \frac{1}{32}, u^{- 1} \leq 32.$ These inequalities also cover the case $Heavy_{B} = \emptyset$ , where the sums are zero.
Fix $x \in Heavy_{B}$ , and define $y_{x} = \frac{p _{x}}{B} .$ Because $B > 0$ and $p_{x} > B$ , $y_{x} \geq 1$ . Also $p_{x} = B y_{x}$ . The pilot count is $J_{x} = J_{0 x} + J_{1 x}, J_{a x} \sim Poi (t_{p} s_{a x}),$ with independent arms and $s_{0 x} + s_{1 x} = p_{x}$ . Since $k_{0} = ⌊ t_{p} B /4 ⌋$ and $t_{p} B \geq 0$ , $k_{0} \leq \frac{t _{p} B}{4} < \frac{t _{p} p _{x}}{4} .$ The standard Poisson lower-tail bound therefore gives $π_{x} = P {J_{x} \leq k_{0}} \leq exp (- \frac{t _{p} p _{x}}{4}) .$ For the light-polynomial second moment, Lemma 16 supplies the factorial-lift bound $E [W_{a, L, x}^{2}] \leq A_{⋆}^{L} (1 + \frac{p _{x}}{B})^{2 L}, a \in {0, 1} .$ Writing $q_{a x} = P_{P} {X = x, A = a, Y = 1}$ , independence of $S_{a x}$ and the factorial counts gives $E [(\frac{S _{a x}}{u} W_{a, L, x})^{2}] = (\frac{q _{a x}}{u} + q_{a x}^{2}) E [W_{a, L, x}^{2}] .$ Since $0 \leq q_{a x} \leq p_{x}$ , and using $(r - s)^{2} \leq 2 r^{2} + 2 s^{2}$ , $Ξ_{x} \leq 4 (\frac{p _{x}}{u} + p_{x}^{2}) A_{⋆}^{L} (1 + \frac{p _{x}}{B})^{2 L} .$ Consequently, $π_{x} Ξ_{x} \leq 4 (\frac{p _{x}}{u} + p_{x}^{2}) A_{⋆}^{L} (1 + y_{x})^{2 L} exp (- \frac{t _{p} p _{x}}{4}) .$
The definition $B = H_{ϵ} L / min (t_{p}, t)$ , together with positivity of $t_{p}, t$ , gives $H_{ϵ} L \leq t_{p} B .$ Therefore $exp (- \frac{t _{p} p _{x}}{4}) \leq exp (- \frac{H _{ϵ} L y _{x}}{4}) .$ The numerical choice of $H_{ϵ}$ in Algorithm 1 includes $15 + \frac{24}{c ^{\circ}} \leq \frac{H _{ϵ}}{4} .$ Using $lo g A_{⋆} \leq 13$ , $lo g (1 + y_{x}) \leq y_{x}$ , and $y_{x} \geq 1$ , $lo g [A_{⋆}^{L} (1 + y_{x})^{2 L} exp (- \frac{H _{ϵ} L y _{x}}{4})] = L lo g A_{⋆} + 2 L lo g (1 + y_{x}) - \frac{H _{ϵ} L y _{x}}{4} \leq 15 L y_{x} - \frac{H _{ϵ} L y _{x}}{4} \leq - \frac{24 L y _{x}}{c ^{\circ}} .$ Thus $A_{⋆}^{L} (1 + y_{x})^{2 L} exp (- \frac{t _{p} p _{x}}{4}) \leq exp (- \frac{24 L y _{x}}{c ^{\circ}}) .$ The calibration predicate also gives $c^{\circ} b_{n} /2 \leq L$ , where $b_{n} = 1 + ⌈ lo g_{2} n ⌉$ . Since $y_{x} \geq 1$ , $12 b_{n} \leq \frac{24 L y _{x}}{c ^{\circ}} .$ Finally $b_{n}$ dominates $lo g n$ for $n \geq 1$ , so $exp (- 12 b_{n}) \leq n^{- 12} .$ Combining the last three displays yields $A_{⋆}^{L} (1 + y_{x})^{2 L} exp (- \frac{t _{p} p _{x}}{4}) \leq n^{- 12} .$ Substituting this into the bound from the previous step gives, for every $x \in Heavy_{B}$ , $π_{x} Ξ_{x} \leq 4 (\frac{p _{x}}{u} + p_{x}^{2}) n^{- 12} .$
Summing the preceding cellwise inequality over $x \in Heavy_{B}$ , $x \in Heavy_{B} \sum π_{x} Ξ_{x} \leq 4 \frac{\sum _{x \in Heavy_{B}} p _{x}}{u} + x \in Heavy_{B} \sum p_{x}^{2} n^{- 12} .$ By the bounds from the first step, $\frac{\sum _{x \in Heavy_{B}} p _{x}}{u} \leq u^{- 1} \leq 32, x \in Heavy_{B} \sum p_{x}^{2} \leq 1.$ Therefore $x \in Heavy_{B} \sum π_{x} Ξ_{x} \leq 4 (32 + 1) n^{- 12} = 132 n^{- 12} .$ This is the claimed bound.

∎

Lemma 9 makes the heavy-cell misclassification contribution negligible on the calibrated scale. Its summand contains both the pilot routing probability and the light-branch second moment, so it is the variance-side counterpart of the heavy-branch mean control.

Lemma 10 controls the contribution from heavy cells near the pilot decision boundary. The product factor $π_{x} η_{x}$ records that only disagreement between the light and heavy branches matters, and the calibrated split in Algorithm 1 makes the summed contribution polynomially small in the labeled sample size.

Lemma 10 [lem:hybrid-heavy-branch-mean-bound] (Heavy branch mean bound).

Let $d$ be a finite alphabet size, let $ϵ > 0$ , and let $P \in M_{d, ϵ}$ as in Definition 6. Let $n \geq 1$ and $m \geq 0$ be integers. Assume:

(Calibration.) The hybrid-estimator calibration predicate $C_{ϵ} (n, m)$ in Algorithm 1 holds.

(Block intensities.) With $M_{0}, n_{p}, n_{f}, m_{p}, m_{f}$ as in Algorithm 1, set $u = \frac{M _{0}}{8}, t_{p} = \frac{n _{p} + m _{p}}{8}, t = \frac{n _{f} + m _{f}}{8} .$

(Cell laws.) For each $x \in [d]$ , let $μ_{x}$ be the product law under which the outcome-block success counts $S_{a x}$ are independent $Poi (u q_{a x})$ variables and the factorial-pool counts $K_{a x}$ are independent $Poi (t s_{a x})$ variables, for $a \in {0, 1}$ .

Let $L = L (n)$ , $B = B (n, m, ϵ)$ , and $k_{0} = k_{0} (n, m, ϵ)$ be the degree, light-cell scale, and pilot threshold from Algorithm 1. For each $x \in [d]$ , let $π_{x} = P {J_{x} \leq k_{0}}, η_{x} = P {J_{x} > k_{0}},$ where $J_{x} = J_{0 x} + J_{1 x}$ has independent components $J_{a x} \sim Poi (t_{p} s_{a x})$ . Let $P_{x}$ be the light polynomial cell contribution and $H_{x}$ the heavy inverse-count cell contribution from Algorithm 1, evaluated under $μ_{x}$ . Then $x \in [d] : p_{x} > B \sum π_{x} η_{x} (E_{μ_{x}} P_{x} - E_{μ_{x}} H_{x})^{2} \leq 282 n^{- 12} .$

⊢ Lean

Proof of Lemma 10.

Let $H = {x \in [d] : p_{x} > B}, E_{x}^{L} = E_{μ_{x}} P_{x}, E_{x}^{H} = E_{μ_{x}} H_{x} .$ The calibration inequalities give $u > 0$ , $t_{p} > 0$ , $t > 0$ , and $B > 0$ . Also $0 \leq p_{x} \leq 1$ for every $x$ , and $\sum_{x} p_{x} = 1$ , hence $x \in H \sum p_{x} \leq 1.$

Fix $x \in H$ . We first establish the calibrated pilot-tail estimate $π_{x} \leq n^{- 12} .$ For $a \in {0, 1}$ , write $s_{a x}$ for the treatment-covariate mass in arm $a$ , so $p_{x} = s_{0 x} + s_{1 x}$ . Under the pilot law, $J_{x} = J_{0 x} + J_{1 x}$ has Poisson mean $λ_{x} = t_{p} (s_{0 x} + s_{1 x}) = t_{p} p_{x},$ because the two pilot-arm counts are independent Poisson variables. The definition of $k_{0}$ gives $k_{0} \leq t_{p} B /4$ , and $x \in H$ gives $B < p_{x}$ . Since $t_{p} > 0$ , $k_{0} < \frac{t _{p} p _{x}}{4} = \frac{λ _{x}}{4} .$ For a Poisson variable with mean $λ_{x}$ , the following lower-tail estimate applies whenever the cutoff is below one quarter of the mean: $P {J_{x} \leq k_{0}} \leq exp {- λ_{x} /4} .$ Indeed, for every $α > 0$ , Markov’s inequality applied to $e^{- α J_{x}}$ gives $P {J_{x} \leq k_{0}} \leq exp {λ_{x} (e^{- α} - 1) + α k_{0}} .$ If $k_{0} = 0$ , this is $e^{- λ_{x}} \leq e^{- λ_{x} /4}$ . If $k_{0} \geq 1$ , take $α = lo g (λ_{x} / k_{0})$ . With $ζ_{x} = k_{0} / λ_{x} \leq 1/4$ , the exponent becomes $λ_{x} {- 1 + ζ_{x} - ζ_{x} lo g ζ_{x}} \leq - λ_{x} /4,$ since $z - z lo g z \leq 1/4 + (lo g 4) /4 < 3/4$ on $0 \leq z \leq 1/4$ . Therefore $π_{x} = P {J_{x} \leq k_{0}} \leq exp {- t_{p} p_{x} /4} .$ Let $h_{n} = 1 + ⌈ lo g_{2} n ⌉$ be the binary-length calibration quantity in Algorithm 1. The tail calibration inequality $15 L + 12 h_{n} \leq \frac{t _{p} B}{4},$ together with $p_{x} > B$ , yields $\frac{t _{p} p _{x}}{4} \geq 12 h_{n} .$ Thus $π_{x} \leq exp {- 12 h_{n}} \leq n^{- 12},$ where the last inequality follows from $n \geq 1$ and $h_{n} \geq lo g n$ .

The heavy inverse-count mean satisfies $∣ E_{x}^{H} ∣ \leq 3 p_{x} .$ To prove this, first write $θ_{x} = p_{x} (μ_{1 x} - μ_{0 x}),$ where, for $a \in {0, 1}$ , $q_{a x}$ is the outcome-marked mass, $s_{a x}$ is the treatment-covariate mass, and the arm outcome regression is characterized by $q_{a x} = s_{a x} μ_{a x}, 0 \leq μ_{a x} \leq 1.$ For the arm-specific inverse-count means set $\overline{H}_{a, x} = E_{μ_{x}} [\frac{S _{a x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1}], a \in {0, 1} .$ By the definition of the heavy cell contribution, $E_{x}^{H} = \overline{H}_{1, x} - \overline{H}_{0, x} .$ We next compute each arm mean. Since $S_{a x}$ is independent of $(K_{0 x}, K_{1 x})$ and $E S_{a x} = u q_{a x}$ , $\overline{H}_{a, x} = q_{a x} E [\frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1}] .$ For the chosen arm $a$ , the fraction inside the expectation is $\frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1} = 1 + \frac{K _{1 - a, x}}{K _{a x} + 1},$ where $1 - a$ denotes the opposite arm. Therefore Lemma 8, applied with this arm and its opposite arm, gives the exact identity $\overline{H}_{a, x} = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} exp {- t s_{a x}} .$ Since $x \in H$ , the cell is occupied. Hence $P \in M_{d, ϵ}$ and Definition 6 and Assumption 1 give, for both arms, $ϵ p_{x} \leq s_{a x} \leq p_{x} .$ Using this, $0 \leq μ_{a x} \leq 1$ , and $t > 0$ , the exact identity yields the armwise envelope $∣ \overline{H}_{a, x} - p_{x} μ_{a x} ∣ = ∣ - (p_{x} - s_{a x}) μ_{a x} exp {- t s_{a x}} ∣ = (p_{x} - s_{a x}) μ_{a x} exp {- t s_{a x}} \leq p_{x} exp {- ϵ t p_{x}} .$ The first equality uses the displayed mean formula, the second uses nonnegativity of every factor, and the last step uses $p_{x} - s_{a x} \leq p_{x}$ and $exp {- t s_{a x}} \leq exp {- ϵ t p_{x}}$ . Hence $∣ E_{x}^{H} - θ_{x} ∣ = (\overline{H}_{1, x} - p_{x} μ_{1 x}) - (\overline{H}_{0, x} - p_{x} μ_{0 x}) \leq ∣ \overline{H}_{1, x} - p_{x} μ_{1 x} ∣ + ∣ \overline{H}_{0, x} - p_{x} μ_{0 x} ∣ \leq 2 p_{x} exp {- ϵ t p_{x}} .$

Since $0 \leq p_{x} \leq 1$ and $μ_{1 x}, μ_{0 x} \in [0, 1]$ , $∣ θ_{x} ∣ \leq p_{x} .$ Also, because $ϵ > 0$ , $t > 0$ , and $p_{x} \geq 0$ , $exp {- ϵ t p_{x}} \leq 1.$ Thus $∣ E_{x}^{H} ∣ \leq ∣ E_{x}^{H} - θ_{x} ∣ + ∣ θ_{x} ∣ \leq 2 p_{x} + p_{x} = 3 p_{x} .$

Now $η_{x} \leq 1$ , $π_{x} \geq 0$ , and $η_{x} \geq 0$ . Using $(E_{x}^{L} - E_{x}^{H})^{2} \leq 2 (E_{x}^{L})^{2} + 2 (E_{x}^{H})^{2}$ and the preceding heavy-mean bound, $(E_{x}^{H})^{2} \leq 9 p_{x}^{2} \leq 9 p_{x},$ where the last inequality uses $0 \leq p_{x} \leq 1$ . Hence $π_{x} η_{x} (E_{x}^{L} - E_{x}^{H})^{2} \leq π_{x} (E_{x}^{L} - E_{x}^{H})^{2} \leq 2 π_{x} (E_{x}^{L})^{2} + 2 π_{x} (E_{x}^{H})^{2} \leq 2 π_{x} (E_{x}^{L})^{2} + 18 p_{x} n^{- 12} .$

Summing this pointwise estimate over $x \in H$ gives $x \in H \sum π_{x} η_{x} (E_{x}^{L} - E_{x}^{H})^{2} \leq 2 x \in H \sum π_{x} (E_{x}^{L})^{2} + 18 n^{- 12} x \in H \sum p_{x} .$

It remains to bound the first sum. For each $x$ , nonnegativity of variance under $μ_{x}$ gives $(E_{x}^{L})^{2} \leq E_{μ_{x}} [P_{x}^{2}] .$ Consequently, $x \in H \sum π_{x} (E_{x}^{L})^{2} \leq x \in H \sum π_{x} E_{μ_{x}} [P_{x}^{2}] .$ The heavy-light second-moment bound in Lemma 9, applied with the same values of $u, t_{p}, t, L, B, k_{0}$ , gives $x \in H \sum π_{x} E_{μ_{x}} [P_{x}^{2}] \leq 132 n^{- 12} .$

Combining the last two displays with $\sum_{x \in H} p_{x} \leq 1$ , $x \in H \sum π_{x} η_{x} (E_{x}^{L} - E_{x}^{H})^{2} \leq 2 \cdot 132 n^{- 12} + 18 n^{- 12} = 282 n^{- 12} .$ This is the desired bound.

∎

The preceding local estimates are assembled into separate variance and mean-bias bounds for the full Poisson hybrid statistic. These two summaries expose exactly which terms enter the final rate calculation.

Lemma 11 [lem:calibrated-hybrid-variance-bound] (Hybrid variance bound).

Let $ϵ$ , the overlap parameter, satisfy $0 < ϵ < 1/2$ . There is a constant $C_{var} > 0$ , depending only on $ϵ$ , such that the following holds. Let $n, m, d$ be integers and let $P$ be an observed-data law on $[d] \times {0, 1}^{2}$ . Assume:

(Sample size.) $n \geq 1$ .

(Model class.) $P \in M_{d, ϵ}$ , as in Definition 6.

(Calibration.) The finite calibration predicate $C_{ϵ} (n, m)$ in Algorithm 1 holds.

Let $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ (n - M_{0}) /2 ⌋, n_{f} = n - M_{0} - n_{p}, m_{p} = ⌊ m /2 ⌋, m_{f} = m - m_{p},$ and set $u = M_{0} /8, t_{p} = (n_{p} + m_{p}) /8, t = (n_{f} + m_{f}) /8.$ Let $L = L (n)$ , $B = B (n, m, ϵ)$ , $k_{0} = k_{0} (n, m, ϵ)$ , and $A_{⋆} = 7056$ be the degree, scale, pilot threshold, and factorial-moment constant from Algorithm 1. Under the independent Poisson count law in which, independently over $x \in [d]$ and $a \in {0, 1}$ , $S_{a x} \sim Poi (u q_{a x}), J_{a x} \sim Poi (t_{p} s_{a x}), K_{a x} \sim Poi (t s_{a x}),$ let $Z = x \in [d] \sum (1 {J_{0 x} + J_{1 x} \leq k_{0}} P_{x} + 1 {J_{0 x} + J_{1 x} > k_{0}} H_{x}),$ where $P_{x}$ and $H_{x}$ are the light-cell polynomial and heavy inverse-count cell contributions specified in Algorithm 1. Then $Var (Z) \leq 4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{var} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 414 n^{- 12} .$

⊢ Lean

Proof of Lemma 11.

Choose $C_{var}$ to be the positive constant supplied, at the present overlap parameter $ϵ$ , by Lemma 2. Fix $n, m, d$ and $P$ satisfying the hypotheses. Write $p_{x} = P (X = x), q_{a x} = P (X = x, A = a, Y = 1), s_{a x} = P (X = x, A = a),$ and let $J_{x} = J_{0 x} + J_{1 x}$ . The calibration predicate in Algorithm 1 gives $u > 0, t_{p} > 0, t > 0, B > 0, L \geq 2, L \leq tB .$ For each $x \in [d]$ , let $Q_{x}$ be the product law of the outcome and factorial counts $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}), a \in {0, 1},$ with all four variables independent. Define $λ_{x} = P {J_{x} \leq k_{0}}, θ_{x} = P {J_{x} > k_{0}} = 1 - λ_{x},$ and, under $Q_{x}$ , $\overset{ˉ}{P}_{x} = E_{Q_{x}} P_{x}, \overset{ˉ}{H}_{x} = E_{Q_{x}} H_{x}, V_{x}^{P} = Var_{Q_{x}} (P_{x}), V_{x}^{H} = Var_{Q_{x}} (H_{x}) .$ The probabilities satisfy $0 \leq λ_{x}, θ_{x} \leq 1$ .

Since the pilot counts are independent of the outcome and factorial counts, the selected cell is a two-branch mixture. For each cell, $Var (1 {J_{x} \leq k_{0}} P_{x} + 1 {J_{x} > k_{0}} H_{x}) = λ_{x} V_{x}^{P} + θ_{x} V_{x}^{H} + λ_{x} θ_{x} (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2} .$ The cells are independent under the product Poisson law, so the variance of the sum is the sum of the cell variances: $Var (Z) = x \in [d] \sum {λ_{x} V_{x}^{P} + θ_{x} V_{x}^{H} + λ_{x} θ_{x} (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2}} .$
Bound the light-branch variance part by splitting the cells at the scale $B$ . Put $Light_{B} = {x \in [d] : p_{x} \leq B}, Heavy_{B} = {x \in [d] : p_{x} > B} .$ For $x \in Light_{B}$ , $λ_{x} \leq 1$ and $V_{x}^{P} \leq E_{Q_{x}} P_{x}^{2} .$ The factorial-lift second-moment estimate in Lemma 16, together with $(r - s)^{2} \leq 2 r^{2} + 2 s^{2}$ , gives $E_{Q_{x}} P_{x}^{2} \leq 4 A_{⋆}^{L} (\frac{p _{x}}{u} + p_{x}^{2}), x \in Light_{B} .$ Indeed, for each arm $a$ , the light factorial weight has second moment at most $A_{⋆}^{L}$ , while $E [(\frac{S _{a x}}{u})^{2}] = \frac{q _{a x}}{u} + q_{a x}^{2} \leq \frac{p _{x}}{u} + p_{x}^{2} .$ Summing over $Light_{B}$ and using $x \in Light_{B} \sum p_{x} \leq min {1, d B}, x \in Light_{B} \sum p_{x}^{2} \leq d B^{2},$ yields $x \in Light_{B} \sum λ_{x} V_{x}^{P} \leq 4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} .$ On the complementary cells, Lemma 9 gives $x \in Heavy_{B} \sum λ_{x} E_{Q_{x}} P_{x}^{2} \leq 132 n^{- 12} .$ Consequently, $x \in [d] \sum λ_{x} V_{x}^{P} \leq 4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + 132 n^{- 12} .$
For the heavy-branch variance part, Lemma 2 applied to the all-heavy statistic gives $Var x \in [d] \sum H_{x} \leq C_{var} (u^{- 1} + t^{- 1}) .$ Independence across cells gives $Var x \in [d] \sum H_{x} = x \in [d] \sum V_{x}^{H},$ and since $0 \leq θ_{x} \leq 1$ , $x \in [d] \sum θ_{x} V_{x}^{H} \leq C_{var} (u^{- 1} + t^{- 1}) .$
It remains to bound the branch-selection mean term. On $Light_{B}$ , the inequality $λ_{x} θ_{x} \leq 1$ and Lemma 7 give $x \in Light_{B} \sum λ_{x} θ_{x} (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2} \leq x \in Light_{B} \sum (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2} \leq \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} .$ On $Heavy_{B}$ , Lemma 10 gives $x \in Heavy_{B} \sum λ_{x} θ_{x} (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2} \leq 282 n^{- 12} .$ Thus $x \in [d] \sum λ_{x} θ_{x} (\overset{ˉ}{P}_{x} - \overset{ˉ}{H}_{x})^{2} \leq \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 282 n^{- 12} .$
Combining the three displayed bounds in the variance decomposition gives $Var (Z) \leq 4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{var} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + (132 + 282) n^{- 12} .$ Since $132 + 282 = 414$ , this is the asserted bound.

∎

Lemma 11 combines the light-cell factorial moment bound, the inverse-count variance contribution, and the pilot-routing error into a single variance inequality. Its display is arranged to match the envelope calculation in Lemma 6 term by term.

Lemma 12 [lem:calibrated-hybrid-mean-bias] (Calibrated hybrid mean bias).

Fix an overlap constant $0 < ϵ < 1/2$ . For every $n, m, d \in N$ and every observed-data law $P$ on $[d] \times {0, 1}^{2}$ , suppose that:

(Sample size.) $1 \leq n$ .

(Model class.) $P \in M_{d, ϵ}$ , as in Definition 6.

(Calibration.) The finite calibration predicate $C_{ϵ} (n, m)$ in Algorithm 1 holds.

Let $M_{0}, n_{p}, n_{f}, m_{p}, m_{f}$ be the block sizes from Algorithm 1, and set $u = \frac{M _{0}}{8}, t_{p} = \frac{n _{p} + m _{p}}{8}, t = \frac{n _{f} + m _{f}}{8} .$ Let $L = L (n)$ , $B = B (n, m, ϵ)$ , and $k_{0} = k_{0} (n, m, ϵ)$ be the degree, light-cell scale, and pilot threshold from Algorithm 1. For each $x \in [d]$ , let the hybrid selected cell mean be the expectation of the cell contribution selected at threshold $k_{0}$ , degree $L$ , scale $B$ , and intensities $(u, t_{p}, t)$ under the independent Poisson count law in Algorithm 1. Then $x \in [d] \sum E_{P} [Z_{x}] - τ (P) \leq \frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12},$ where $τ (P)$ is the average treatment effect functional in Definition 8.

⊢ Lean

Proof of Lemma 12.

For $x \in [d]$ and $a \in {0, 1}$ , write $p_{x} = P {X = x}, s_{a x} = P {X = x, A = a}, q_{a x} = P {X = x, A = a, Y = 1},$ and define the totalized outcome regression by the paper’s total real-division convention $μ_{a x} := \frac{q _{a x}}{s _{a x}},$ so that $μ_{a x} = 0$ when $s_{a x} = 0$ , because real division is totalized with $r /0 = 0$ . Put $ϑ_{x} := p_{x} (μ_{1 x} - μ_{0 x}) .$ Then $τ (P) = \sum_{x} ϑ_{x}$ , $\sum_{x} p_{x} = 1$ , and $∣ ϑ_{x} ∣ \leq p_{x},$ because $p_{x} \geq 0$ and the two binary outcome regressions lie in $[0, 1]$ . Write $b_{n}$ for the binary-length calibration quantity appearing in $C_{ϵ} (n, m)$ . Throughout this proof, $lo g$ denotes the natural logarithm; since $b_{n} = 1 + ⌈ lo g_{2} n ⌉$ , $n \geq 1$ implies $lo g n = (lo g 2) lo g_{2} n \leq b_{n}$ , and hence $exp (- 12 b_{n}) \leq n^{- 12}$ . We record the positivity consequences of the finite inequalities defining $C_{ϵ} (n, m)$ in Algorithm 1. Since $24 \leq n$ , we have $n > 0$ and $n + m > 0$ , so the denominators in the block-proportion bounds are positive. Those bounds give $\frac{1}{32} \leq \frac{u}{n}, \frac{1}{32} \leq \frac{t _{p}}{n + m}, \frac{1}{32} \leq \frac{t}{n + m},$ and hence $u > 0$ , $t_{p} > 0$ , and $t > 0$ . The scale inequality $8 \leq t_{p} B$ , together with $t_{p} > 0$ , then gives $B > 0$ . The calibration-scale definition in Definition 13 makes $L = L (n)$ a maximum whose first entry is $2$ ; in particular $L \geq 2$ . The model condition in Definition 6 places $P$ in the occupied-cell overlap class, and Assumption 1 gives, whenever $p_{x} > 0$ , $ϵ p_{x} \leq s_{0 x}, ϵ p_{x} \leq s_{1 x} .$ When $p_{x} = 0$ , the identity $s_{0 x} + s_{1 x} = p_{x}$ and nonnegativity of the two cell masses imply $s_{0 x} = s_{1 x} = 0$ , so the same inequalities remain valid.

For each cell, let $J_{0 x}^{p}$ and $J_{1 x}^{p}$ be the two independent pilot arm counts, with $J_{0 x}^{p} \sim Poi (t_{p} s_{0 x}), J_{1 x}^{p} \sim Poi (t_{p} s_{1 x}), J_{x} := J_{0 x}^{p} + J_{1 x}^{p} .$ Define $π_{x} := Pr {J_{x} \leq k_{0}}, η_{x} := Pr {J_{x} > k_{0}} .$ Thus $0 \leq π_{x}, η_{x} \leq 1$ and $η_{x} = 1 - π_{x}$ . Let $Q_{x}$ be the independent product law of four armwise Poisson counts $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}), a \in {0, 1},$ with all four variables mutually independent. These are ordinary nonnegative Poisson intensities, because $u, t > 0$ and the cell masses $q_{a x}$ and $s_{a x}$ are nonnegative. Define the light and heavy branch means $Λ_{x} := E_{Q_{x}} [\frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}],$ and $Γ_{x} := E_{Q_{x}} [\frac{S _{1 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{1 x} + 1} - \frac{S _{0 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{0 x} + 1}] .$ Let $M_{x}$ denote the selected cell mean under the independent Poisson count law described in the lemma: the pilot pool selects the light branch on ${J_{x} \leq k_{0}}$ and the heavy branch on ${J_{x} > k_{0}}$ , while the outcome and factorial pools are integrated under $Q_{x}$ . Thus $M_{x} = E [1_{{J_{x} \leq k_{0}}} (\frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}) + 1_{{J_{x} > k_{0}}} (\frac{S _{1 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{1 x} + 1} - \frac{S _{0 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{0 x} + 1})] .$ Independence of the pilot pool from the two evaluation pools gives the exact identity $M_{x} = π_{x} Λ_{x} + η_{x} Γ_{x} .$

We now derive the first-moment identity from the armwise Poisson factorial moments, using the factorial-lift definition in Definition 24. For $a \in {0, 1}$ , set $Λ_{a, x} := E_{Q_{x}} [\frac{S _{a x}}{u} W_{a, L, x}], Λ_{x} = Λ_{1, x} - Λ_{0, x} .$ Under $Q_{x}$ , the outcome count $S_{a x}$ is independent of the factorial-pool counts, and $E_{Q_{x}} S_{a x} = u q_{a x} .$ Write the light polynomial as in Definition 24: $W_{a, L} (r_{0}, r_{1}) = ρ_{0}, ρ_{1} \sum c_{ρ_{0}, ρ_{1}}^{(a)} r_{0}^{ρ_{0}} r_{1}^{ρ_{1}},$ with only finitely many nonzero coefficients. Its factorial lift is $W_{a, L, x} (K_{0 x}, K_{1 x}) = ρ_{0}, ρ_{1} \sum c_{ρ_{0}, ρ_{1}}^{(a)} \frac{( K _{0 x} ) _{ρ_{0}} ( K _{1 x} ) _{ρ_{1}}}{t ^{ρ_{0} + ρ_{1}}},$ where $(k)_{ρ}$ is the falling factorial. If $K \sim Poi (t s)$ with $s \geq 0$ , then, for every $ρ \geq 0$ , $E (K)_{ρ} = ℓ = ρ \sum \infty \frac{ℓ !}{( ℓ - ρ )!} e^{- t s} \frac{( t s ) ^{ℓ}}{ℓ !} = (t s)^{ρ} j = 0 \sum \infty e^{- t s} \frac{( t s ) ^{j}}{j !} = (t s)^{ρ} .$ Since $t > 0$ and the two factorial-pool counts are independent Poisson variables with means $t s_{0 x}$ and $t s_{1 x}$ , this gives $E_{Q_{x}} \frac{( K _{0 x} ) _{ρ_{0}} ( K _{1 x} ) _{ρ_{1}}}{t ^{ρ_{0} + ρ_{1}}} = s_{0 x}^{ρ_{0}} s_{1 x}^{ρ_{1}} .$ Termwise summation over the finite coefficient expansion yields the first-moment identity $E_{Q_{x}} W_{a, L, x} = W_{a, L} (s_{0 x}, s_{1 x}),$ and therefore $Λ_{a, x} = q_{a x} W_{a, L} (s_{0 x}, s_{1 x}) .$ Using $s_{0 x} + s_{1 x} = p_{x}$ and $W_{a, L} (s_{0 x}, s_{1 x}) = \frac{p _{x}}{B} G_{L} (\frac{s _{a x}}{B}),$ we obtain $Λ_{x} = q_{1 x} \frac{p _{x}}{B} G_{L} (\frac{s _{1 x}}{B}) - q_{0 x} \frac{p _{x}}{B} G_{L} (\frac{s _{0 x}}{B}) .$

Consider first a light cell, $p_{x} \leq B$ . If $p_{x} = 0$ , then $s_{0 x} = s_{1 x} = q_{0 x} = q_{1 x} = 0$ , so $Λ_{x} = ϑ_{x} = 0$ and the light-branch bias bound is immediate. It remains in this paragraph to handle the case $p_{x} > 0$ . For $a \in {0, 1}$ , set $z_{a x} := \frac{s _{a x}}{B}, E_{a x} := E_{L} (z_{a x}) .$ The overlap inequalities give $s_{a x} \geq ϵ p_{x} > 0$ for $a = 0, 1$ , so $z_{a x} > 0$ ; the light-cell condition $p_{x} \leq B$ , together with $s_{a x} \leq p_{x}$ and $B > 0$ , gives $z_{a x} \leq 1$ . Hence $z_{a x} \in (0, 1]$ . The factorial-moment calculation above has already identified the light-pool first moment. Since $L \geq 2$ and $z_{a x} \in (0, 1]$ , the interval part of Lemma 16 applies and gives $z_{a x} G_{L} (z_{a x}) + E_{a x} = 1, 0 \leq E_{a x}, E_{a x} \leq \frac{1}{L ^{2} z _{a x}} .$ Since $q_{a x} = s_{a x} μ_{a x}$ , the displayed formula for $Λ_{x}$ becomes, in the case $p_{x} > 0$ , $Λ_{x} = p_{x} μ_{1 x} (1 - E_{1 x}) - p_{x} μ_{0 x} (1 - E_{0 x}),$ and therefore $∣ Λ_{x} - ϑ_{x} ∣ \leq p_{x} μ_{1 x} E_{1 x} + p_{x} μ_{0 x} E_{0 x} \leq \frac{p _{x} B}{L ^{2} s _{1 x}} + \frac{p _{x} B}{L ^{2} s _{0 x}} \leq \frac{2 B}{ϵ L ^{2}} .$ Together with the already settled case $p_{x} = 0$ , this proves the light-branch bound for every light cell.

For the heavy branch, decompose the branch mean armwise by setting $Γ_{a, x} := E_{Q_{x}} [\frac{S _{a x}}{u} (1 + \frac{K _{1 - a, x}}{K _{a x} + 1})], a \in {0, 1},$ so that $Γ_{x} = Γ_{1, x} - Γ_{0, x}$ . By independence of $S_{a x}$ from the factorial-pool counts and $E_{Q_{x}} S_{a x} = u q_{a x}$ , $Γ_{a, x} = q_{a x} E [1 + \frac{K _{1 - a, x}}{K _{a x} + 1}] .$ Substituting the cell quantities $p_{x} = s_{0 x} + s_{1 x}$ , $s_{a x}$ , $q_{a x}$ , and $μ_{a x} = q_{a x} / s_{a x}$ in Lemma 8, and using the total real-division simplification $u / u = 1$ that follows from $u > 0$ , gives $q_{a x} E [1 + \frac{K _{1 - a, x}}{K _{a x} + 1}] = p_{x} μ_{a x} - (p_{x} - s_{a x}) μ_{a x} exp (- t s_{a x}),$ whose left side is $Γ_{a, x}$ . The totalized zero-arm cases are covered as follows: if $s_{a x} = 0$ , then $0 \leq q_{a x} \leq s_{a x}$ gives $q_{a x} = 0$ , so $Γ_{a, x} = 0$ , while $μ_{a x} = q_{a x} / s_{a x} = 0$ and the right side is also zero. If $s_{a x} > 0$ , the quotient is the ordinary outcome regression and the same identity applies directly. Thus, for each arm, $∣ Γ_{a, x} - p_{x} μ_{a x} ∣ = (p_{x} - s_{a x}) μ_{a x} exp (- t s_{a x}) .$ If $p_{x} = 0$ , this expression is zero. If $p_{x} > 0$ , occupied-cell overlap gives $s_{a x} \geq ϵ p_{x}$ ; together with $0 \leq p_{x} - s_{a x} \leq p_{x}$ and $0 \leq μ_{a x} \leq 1$ , this yields $∣ Γ_{a, x} - p_{x} μ_{a x} ∣ \leq p_{x} exp (- ϵ t p_{x}) .$ Summing the two armwise errors gives $∣ Γ_{x} - ϑ_{x} ∣ \leq 2 p_{x} exp (- ϵ t p_{x}) .$

We next derive the pilot-selection estimate used on light cells. Since the two pilot arm counts are independent Poisson variables with means $t_{p} s_{0 x}$ and $t_{p} s_{1 x}$ , their sum satisfies $J_{x} = J_{0 x}^{p} + J_{1 x}^{p} \sim Poi (t_{p} (s_{0 x} + s_{1 x})) = Poi (t_{p} p_{x}) .$ Let $λ_{x}^{p} := t_{p} p_{x} .$ For any $α_{tail} > 0$ , the exponential Markov bound applied to $(1 + α_{tail})^{J_{x}}$ gives $Pr {J_{x} > k_{0}} exp (- α_{tail} λ_{x}^{p}) \leq \frac{E ( 1 + α _{tail} ) ^{J_{x}}}{( 1 + α _{tail} ) ^{k_{0}}} exp (- α_{tail} λ_{x}^{p}) = (1 + α_{tail})^{- k_{0}},$ because the Poisson probability generating function gives $E (1 + α_{tail})^{J_{x}} = exp (α_{tail} λ_{x}^{p})$ . Taking $α_{tail} = ϵ t / t_{p}$ , which is positive since the calibration predicate gives $t_{p} > 0$ and $t > 0$ , yields $η_{x} exp (- ϵ t p_{x}) \leq (1 + ϵ \frac{t}{t _{p}})^{- k_{0}} .$ By $\overset{ϵ}{ˉ} \leq ϵ$ from Definition 13 and $t / t_{p} \geq 0$ , $(1 + \overset{ϵ}{ˉ} \frac{t}{t _{p}})^{k_{0}} \leq (1 + ϵ \frac{t}{t _{p}})^{k_{0}} .$ The calibration inequality in Algorithm 1 gives $n^{12} \leq (1 + \overset{ϵ}{ˉ} \frac{t}{t _{p}})^{k_{0}},$ and hence, with positive denominators, $(1 + ϵ \frac{t}{t _{p}})^{- k_{0}} \leq n^{- 12} .$ Combining these estimates with the mixture identity yields, for $p_{x} \leq B$ , $∣ M_{x} - ϑ_{x} ∣ \leq \frac{2 B}{ϵ L ^{2}} + 2 p_{x} n^{- 12} .$

Now consider a heavy cell, $B < p_{x}$ . Put $λ_{x}^{p} := t_{p} p_{x} .$ We derive the light-pilot lower-tail estimate directly. By the definition above, $J_{x} = J_{0 x}^{p} + J_{1 x}^{p}$ , and independence of the two pilot arm counts gives $J_{x} \sim Poi (t_{p} (s_{0 x} + s_{1 x})) = Poi (λ_{x}^{p}) .$ Moreover, Algorithm 1 defines the pilot threshold by $k_{0} = ⌊ \frac{t _{p} B}{4} ⌋ .$ Because $t_{p} B /4 \geq 0$ , the defining floor inequality, after casting the natural-valued threshold $k_{0}$ to the reals, gives $k_{0} \leq t_{p} B /4$ . The heavy-cell inequality $B < p_{x}$ , multiplied by the positive number $t_{p} /4$ , gives $t_{p} B /4 < t_{p} p_{x} /4 = λ_{x}^{p} /4$ . Thus $k_{0} \leq \frac{t _{p} B}{4} < \frac{λ _{x}^{p}}{4} .$ With $θ = lo g 2$ , apply exponential Markov directly to $e^{- θ J_{x}}$ . Since the event $J_{x} \leq k_{0}$ is the same as $e^{- θ J_{x}} \geq e^{- θ k_{0}}$ , with the natural number $k_{0}$ cast to a real in the exponential, the floor inequality above gives $π_{x} = Pr {J_{x} \leq k_{0}} \leq exp (θ k_{0} + λ_{x}^{p} (e^{- θ} - 1)) \leq exp (\frac{θ λ _{x}^{p}}{4} + λ_{x}^{p} (e^{- θ} - 1)) = exp ((\frac{θ}{4} - \frac{1}{2}) λ_{x}^{p}) \leq exp (- \frac{λ _{x}^{p}}{4}),$ since $e^{- θ} = 1/2$ and $θ \leq 1$ . The tail-calibration inequalities in Algorithm 1 include $15 L + 12 b_{n} \leq \frac{t _{p} B}{4},$ so $12 b_{n} \leq 15 L + 12 b_{n} \leq \frac{t _{p} B}{4} < \frac{λ _{x}^{p}}{4} .$ Consequently $π_{x} \leq exp (- 12 b_{n}) \leq n^{- 12},$ where the last comparison is the binary-length consequence recorded at the start of the proof.

We next prove the auxiliary estimate for the wrong-light contribution on the same heavy cell: $π_{x} ∣ Λ_{x} ∣ \leq p_{x} n^{- 12} .$ Put $y_{x} := \frac{p _{x}}{B},$ so $y_{x} \geq 1$ . First we bound the absolute light-branch mean. The first-moment identity gives $Λ_{x} = q_{1 x} \frac{p _{x}}{B} G_{L} (\frac{s _{1 x}}{B}) - q_{0 x} \frac{p _{x}}{B} G_{L} (\frac{s _{0 x}}{B}) .$ For each arm, $0 \leq s_{a x} \leq p_{x}$ and $0 \leq q_{a x} \leq p_{x}$ . If $G_{L} (z) = ρ = 0 \sum L - 2 g_{ρ} z^{ρ},$ then the degree bound in Lemma 5 justifies the displayed finite expansion with no terms beyond degree $L - 2$ , while Lemma 16 gives $2∥ G_{L} ∥_{1} \leq 1 4^{L}$ . Since $0 \leq s_{a x} / B \leq y_{x}$ and $y_{x} \geq 1$ , each displayed exponent satisfies $(\frac{s _{a x}}{B})^{ρ} \leq y_{x}^{ρ} \leq (1 + y_{x})^{ρ} \leq (1 + y_{x})^{L} .$ Multiplying this monomial bound by $∣ g_{ρ} ∣$ and summing over the displayed finite support gives $G_{L} (\frac{s _{a x}}{B}) \leq ρ = 0 \sum L - 2 ∣ g_{ρ} ∣ (\frac{s _{a x}}{B})^{ρ} \leq ∥ G_{L} ∥_{1} (1 + y_{x})^{L} .$ Also $y_{x} \leq (1 + y_{x})^{L}$ , because $L \geq 2$ and $y_{x} \geq 0$ . Therefore $∣ Λ_{x} ∣ \leq 2 p_{x} y_{x} ∥ G_{L} ∥_{1} (1 + y_{x})^{L} \leq p_{x} 1 4^{L} (1 + y_{x})^{2 L} .$ This coefficient-envelope argument uses the exact first-moment identity above, together with the degree and coefficient bounds for $G_{L}$ , and yields the displayed bound without invoking any second-moment estimate.

The lower-tail calculation above, kept in exponential form, gives $π_{x} \leq exp (- \frac{t _{p} p _{x}}{4}) .$ The explicit scale definition in Definition 13 gives $B = \frac{H _{ϵ} L}{min ( t _{p} , t )} .$ Since $t_{p} > 0$ and $t > 0$ , their minimum is positive and the scale identity gives $H_{ϵ} L = B min (t_{p}, t)$ . As $min (t_{p}, t) \leq t_{p}$ and $B > 0$ , multiplication preserves the inequality, so $H_{ϵ} L \leq t_{p} B$ . Thus, with $p_{x} = B y_{x}$ , $exp (- \frac{t _{p} p _{x}}{4}) \leq exp (- \frac{H _{ϵ} L y _{x}}{4}) .$ The environment Definition 13 gives $A_{⋆} = 7056$ , $c^{\circ} = 1/256$ , and $\frac{H _{ϵ}}{4} \geq 15 + \frac{24}{c ^{\circ}} .$ Elementary arithmetic applied to the displayed value of $A_{⋆}$ gives $14 \leq A_{⋆} < e^{13}$ . Using these consequences, $y_{x} \geq 1$ , and $lo g (1 + y_{x}) \leq y_{x}$ , we obtain $L lo g A_{⋆} + 2 L lo g (1 + y_{x}) - \frac{H _{ϵ} L y _{x}}{4} \leq 13 L y_{x} + 2 L y_{x} - (15 + \frac{24}{c ^{\circ}}) L y_{x} = - \frac{24 L y _{x}}{c ^{\circ}} .$ Exponentiating gives the calibrated absorption bound $A_{⋆}^{L} (1 + y_{x})^{2 L} exp (- \frac{H _{ϵ} L y _{x}}{4}) \leq exp (- \frac{24 L y _{x}}{c ^{\circ}}) .$ Combining the envelope for $Λ_{x}$ , the pilot-tail display, and $1 4^{L} \leq A_{⋆}^{L}$ , $π_{x} ∣ Λ_{x} ∣ \leq p_{x} exp (- \frac{24 L y _{x}}{c ^{\circ}}) .$ Finally, the degree lower bound $c^{\circ} b_{n} /2 \leq L$ in $C_{ϵ} (n, m)$ , listed in Algorithm 1, together with $y_{x} \geq 1$ , implies $12 b_{n} \leq \frac{24 L y _{x}}{c ^{\circ}} .$ The binary-length consequence recorded at the start of the proof gives $exp (- \frac{24 L y _{x}}{c ^{\circ}}) \leq exp (- 12 b_{n}) \leq n^{- 12} .$ Combining the last two displays proves $π_{x} ∣ Λ_{x} ∣ \leq p_{x} n^{- 12} .$

The intrinsic heavy-branch exponential is also absorbed on a heavy cell: $exp (- ϵ t p_{x}) \leq n^{- 12} .$ Indeed, $B < p_{x}$ , $\overset{ϵ}{ˉ} \leq ϵ$ , and the tail-calibration inequality in Algorithm 1, $12 b_{n} \leq \overset{ϵ}{ˉ} tB,$ give $12 b_{n} \leq ϵ t p_{x}$ , and hence, by the binary-length consequence recorded at the start of the proof, $exp (- ϵ t p_{x}) \leq exp (- 12 b_{n}) \leq n^{- 12} .$ Therefore $η_{x} ∣ Γ_{x} - ϑ_{x} ∣ \leq 2 p_{x} n^{- 12} .$ Together with $π_{x} ∣ ϑ_{x} ∣ \leq p_{x} n^{- 12}$ , the decomposition $M_{x} - ϑ_{x} = π_{x} Λ_{x} - π_{x} ϑ_{x} + η_{x} (Γ_{x} - ϑ_{x})$ gives, for $B < p_{x}$ , $∣ M_{x} - ϑ_{x} ∣ \leq 4 p_{x} n^{- 12} .$

The alternatives $p_{x} \leq B$ and $B < p_{x}$ are disjoint and exhaustive, so they partition $[d]$ . Summing the two cellwise bounds over these two regimes, $x \in [d] \sum M_{x} - τ (P) = x \in [d] \sum (M_{x} - ϑ_{x}) \leq x \in [d] \sum ∣ M_{x} - ϑ_{x} ∣ \leq x : p_{x} \leq B \sum (\frac{2 B}{ϵ L ^{2}} + 2 p_{x} n^{- 12}) + x : B < p_{x} \sum 4 p_{x} n^{- 12} \leq \frac{2 d B}{ϵ L ^{2}} + 4 n^{- 12} x \in [d] \sum p_{x} = \frac{2 d B}{ϵ L ^{2}} + 4 n^{- 12} \leq \frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12} .$ This is the claimed bound.

∎

Lemma 12 gives the matching aggregate bias control for the selected cell means. The bound contains the light-cell Chebyshev approximation term and a polynomially small pilot-boundary residual, which together form the squared bias term in Lemma 6.

Lemma 6 is the bookkeeping step that turns the local hybrid analysis into the global rate scale. Its left-hand side collects the light-cell factorial moment contribution, the heavy-cell inverse-count contribution, the Chebyshev approximation residual, the pilot-boundary term, and the finite prefix terms, while the right-hand side matches the two summands in Definition 12.

The calibrated hybrid statistic combines the light-cell polynomial with the all-heavy inverse-count weight. The construction below is the Poissonized counterpart of the finite-sample mixed estimator introduced in Algorithm 1.

Definition 25 [synth_9] (Calibrated Poisson hybrid statistic).

Use the split sizes, intensities, constants, dyadic overlap certificate, and calibration predicate $C_{ϵ} (n, m)$ from Algorithm 1. In the independent Poisson experiment with outcome-prefix length $D_{0} \sim Poi (u)$ , pilot-prefix length $D_{p} \sim Poi (t_{p})$ , and factorial-prefix length $D_{f} \sim Poi (t)$ , define $Z^{mix}$ as follows. If $C_{ϵ} (n, m)$ is false, set $Z^{mix} := 0.$ If $C_{ϵ} (n, m)$ is true, return zero on the event that any Poisson prefix exceeds its corresponding pool. On the complementary valid-prefix event, let $J_{x}$ be the pilot count in cell $x$ , let $K_{a x}$ be the factorial-pool count in arm $a$ and cell $x$ , and let $S_{a x}$ be the outcome-block success count in arm $a$ and cell $x$ . With $H_{L} (z) = \frac{1 - T _{L} ( 1 - 2 z )}{2 L ^{2}}, E_{L} (z) = \frac{H _{L} ( z )}{z}, G_{L} (z) = \frac{1 - E _{L} ( z )}{z},$ under polynomial continuation at zero, expand $W_{a, L} (s_{0}, s_{1}) = \frac{s _{0} + s _{1}}{B} G_{L} (s_{a} / B) = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} s_{0}^{r_{0}} s_{1}^{r_{1}} .$ For falling factorials $(k)_{r}$ , put $W_{a, L, x} = r_{0}, r_{1} \sum c_{a, r_{0}, r_{1}} \frac{( K _{0 x} ) _{r_{0}} ( K _{1 x} ) _{r_{1}}}{t ^{r_{0} + r_{1}}}, W_{a, x}^{H} = \frac{K _{0 x} + K _{1 x} + 1}{K _{a x} + 1},$ and define the cell contribution $Z_{x} = 1 {J_{x} \leq k_{0}} (\frac{S _{1 x}}{u} W_{1, L, x} - \frac{S _{0 x}}{u} W_{0, L, x}) + 1 {J_{x} > k_{0}} (\frac{S _{1 x}}{u} W_{1, x}^{H} - \frac{S _{0 x}}{u} W_{0, x}^{H}) .$ The calibrated Poisson hybrid statistic is the clipped aggregate $Z^{mix} := clip_{[- 1, 1]} (x = 1 \sum d Z_{x})$ on the valid-prefix event, together with the preceding zero values on invalid prefixes or failed calibration.

The pilot threshold in Definition 25 chooses between two weight regimes at the cell level. Cells with small pilot counts use the bounded-degree Chebyshev lift, while cells with larger pilot counts use the inverse-count weight from Lemma 2; clipping keeps the aggregate on the natural scale of the treatment effect.

Lemma 13 [lem:poisson-hybrid-risk] (Hybrid Poisson risk).

Fix an overlap constant $ϵ$ with $0 < ϵ < 1/2$ . There is a constant $C_{ϵ} \in (0, \infty)$ , depending only on $ϵ$ , such that the following holds for every $n, m, d$ and every observed-data law $P$ :

(Sample size.) $n \geq 1$ .

(Alphabet size.) $d \geq 2$ .

(Model class.) $P \in M_{d, ϵ}$ , as in Definition 6.

Let $M_{0} = ⌊ n /3 ⌋$ , $n_{p} = ⌊(n - M_{0}) /2 ⌋$ , $n_{f} = n - M_{0} - n_{p}$ , $m_{p} = ⌊ m /2 ⌋$ , and $m_{f} = m - m_{p}$ , and set $u = \frac{M _{0}}{8}, t_{p} = \frac{n _{p} + m _{p}}{8}, t = \frac{n _{f} + m _{f}}{8} .$ In the independent Poisson count experiment with these intensities, let $Z^{mix}$ be the calibrated hybrid statistic when the finite calibration predicate $C_{ϵ} (n, m)$ holds and the constant $0$ otherwise. Then its mean-squared error satisfies $E_{P} [(Z^{mix} - τ (P))^{2}] \leq C_{ϵ} r_{ϵ} (n, m, d),$ where $r_{ϵ} (n, m, d)$ is the frontier rate in Definition 12, equivalently $r_{ϵ} (n, m, d) = min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} lo g ^{2} ( e n )}} .$

⊢ Lean

Proof of Lemma 13.

Fix $ϵ \in (0, 1/2)$ . Throughout, let $M_{0}, n_{p}, n_{f}, m_{p}, m_{f}, u, t_{p}, t, L, B, k_{0}$ be the block sizes, Poisson intensities, degree, scale, and pilot threshold from Algorithm 1. Write $rad_{n, m, d} = \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} lo g ^{2} ( e n )},$ so that $r_{ϵ} (n, m, d) = min {1, rad_{n, m, d}}$ by Definition 12.

$1.$ First record the uniform bounded-loss estimate. For $P \in M_{d, ϵ}$ , the functional $τ (P)$ lies in $[- 1, 1]$ , since it is a cell-mass average of differences of binary outcome regressions. On the calibrated branch, the hybrid statistic is clipped to $[- 1, 1]$ , so its squared distance from $τ (P)$ is at most $4$ pointwise. On the uncalibrated branch, the risk term is the deterministic quantity $τ (P)^{2}$ , which is at most $1$ and hence at most $4$ . Therefore, for all $n, m, d$ , $E_{P} [(Z^{mix} - τ (P))^{2}] \leq 4.$

$2.$ We next record the calibrated variance envelope used when $C_{ϵ} (n, m)$ holds. For each $x \in [d]$ , let $S_{a x}$ , $J_{a x}$ , and $K_{a x}$ be the independent outcome-block, pilot-pool, and factorial-pool Poisson counts from Algorithm 1, and put $J_{x} = J_{0 x} + J_{1 x}$ . Let $P_{x}$ and $H_{x}$ be the light-cell polynomial and heavy inverse-count cell contributions from Algorithm 1, and define the unclipped selected hybrid sum $Z = x = 1 \sum d (1 {J_{x} \leq k_{0}} P_{x} + 1 {J_{x} > k_{0}} H_{x}) .$ By Lemma 11, there is a constant $C_{var} > 0$ , depending only on $ϵ$ , such that, whenever $n \geq 1$ , $P \in M_{d, ϵ}$ , and $C_{ϵ} (n, m)$ holds, $Var_{P} (Z) \leq 4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{var} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 414 n^{- 12} .$

$3.$ Still under $C_{ϵ} (n, m)$ , projection onto $[- 1, 1]$ cannot increase squared distance from $τ (P) \in [- 1, 1]$ . Hence $E_{P} [(Z^{mix} - τ (P))^{2}] \leq E_{P} [(Z - τ (P))^{2}] = Var_{P} (Z) + (E_{P} Z - τ (P))^{2} .$ For the same hybrid branches, Lemma 12 gives $E_{P} Z - τ (P) \leq \frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12} .$ Consequently, $E_{P} [(Z^{mix} - τ (P))^{2}] \leq Var_{P} (Z) + (\frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12})^{2} .$

$4.$ Apply Lemma 6 with $C_{0} = C_{var}$ . That result gives a constant depending on $ϵ$ and $C_{var}$ ; since Step 2 chose $C_{var}$ depending only on $ϵ$ , we may write this constant as $C_{abs} > 0$ , depending only on $ϵ$ . Under $C_{ϵ} (n, m)$ , $4 A_{⋆}^{L} {\frac{min { 1 , d B }}{u} + d B^{2}} + C_{var} (u^{- 1} + t^{- 1}) + \frac{8 d B ^{2}}{ϵ ^{2} L ^{4}} + \frac{4}{ϵ t} + 414 n^{- 12} + (\frac{2 d B}{ϵ L ^{2}} + 6 n^{- 12})^{2} \leq C_{abs} rad_{n, m, d} .$ Set $C_{cal} = max {4, C_{abs}} .$ If $1 \leq rad_{n, m, d}$ , then $r_{ϵ} (n, m, d) = 1$ , and Step 1 gives $E_{P} [(Z^{mix} - τ (P))^{2}] \leq 4 \leq C_{cal} r_{ϵ} (n, m, d) .$ If $rad_{n, m, d} \leq 1$ , then $r_{ϵ} (n, m, d) = rad_{n, m, d}$ . Steps 2 and 3, followed by the absorption display, give $E_{P} [(Z^{mix} - τ (P))^{2}] \leq C_{abs} rad_{n, m, d} \leq C_{cal} r_{ϵ} (n, m, d) .$ Thus the desired bound holds on the calibrated branch with constant $C_{cal}$ .

$5.$ It remains to cover the finite set of uncalibrated sample sizes. The uniform calibration-failure bound gives an integer $n_{ϵ}^{†} \geq 1$ , depending only on $ϵ$ , such that $\neg C_{ϵ} (n, m) ⟹ n < n_{ϵ}^{†} for all n, m .$ Now set $C_{ϵ} = max {C_{cal}, 4 n_{ϵ}^{†}} .$ If $C_{ϵ} (n, m)$ holds, Step 4 gives the result because $C_{cal} \leq C_{ϵ}$ . If $C_{ϵ} (n, m)$ fails, then $n < n_{ϵ}^{†}$ , so $n \leq n_{ϵ}^{†}$ . Since $n \geq 1$ , the frontier rate is positive and dominates $n^{- 1}$ : the second entry in the minimum is $\frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} lo g ^{2} ( e n )} \geq \frac{1}{n},$ and $1/ n \leq 1$ , so $\frac{1}{n} \leq r_{ϵ} (n, m, d) .$ Multiplying by the positive number $n$ gives $1 \leq n r_{ϵ} (n, m, d)$ . Since $n \leq n_{ϵ}^{†}$ and $r_{ϵ} (n, m, d) > 0$ , $1 \leq n r_{ϵ} (n, m, d) \leq n_{ϵ}^{†} r_{ϵ} (n, m, d) .$ Combining this with Step 1 gives $E_{P} [(Z^{mix} - τ (P))^{2}] \leq 4 \leq 4 n_{ϵ}^{†} r_{ϵ} (n, m, d) \leq C_{ϵ} r_{ϵ} (n, m, d) .$ This proves the asserted bound for every $n, m, d$ and every $P \in M_{d, ϵ}$ .

∎

Lemma 13 establishes the Poisson risk bound at exactly the split sizes and intensities used by the finite estimator. The bound has the same two components as the rate in Definition 12: labeled outcomes contribute the $1/ n$ term, and treatment–covariate information contributes the alphabet-dependent term through the pilot and factorial pools.

Uniform finite-calibration input.

The uncalibrated branch uses the formally checked helper $calibrationPredicate_failure_bounded$ : for fixed $0 < ϵ < 1/2$ , there is $n_{ϵ}^{†} \geq 1$ such that $\neg C_{ϵ} (n, m) ⟹ n < n_{ϵ}^{†} for every m .$ This is the contrapositive of eventual validity of all finitely many calibration inequalities, uniformly over the auxiliary sample size. It reduces calibration failure to finitely many labeled sizes, where clipping gives risk at most $4$ and $r_{ϵ} (n, m, d) \geq 1/ n$ , so $4 \leq 4 n_{ϵ}^{†} r_{ϵ} (n, m, d)$ .

It remains to connect the independent Poisson count analysis to the fixed sample sizes in the annotation experiment. The next definition packages the three independent Poisson pools used in that transfer.

Definition 26 [synth_6] (Three-pool Poisson count law).

Fix an observed-data law $P$ , intensities $u, t_{p}, t \geq 0$ , and write $q_{a x} = P (X = x, A = a, Y = 1), s_{a x} = P (X = x, A = a) .$ The law $P_{P, u, t_{p}, t}^{Pois}$ is the joint distribution of independent count triples $(S_{a x}, J_{a x}, K_{a x})_{a \in {0, 1}, x \in [d]},$ where $S_{a x} \sim Poi (u q_{a x}), J_{a x} \sim Poi (t_{p} s_{a x}), K_{a x} \sim Poi (t s_{a x}) .$ Here $S_{a x}$ is the outcome-success count from the outcome pool, $J_{a x}$ is the pilot count, and $K_{a x}$ is the factorial-pool treatment-covariate count. The product law makes all these counts independent across cells, arms, and pools.

The three-pool law in Definition 26 mirrors the deterministic sample split in Algorithm 1. Each pool has its own intensity, so the Poisson calculation can track outcome information, pilot classification, and polynomial weight estimation separately.

Lemma 14 [lem:fixed-sample-hybrid-transfer] (Fixed-sample hybrid transfer).

There is a universal constant $C > 0$ such that the following holds. Let

(Overlap level.) The overlap level $ϵ$ satisfies $0 < ϵ < 1/2$ .

(Sample sizes and alphabet.) $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ .

(Model class.) $P \in M_{d, ϵ}$ , with $M_{d, ϵ}$ as in Definition 6.

For the mixed estimator $τ_{n, m, d}^{mix}$ in Algorithm 1, the estimator is measurable as a function of the annotation sample and $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq 2 R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{C}{n} .$ Here $τ (P)$ is the average treatment effect functional in Definition 8. Writing $M_{0} = ⌊ n /3 ⌋$ , $n_{p} = ⌊(n - M_{0}) /2 ⌋$ , $n_{f} = n - M_{0} - n_{p}$ , $m_{p} = ⌊ m /2 ⌋$ , $m_{f} = m - m_{p}$ , $M_{p} = n_{p} + m_{p}$ , $M_{f} = n_{f} + m_{f}$ , $u = M_{0} /8$ , $t_{p} = M_{p} /8$ , and $t = M_{f} /8$ , the Poisson hybrid risk is $R_{hyb}^{Pois} (P; n, m, ϵ) = ⎩ ⎨ ⎧ \int {Z^{mix} (K) - τ (P)}^{2} d P_{P, u, t_{p}, t}^{Pois} (K), τ (P)^{2}, C_{ϵ} (n, m), otherwise,$ where $Z^{mix}$ is the calibrated hybrid statistic of Definition 25.

⊢ Lean

Proof of Lemma 14.

Take $C = 384$ . Fix $0 < ϵ < 1/2$ , $n \geq 1$ , $m \geq 0$ , $d \geq 2$ , and $P \in M_{d, ϵ}$ . The estimator is measurable because it is either the constant zero map or, on the calibrated branch of Algorithm 1, a finite linear combination of measurable prefix-count statistics. It remains to prove the displayed risk bound.

1. Suppose first that $C_{ϵ} (n, m)$ holds. Put $M_{0} = ⌊ n /3 ⌋, M_{p} = n_{p} + m_{p}, M_{f} = n_{f} + m_{f},$ and $u = \frac{M _{0}}{8}, t_{p} = \frac{M _{p}}{8}, t = \frac{M _{f}}{8} .$ Let $P_{3}$ be the three-pool product law whose complete-record coordinate has law $P$ on $[d] \times {0, 1} \times {0, 1}$ , while the two shared coordinates have law $P_{X A}$ on $[d] \times {0, 1}$ . Let $λ = (u, t_{p}, t)$ and let the capacity vector be $M = (M_{0}, M_{p}, M_{f})$ . For a three-prefix array $s = (s_{0}, s_{p}, s_{f}),$ define $Ψ_{n, m, d, ϵ} (s) = Z_{n, m, ϵ, u, t}^{mix} (hist (s_{0}), hist (s_{p}), hist (s_{f})),$ where the first histogram records complete outcome-labeled observations and the last two record treatment-covariate observations. This prefix statistic is measurable and satisfies $- 1 \leq Ψ_{n, m, d, ϵ} (s) \leq 1 for every s, - 1 \leq τ (P) \leq 1,$ the first bound coming from clipping in Algorithm 1 and the second from Definition 8 together with occupied-cell overlap in Definition 6.

2. For fixed pools $p = (p_{0}, p_{p}, p_{f}) \in ([d] \times {0, 1}^{2})^{M_{0}} \times ([d] \times {0, 1})^{M_{p}} \times ([d] \times {0, 1})^{M_{f}},$ define the Rao–Blackwellized prefix statistic by $RB (p) = r_{0}, r_{p}, r_{f} \geq 0 \sum Pr {Poi (u) = r_{0}} Pr {Poi (t_{p}) = r_{p}} Pr {Poi (t) = r_{f}} Ψ^{\circ} (p; r_{0}, r_{p}, r_{f}),$ where $Ψ^{\circ} (p; r_{0}, r_{p}, r_{f}) = {Ψ_{n, m, d, ϵ} (p_{0} [1 : r_{0}], p_{p} [1 : r_{p}], p_{f} [1 : r_{f}]), 0, r_{0} \leq M_{0}, r_{p} \leq M_{p}, r_{f} \leq M_{f}, otherwise.$ Under $C_{ϵ} (n, m)$ , expanding the finite sums in Algorithm 1 gives the pointwise identity $τ_{n, m, d}^{mix} (L_{n}, U_{m}) = RB (P (L_{n}, U_{m})),$ where $P (L_{n}, U_{m})$ is the deterministic three-pool split formed from the first $M_{0}$ complete labeled records, the next $n_{p}$ labeled records projected to $(X, A)$ together with the first $m_{p}$ auxiliary records, and the remaining $n_{f}$ projected labeled records together with the remaining $m_{f}$ auxiliary records. By Definition 9, this split has law equal to the fixed-pool law with coordinate law $P_{3}$ and capacities $M$ . Hence $E [{τ_{n, m, d}^{mix} - τ (P)}^{2}] = \int {RB (p) - τ (P)}^{2} d P_{fix} (p) .$

3. The finite-pool-to-Poisson comparison used here is the following. For any prefix statistic with values in an interval $[a, b]$ , any target and overflow value in $[a, b]$ , and independent Poisson requested lengths with means $λ$ , $\int {RB (p) - θ}^{2} d P_{fix} (p) \leq \int {Ψ (s) - θ}^{2} d P_{Pois} (s) + (b - a)^{2} ι \in {0, p, f} \sum Pr {Poi (λ_{ι}) > M_{ι}} .$ Indeed, $RB$ is the conditional expectation of the capped randomized-prefix statistic given the fixed pools, so Jensen’s inequality bounds its squared risk by the capped randomized-prefix risk. Off the overflow event the capped and uncapped prefixes agree; on overflow both the capped value and the target lie in $[a, b]$ , so the extra squared loss is bounded by $(b - a)^{2}$ times the overflow probability. Applying this with $a = - 1$ , $b = 1$ , $θ = τ (P)$ , overflow value $0$ , and $Ψ = Ψ_{n, m, d, ϵ}$ gives $\int {RB (p) - τ (P)}^{2} d P_{fix} (p) \leq \int {Ψ_{n, m, d, ϵ} (s) - τ (P)}^{2} d P_{Pois} (s) + 4 [π_{0} + π_{p} + π_{f}],$ where $π_{0} = Pr {Poi (u) > M_{0}}, π_{p} = Pr {Poi (t_{p}) > M_{p}}, π_{f} = Pr {Poi (t) > M_{f}} .$ The image of the independent Poisson-prefix law under the three histograms is exactly $P_{P, u, t_{p}, t}^{Pois}$ , and therefore $\int {Ψ_{n, m, d, ϵ} (s) - τ (P)}^{2} d P_{Pois} (s) = R_{hyb}^{Pois} (P; n, m, ϵ)$ on the calibrated branch.

4. It remains to bound the three overflow probabilities. The calibration predicate gives $24 \leq n$ . From the definitions of the integer blocks, $n \leq 4 M_{0}, n \leq 4 M_{p}, n \leq 4 M_{f},$ and all three capacities are positive. For any positive integer $M_{∙}$ satisfying $n \leq 4 M_{∙}$ , $Pr {Poi (M_{∙} /8) > M_{∙}} \leq \frac{8}{M _{∙}} \leq \frac{32}{n} .$ To prove the first inequality, let $X_{∙} \sim Poi (M_{∙} /8)$ . The overflow event is contained in the self-normalized Bernstein event at parameter $M_{∙} /4$ , because $2 \cdot \frac{M _{∙}}{8} \cdot \frac{M _{∙}}{4} = \frac{M _{∙}}{4}$ and $X_{∙} > M_{∙}$ implies $X_{∙} - \frac{M _{∙}}{8} > \frac{7 M _{∙}}{8} > \frac{M _{∙}}{4} + 2 \cdot \frac{M _{∙}}{4} .$ The Poisson Bernstein bound for this larger event gives $Pr {Poi (M_{∙} /8) > M_{∙}} \leq exp (- M_{∙} /4) = \frac{1}{exp ( M _{∙} /4 )} \leq \frac{4}{M _{∙}} \leq \frac{8}{M _{∙}},$ where $1 + M_{∙} /4 \leq exp (M_{∙} /4)$ gives the penultimate inequality. Consequently $π_{0} + π_{p} + π_{f} \leq \frac{96}{n},$ and the preceding display yields $E [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{384}{n} .$ Since $R_{hyb}^{Pois} (P; n, m, ϵ) \geq 0$ , this is at most $2 R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{384}{n} .$

5. If $C_{ϵ} (n, m)$ fails, then Algorithm 1 sets $τ_{n, m, d}^{mix} = 0$ , while the definition of the hybrid Poisson risk gives $R_{hyb}^{Pois} (P; n, m, ϵ) = τ (P)^{2} .$ Because the annotation law is a probability measure, $E [{τ_{n, m, d}^{mix} - τ (P)}^{2}] = τ (P)^{2} \leq 2 τ (P)^{2} + \frac{384}{n} = 2 R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{384}{n},$ using $n \geq 1$ . The two cases prove the asserted bound with the universal constant $C = 384$ .

∎

Lemma 14 converts the Poisson risk calculation into the fixed- $n, m$ annotation experiment of Definition 9. The additional $C / n$ term is on the labeled-outcome scale already present in the rate, so combining Lemmas 13 and 14 yields the finite-sample upper-bound ingredient used in Theorem 2.

The final calibration statements make the Chebyshev constants explicit. The first controls the coefficient norm of the shifted quotient, and the second records the numerical bounds used in the finite calibration predicate of Algorithm 1.

Lemma 15 [lem:shifted-chebyshev-quotient-coefficient-bound] (Shifted quotient coefficient bound).

Let $L$ be a positive integer, and let $G_{L}$ be the polynomial continuation at zero of $(1 - E_{L} (z)) / z$ from Algorithm 1. For a real polynomial $p (z) = \sum_{i} a_{i} z^{i}$ , write $∥ p ∥_{1} = i \sum ∣ a_{i} ∣$ for its coefficient $ℓ^{1}$ -norm. Then $∥ G_{L} ∥_{1} \leq 6^{L} .$

⊢ Lean

Proof of Lemma 15.

Let $L \geq 1$ . We use the notation for $H_{L}, E_{L}, G_{L}$ from Algorithm 1. The phrase “polynomial continuation at zero” is used through the quotient identities $z E_{L} (z) = H_{L} (z), z G_{L} (z) = 1 - E_{L} (z),$ with equality in $R [z]$ ; the cancellations below use only that multiplication by the nonzero indeterminate $z$ is injective on $R [z]$ .

Define, for every $α \in N$ , $θ_{L, α} = (- 1)^{α} \frac{2 ^{2 α + 3}}{L ( L + α + 2 )} (2 α + 4 L + α + 2),$ and put $G_{L} (z) = 0 \leq α < L - 1 \sum θ_{L, α} z^{α} .$ We first derive the shifted-Chebyshev coefficient expansion used below. Write $Q_{r} (z) = T_{r} (1 - 2 z)$ . The initial polynomials and the defining Chebyshev recurrence give $Q_{0} (z) = 1, Q_{1} (z) = 1 - 2 z, Q_{r + 2} (z) = 2 (1 - 2 z) Q_{r + 1} (z) - Q_{r} (z) .$ For $r \in N$ and $ν \in Z$ , define $β_{r, ν} := ⎩ ⎨ ⎧ 1, (- 1)^{ν} \frac{r}{r + ν} (2 ν r + ν) 4^{ν}, 0, ν = 0, 1 \leq ν \leq r, ν < 0 or r < ν .$ Comparing coefficients in the displayed recurrence reduces the induction to $β_{r + 2, ν} = 2 β_{r + 1, ν} - 4 β_{r + 1, ν - 1} - β_{r, ν} .$ For $1 \leq ν \leq r$ , write $A_{r, ν} := \frac{r}{r + ν} (2 ν r + ν) .$ Set $A_{r, 0} := 1$ . For $ν \geq 1$ , the factorial form is $A_{r, ν} = \frac{r ( r + ν - 1 )!}{( 2 ν )! ( r - ν )!},$ After cancelling the common factor $(- 1)^{ν} 4^{ν}$ , the coefficient recurrence is $A_{r + 2, ν} = 2 A_{r + 1, ν} + A_{r + 1, ν - 1} - A_{r, ν} .$ For $1 \leq ν \leq r$ , multiply this equality by $(2 ν)! (r + 2 - ν)! / (r + ν - 1)!$ . It becomes $(r + 2) (r + ν + 1) (r + ν) = 2 (r + 1) (r + ν) (r + 2 - ν) + (r + 1) (2 ν) (2 ν - 1) - r (r + 2 - ν) (r + 1 - ν),$ and expanding both sides proves it. At $ν = 0$ , the term $β_{r + 1, - 1}$ is zero by the outside-range convention, and the identity becomes $1 = 2 \cdot 1 - 0 - 1$ . At $ν = r + 1$ and $ν = r + 2$ , the same zero convention leaves, respectively, $(- 1)^{r + 1} (r + 2) 4^{r + 1} (- 1)^{r + 2} \frac{4 ^{r + 2}}{2} = 2 ((- 1)^{r + 1} \frac{4 ^{r + 1}}{2}) - 4 ((- 1)^{r} (r + 1) 4^{r}), = - 4 ((- 1)^{r + 1} \frac{4 ^{r + 1}}{2}),$ so the recurrence holds there as well. Since the formula agrees with $Q_{0}, Q_{1}$ , induction proves, for every real $z$ , $T_{L} (1 - 2 z) = 1 + 1 \leq ν \leq L \sum (- 1)^{ν} \frac{L}{L + ν} (2 ν L + ν) 4^{ν} z^{ν} .$ The $ν = 1$ term is $- 2 L^{2} z$ . For the remaining terms write $ν = α + 2$ . Then $(- 1)^{α + 2} \frac{L}{L + α + 2} (2 α + 4 L + α + 2) 4^{α + 2} z^{α + 2} = 2 L^{2} θ_{L, α} z^{α + 2},$ so, with $Q_{L} (z) = T_{L} (1 - 2 z)$ , $Q_{L} (z) = 1 - 2 L^{2} z + 2 L^{2} z^{2} G_{L} (z) .$ Using these quotient identities, since $Q_{L} (0) = 1$ , the polynomial $H_{L} (z) = \frac{1 - Q _{L} ( z )}{2 L ^{2}}$ has zero constant term. Moreover $Q_{L}^{'} (0) = - 2 T_{L}^{'} (1) = - 2 L^{2}$ , so $H_{L}^{'} (0) = 1$ , and hence the quotient identity for $E_{L}$ gives $E_{L} (0) = 1$ . Thus $1 - E_{L}$ has zero constant term, as required for the quotient identity defining $G_{L}$ . Using the displayed formula for $Q_{L}$ , $H_{L} (z) = \frac{1 - Q _{L} ( z )}{2 L ^{2}} = z - z^{2} G_{L} (z) = z (1 - z G_{L} (z)) .$ Thus $z E_{L} (z) = z (1 - z G_{L} (z)) .$ Because multiplication by the nonzero indeterminate $z$ is cancellative in $R [z]$ , $E_{L} (z) = 1 - z G_{L} (z) .$ Consequently $z G_{L} (z) = 1 - E_{L} (z) = z G_{L} (z),$ and the same cancellation gives $G_{L} (z) = G_{L} (z)$ as polynomials.
Since $G_{L} (z) = 0 \leq α < L - 1 \sum θ_{L, α} z^{α},$ every coefficient of $G_{L}$ outside the range $0 \leq α < L - 1$ is zero. Hence $∥ G_{L} ∥_{1} = ∥ G_{L} ∥_{1} = α \in supp (G_{L}) \sum ∣ [z^{α}] G_{L} ∣ \leq 0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣.$
The last sum is the positive coefficient sum at $1$ : $0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣ = 0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣ 1^{α} .$ The coefficient formula also gives $∣ θ_{L, α} ∣ = \frac{2 ^{2 α + 3}}{L ( L + α + 2 )} (2 α + 4 L + α + 2),$ because $L > 0$ and all factors except $(- 1)^{α}$ are nonnegative.
It remains to bound this positive coefficient sum. Write $Θ_{L}^{+} = 0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣.$ This is the positive coefficient sum at the argument $1$ . The sign identity $(- 1)^{α} θ_{L, α} = ∣ θ_{L, α} ∣$ gives $Θ_{L}^{+} = 0 \leq α < L - 1 \sum θ_{L, α} (- 1)^{α} = G_{L} (- 1) .$ Substituting $z = - 1$ in the identity $Q_{L} (z) = 1 - 2 L^{2} z + 2 L^{2} z^{2} G_{L} (z)$ and using $Q_{L} (- 1) = T_{L} (3)$ gives $T_{L} (3) = 1 + 2 L^{2} + 2 L^{2} Θ_{L}^{+} .$ Since $Θ_{L}^{+} \geq 0$ and $L \geq 1$ , $Θ_{L}^{+} \leq 2 L^{2} Θ_{L}^{+} \leq 1 + 2 L^{2} + 2 L^{2} Θ_{L}^{+} = T_{L} (3) .$ It remains to bound $T_{L} (3)$ . Put $a_{ν} = T_{ν} (3)$ . The Chebyshev recurrence gives $a_{0} = 1, a_{1} = 3, a_{ν + 2} = 6 a_{ν + 1} - a_{ν} .$ Let $λ_{+} = 3 + 8$ and $λ_{-} = 3 - 8$ . Since $λ_{+} + λ_{-} = 6$ and $λ_{+} λ_{-} = 1$ , both roots satisfy $x^{2} = 6 x - 1$ . Consequently $b_{ν} := \frac{λ _{+}^{ν} + λ _{-}^{ν}}{2}$ satisfies $b_{ν + 2} = 6 b_{ν + 1} - b_{ν}$ , while $b_{0} = 1 = a_{0}, b_{1} = \frac{λ _{+} + λ _{-}}{2} = 3 = a_{1} .$ Induction in the common recurrence gives $a_{ν} = b_{ν}$ , so $a_{ν} = \frac{( 3 + 8 ) ^{ν} + ( 3 - 8 ) ^{ν}}{2},$ and in particular $a_{ν} \geq 0$ , because $2 < 8 < 3$ makes both $λ_{+}$ and $λ_{-}$ positive. A two-step induction now gives $a_{ν} \leq 6^{ν}$ : the cases $ν = 0, 1$ are immediate, and if the bound holds for $ν$ and $ν + 1$ , then $a_{ν + 2} = 6 a_{ν + 1} - a_{ν} \leq 6 a_{ν + 1} \leq 6 \cdot 6^{ν + 1} = 6^{ν + 2} .$ Thus $T_{L} (3) = a_{L} \leq 6^{L}$ , and therefore $0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣ = Θ_{L}^{+} \leq 6^{L} .$
Combining the preceding inequalities yields $∥ G_{L} ∥_{1} \leq 0 \leq α < L - 1 \sum ∣ θ_{L, α} ∣ \leq 6^{L},$ which proves the claim.

∎

Lemma 15 gives a direct coefficient bound for the quotient polynomial $G_{L}$ . This estimate is the coefficient-growth input behind the explicit factorial second-moment calibration.

Lemma 16 [lem:explicit-chebyshev-calibration] (Explicit Chebyshev calibration).

Let $T_{j}$ be the Chebyshev polynomial of degree $j$ , set $Q_{j} (z) = T_{j} (1 - 2 z)$ , and let $∥ \cdot ∥_{1}$ denote coefficient $ℓ^{1}$ -norm. For every integer $L \geq 2$ , the polynomial continuations $E_{L}$ and $G_{L}$ satisfy the following calibration.

(Coefficient bounds.) $∥ Q_{L} ∥_{1} \leq 7^{L}, 2∥ G_{L} ∥_{1} \leq 1 4^{L} .$

(Factorial-lift second moment.) For every $a \in {0, 1}$ , every $t > 0$ , $B > 0$ , and every $s_{0}, s_{1} \geq 0$ with $L \leq tB$ , set $p = s_{0} + s_{1}$ . If $K_{b} \sim Poi (t s_{b})$ independently and $W_{a, L}$ denotes the factorial lift of the light-cell polynomial based on $G_{L}$ , then $E W_{a, L}^{2} \leq 705 6^{L} (1 + \frac{p}{B})^{2 L},$ and, whenever $p \leq B$ , $E W_{a, L}^{2} \leq 705 6^{L} .$

(Interval identities.) For every $z \in [0, 1]$ , $0 \leq E_{L} (z), E_{L} (z) \leq 1, z G_{L} (z) + E_{L} (z) = 1.$ Moreover, for every $z \in (0, 1]$ , $E_{L} (z) \leq \frac{1}{L ^{2} z} .$

⊢ Lean

Proof of Lemma 16.

Fix $L \geq 2$ , so $L > 0$ .

Write $Q_{L} (z) = 1 + j = 1 \sum L q_{L, j} z^{j}, q_{L, j} = [z^{j}] Q_{L} .$ The shifted Chebyshev expansion used for $Q_{L} (z) = T_{L} (1 - 2 z)$ gives $∥ Q_{L} ∥_{1} = 1 + j = 1 \sum L ∣ q_{L, j} ∣ = T_{L} (3) .$ The standard growth bound for Chebyshev polynomials at $3$ gives $T_{L} (3) \leq 6^{L}$ , and hence $∥ Q_{L} ∥_{1} \leq 6^{L} \leq 7^{L} .$ For the quotient polynomial, Lemma 15 gives $∥ G_{L} ∥_{1} \leq 6^{L} .$ Since $L \geq 2$ , $2∥ G_{L} ∥_{1} \leq 2 \cdot 6^{L} \leq 2^{L} 7^{L} = 1 4^{L} .$ This proves the two coefficient bounds.
Fix $a \in {0, 1}$ , $t > 0$ , $B > 0$ , and $s_{0}, s_{1} \geq 0$ , and assume $L \leq tB$ . Put $D = tB, p = s_{0} + s_{1}, η = 1 + \frac{p}{B} .$ Then $D > 0$ , $D (s_{b} / B) = t s_{b}$ for $b = 0, 1$ , and $η \geq 1$ . Let $I_{L} = {j \in N : [z^{j}] G_{L} \neq = 0} .$ To identify this index set, write $Q_{L} (z) = 1 + q_{L, 1} z + r = 2 \sum L q_{L, r} z^{r}, q_{L, 1} = - 2 L^{2} .$ The definitions of $H_{L}$ and $E_{L}$ give the scaled identity $z E_{L} (z) = \frac{1 - Q _{L} ( z )}{2 L ^{2}} .$ Using the displayed value of $q_{L, 1}$ , this identity gives $E_{L} (z) = 1 - r = 2 \sum L \frac{q _{L, r}}{2 L ^{2}} z^{r - 1},$ and the quotient definition of $G_{L}$ gives $G_{L} (z) = \frac{1 - E _{L} ( z )}{z} = r = 2 \sum L \frac{q _{L, r}}{2 L ^{2}} z^{r - 2} .$ Thus $I_{L} \subseteq {0, \dots, L - 2},$ with the assertion empty if the displayed sum has no nonzero terms, and every $j \in I_{L}$ satisfies $j + 1 \leq L$ . For $K = (K_{0}, K_{1})$ , with $K_{a} = K_{1}$ when $a = 1$ and $K_{a} = K_{0}$ when $a = 0$ , define $X_{j} (K) = \frac{( K _{a} ) _{j + 1}}{D ^{j + 1}} + \frac{K _{1 - a} ( K _{a} ) _{j}}{D ^{j + 1}}, j \in I_{L} .$ If $I_{L}$ is empty, the following sum is zero. With $K_{b} \sim Poi (t s_{b})$ independently, the identities $D (s_{b} / B) = t s_{b}$ give the factorial lift as $W_{a, L} = j \in I_{L} \sum [z^{j}] G_{L} X_{j} (K) .$

For the moment bound, first use the one-dimensional Poisson falling-factorial estimate: if $N_{z} \sim Poi (D z)$ , $z \geq 0$ , and $r, v \leq L$ , then $E \frac{( N _{z} ) _{r} ( N _{z} ) _{v}}{D ^{r + v}} = h = 0 \sum m i n (r, v) (h r) (h v) h! \frac{( D z ) ^{r + v - h}}{D ^{r + v}} .$ For $0 \leq h \leq min (r, v)$ , the corresponding summand satisfies $(h r) (h v) h! \frac{( D z ) ^{r + v - h}}{D ^{r + v}} \leq \frac{( r v ) ^{h}}{h ! D ^{h}} (1 + z)^{r + v} \leq \frac{L ^{h}}{h !} (1 + z)^{r + v},$ because $r v / D \leq L$ , which follows from $r, v \leq L$ and $L \leq D$ , and because $z^{r + v - h} \leq (1 + z)^{r + v}$ . Therefore $E \frac{( N _{z} ) _{r} ( N _{z} ) _{v}}{D ^{r + v}} \leq h = 0 \sum L \frac{L ^{h}}{h !} (1 + z)^{r + v} \leq 3^{L} (1 + z)^{r + v},$ where the last inequality is the elementary finite-series bound $\sum_{h = 0}^{L} L^{h} / h! \leq 3^{L}$ .

This estimate gives the same-arm square bound. For every $0 \leq r \leq L$ , integrating out the unused arm in the product law and applying the preceding estimate with $z = s_{a} / B$ and orders $(r, r)$ gives $E (\frac{( K _{a} ) _{r}}{D ^{r}})^{2} \leq 3^{L} (1 + \frac{s _{a}}{B})^{2 r} .$ Since $1 + s_{a} / B \leq η$ , $η \geq 1$ , and $2 r \leq 2 L$ , $E (\frac{( K _{a} ) _{r}}{D ^{r}})^{2} \leq 3^{L} η^{2 L} .$ In particular, for $j \in I_{L}$ , using $j + 1 \leq L$ , $E (\frac{( K _{a} ) _{j + 1}}{D ^{j + 1}})^{2} \leq 3^{L} η^{2 L} .$

The cross-arm square bound follows by the product structure. For $j \in I_{L}$ , independence and $(K_{1 - a})_{1} = K_{1 - a}$ give $E (\frac{K _{1 - a} ( K _{a} ) _{j}}{D ^{j + 1}})^{2} = E (\frac{( K _{a} ) _{j}}{D ^{j}})^{2} E (\frac{( K _{1 - a} ) _{1}}{D})^{2} .$ Applying the one-dimensional estimate to arm $a$ with orders $(j, j)$ and to arm $1 - a$ with orders $(1, 1)$ yields $E (\frac{K _{1 - a} ( K _{a} ) _{j}}{D ^{j + 1}})^{2} \leq 3^{L} (1 + \frac{s _{a}}{B})^{2 j} \cdot 3^{L} (1 + \frac{s _{1 - a}}{B})^{2} .$ Now $1 + s_{a} / B \leq η$ , $1 + s_{1 - a} / B \leq η$ , $η \geq 1$ , and $2 j + 2 \leq 2 L$ . Hence $(1 + \frac{s _{a}}{B})^{2 j} (1 + \frac{s _{1 - a}}{B})^{2} \leq η^{2 j + 2} \leq η^{2 L},$ and therefore $E (\frac{K _{1 - a} ( K _{a} ) _{j}}{D ^{j + 1}})^{2} \leq 9^{L} η^{2 L} .$ Since $(ξ + υ)^{2} \leq 2 ξ^{2} + 2 υ^{2}$ , $E X_{j} (K)^{2} \leq 2 \cdot 3^{L} η^{2 L} + 2 \cdot 9^{L} η^{2 L} \leq 3 6^{L} η^{2 L} = (6^{L} η^{L})^{2} .$ The finite-sum $L^{2}$ inequality $E j \in I_{L} \sum c_{j} X_{j}^{2} \leq j \in I_{L} \sum ∣ c_{j} ∣^{2} j \in I_{L} max E X_{j}^{2}$ therefore yields $E W_{a, L}^{2} \leq ∥ G_{L} ∥_{1}^{2} (6^{L} η^{L})^{2} \leq (6^{L})^{2} (6^{L} η^{L})^{2} = 129 6^{L} η^{2 L} .$ Because $1296 \leq 7056$ , $E W_{a, L}^{2} \leq 705 6^{L} (1 + \frac{p}{B})^{2 L} .$ If $p \leq B$ , then $η \leq 2$ , and hence $E W_{a, L}^{2} \leq 129 6^{L} 2^{2 L} = 518 4^{L} \leq 705 6^{L} .$ This proves both second-moment assertions.
Let $z \in [0, 1]$ . The scaled identity for the continuation is $z E_{L} (z) = (2 L^{2})^{- 1} {1 - Q_{L} (z)} = \frac{1 - Q _{L} ( z )}{2 L ^{2}} .$ At $z = 0$ , the derivative identity $T_{L}^{'} (1) = L^{2}$ gives $E_{L} (0) = 1$ . For $z > 0$ , put $θ = arccos (1 - 2 z) .$ Then $Q_{L} (z) = cos (L θ)$ , so $∣ Q_{L} (z) ∣ \leq 1$ . Hence $1 - Q_{L} (z) \geq 0$ , and the scaled identity gives $E_{L} (z) \geq 0$ . Also $1 - Q_{L} (z) = 1 - cos (L θ) \leq L^{2} (1 - cos θ) = 2 L^{2} z,$ so $z E_{L} (z) \leq z$ , whence $E_{L} (z) \leq 1$ . Finally, $∣ Q_{L} (z) ∣ \leq 1$ gives $1 - Q_{L} (z) \leq 2$ , and the normalization by $2 L^{2}$ in the scaled identity gives $z E_{L} (z) = \frac{1 - Q _{L} ( z )}{2 L ^{2}} \leq \frac{1}{L ^{2}},$ and division by $z > 0$ gives $E_{L} (z) \leq \frac{1}{L ^{2} z} .$ The quotient definition gives the polynomial identity $z G_{L} (z) + E_{L} (z) = 1$ for every real $z$ , and therefore on $[0, 1]$ .

∎

Lemma 16 supplies the numerical coefficient bounds used to instantiate Lemma 5 in the calibrated estimator. In particular, the constant $7056$ gives the explicit factorial-lift second-moment scale that appears in the finite calibration of Algorithm 1.

Common-marginal lower bounds

The converse in Theorem 3 rests on priors whose treatment–covariate marginal is shared across branches while their average treatment effect centers are separated. This appendix collects the finite approximation and transfer statements that make that construction compatible with the annotation experiment in Definition 9. The first object is a two-cell calculation that isolates the statistical issue in its smallest form.

Definition 27 [def:finite-small-instance] (Finite small instance

C_{small}

Let $ϵ = \frac{1}{4}, d = 2, p_{1} = p_{2} = \frac{1}{2}, e_{1} = e_{2} = \frac{1}{2}, μ_{01} = μ_{02} = 0.$ For $n = m = 1$ , independence of the samples gives $E [21 {L_{1} = ((1, 1, 1))} (1 {U_{1} = ((1, 0))} + 1 {U_{1} = ((1, 1))})] = 2 q_{11} p_{1},$ where $q_{11}$ is the outcome-marked mass for treatment-covariate cell $(1, 1)$ . At $μ_{11} = 1$ , this equals $1/4$ , while the treated-cell target is $p_{1} μ_{11} = \frac{1}{2} .$ Define $Π_{0}^{small}$ to put unit mass on the law with $μ_{11} = μ_{12} = 0,$ and define $Π_{1}^{small}$ to put unit mass on the law with $μ_{11} = 1, μ_{12} = 0.$ Under both priors, $P_{X A} (x, a) = \frac{1}{4} for all (x, a),$ whereas $τ (P) = 0$ under $Π_{0}^{small}$ , and $τ (P) = \frac{1}{2}$ under $Π_{1}^{small}$ . Consequently, the auxiliary laws agree for every $m$ , and the total variation distance between the two labeled laws is $1 - (\frac{3}{4})^{n} .$ These identities constitute $C_{small}$ .

⊢ Lean

The instance in Definition 27 makes concrete the role of outcome labels. The two priors $Π_{0}^{small}$ and $Π_{1}^{small}$ share the same treatment–covariate marginal $P_{X A}$ , so auxiliary observations identify the common table while the target separation remains attached to the outcome-marked mass $q_{a x}$ . The calculation $C_{small}$ is finite, but it has the same common-marginal structure used in the high-dimensional lower bound.

The next statement supplies the approximation-duality ingredient. It constructs a signed finite measure on a short interval whose first $L$ moments vanish while the inverse-weight functional retains a fixed gap. This is the same polynomial approximation mechanism that appears in nonsmooth functional lower bounds for discrete distributions (Jiao et al., 2015; Wu et al., 2016; Valiant et al., 2017; Orlitsky et al., 2016).

Lemma 17 [lem:finite-alternation-duality] (Finite alternation duality).

Fix an overlap constant $ϵ$ with $0 < ϵ < 1/2$ , and set $κ = \frac{1 - 2 ϵ}{ϵ} .$ There exist constants $γ_{ϵ} > 0$ and $c_{ϵ} > 0$ such that, for every integer $L \geq 2$ and every $B > 0$ , with $a = \frac{γ _{ϵ} B}{L ^{2}},$ one has $a / κ \leq B$ , and there are distinct points $p_{0}, \dots, p_{L + 1} \in [a / κ, B]$ and real weights $w_{0}, \dots, w_{L + 1}$ satisfying $i = 0 \sum L + 1 ∣ w_{i} ∣ = 1, i = 0 \sum L + 1 w_{i} p_{i}^{j} = 0 for every 0 \leq j \leq L,$ and $c_{ϵ} \leq i = 0 \sum L + 1 w_{i} \frac{p _{i}}{p _{i} + a} .$

⊢ Lean

Proof of Lemma 17.

Set $D_{ϵ} = \frac{κ}{( 1 + κ ) ( 2 + κ )}, γ_{ϵ} = min {\frac{κ}{4}, \frac{D _{ϵ} κ}{32}}, c_{ϵ} = \frac{D _{ϵ}}{8} .$ Since $0 < ϵ < 1/2$ , we have $κ > 0$ . Hence $D_{ϵ} > 0$ , $γ_{ϵ} > 0$ , and $c_{ϵ} > 0$ . Also $D_{ϵ} < 1$ , because $κ < (1 + κ) (2 + κ)$ for $κ > 0$ .
Fix $L \geq 2$ and $B > 0$ , and put $a = \frac{γ _{ϵ} B}{L ^{2}} .$ Then $a > 0$ . Since $γ_{ϵ} \leq κ /4$ and $L^{2} \geq 4$ , $\frac{a}{κ} = \frac{γ _{ϵ} B}{κ L ^{2}} \leq B .$ Moreover, $\frac{a}{κ} \leq \frac{B}{16},$ because $γ_{ϵ} \leq κ /4$ and $L^{2} \geq 4$ . In particular $a / κ < B$ .
Let $I = [\frac{a}{κ}, B], f (p) = \frac{p}{p + a} .$ The pole $- a$ lies to the left of $I$ , since $a > 0$ and $a / κ > 0$ . Thus $f$ is continuous on $I$ . Let $E$ be the best uniform error for approximating $f$ on $I$ by polynomials of degree at most $L$ . The finite alternation theorem supplies an alternation dual certificate: distinct nodes in $I$ , normalized weights annihilating all monomials of degree at most $L$ , and an associated degree- $L$ best approximating polynomial.
We claim $c_{ϵ} \leq E .$ Suppose instead that $E < c_{ϵ}$ . Define $p_{-} = \frac{a}{κ}, p_{+} = \frac{2 a}{κ} .$ Then $p_{-} < p_{+}$ , and both points belong to $I$ . Indeed $p_{-} = a / κ$ , while $p_{+} = \frac{2 γ _{ϵ} B}{κ L ^{2}} \leq B$ because $γ_{ϵ} \leq κ$ and $L^{2} \geq 4$ . At these two points, $f (p_{-}) = \frac{1}{1 + κ}, f (p_{+}) = \frac{2}{2 + κ},$ so $f (p_{+}) - f (p_{-}) = \frac{κ}{( 1 + κ ) ( 2 + κ )} = D_{ϵ} .$ Let $Q$ be the best approximating polynomial from the alternation certificate. For every $p \in I$ , $∣ f (p) - Q (p) ∣ \leq E .$ Consequently, $D_{ϵ} - 2 E \leq Q (p_{+}) - Q (p_{-}) .$ By the mean-value theorem there is a point $ξ \in [p_{-}, p_{+}] \subset I$ such that $Q^{'} (ξ) = \frac{Q ( p _{+} ) - Q ( p _{-} )}{p _{+} - p _{-}},$ and therefore $\frac{D _{ϵ} - 2 E}{p _{+} - p _{-}} \leq ∣ Q^{'} (ξ) ∣.$ Since $E < c_{ϵ} = D_{ϵ} /8$ , the numerator is positive.
On $I$ , the target satisfies $0 \leq f (p) \leq 1$ . Combining this with $∣ f (p) - Q (p) ∣ \leq E$ gives $∣ Q (p) ∣ \leq 1 + E (p \in I),$ or equivalently $∥ Q ∥_{\infty, I} \leq 1 + E .$ Markov’s inequality on the interval $I = [a / κ, B]$ , applied to the degree- $L$ polynomial $Q$ , gives $∥ Q^{'} ∥_{\infty, I} \leq \frac{2 L ^{2}}{B - a / κ} ∥ Q ∥_{\infty, I} \leq \frac{2 L ^{2}}{B - a / κ} (1 + E) .$ Because $a / κ \leq B /16$ , we have $B /2 \leq B - a / κ$ . Also $E < c_{ϵ} < 1$ . Hence $∣ Q^{'} (ξ) ∣ \leq \frac{2 L ^{2}}{B - a / κ} (1 + E) \leq \frac{2 L ^{2}}{B /2} \cdot 2 = \frac{8 L ^{2}}{B} .$ Combining this with the previous lower bound and using $p_{+} - p_{-} = \frac{a}{κ} = \frac{γ _{ϵ} B}{κ L ^{2}}$ yields $\frac{( D _{ϵ} - 2 E ) κ L ^{2}}{γ _{ϵ} B} \leq \frac{8 L ^{2}}{B} .$ After cancellation, $(D_{ϵ} - 2 E) κ \leq 8 γ_{ϵ} .$ But $E < D_{ϵ} /8$ , so $D_{ϵ} - 2 E > 3 D_{ϵ} /4$ , while $8 γ_{ϵ} \leq \frac{D _{ϵ} κ}{4} .$ This is impossible. Therefore $c_{ϵ} \leq E$ .
Let the alternation certificate have nodes $p_{0}, \dots, p_{L + 1} \in I$ and weights $α_{0}, \dots, α_{L + 1}$ . They satisfy $i = 0 \sum L + 1 ∣ α_{i} ∣ = 1, i = 0 \sum L + 1 α_{i} p_{i}^{j} = 0 (0 \leq j \leq L),$ and $i = 0 \sum L + 1 α_{i} f (p_{i}) = E .$ Choose $s = {1, - 1, if \sum_{i} α_{i} f (p_{i}) \geq 0, if \sum_{i} α_{i} f (p_{i}) < 0, w_{i} = s α_{i} .$ Then the nodes remain distinct and lie in $I$ , and $i = 0 \sum L + 1 ∣ w_{i} ∣ = i = 0 \sum L + 1 ∣ α_{i} ∣ = 1.$ For every $0 \leq j \leq L$ , $i = 0 \sum L + 1 w_{i} p_{i}^{j} = s i = 0 \sum L + 1 α_{i} p_{i}^{j} = 0.$ Finally, $i = 0 \sum L + 1 w_{i} \frac{p _{i}}{p _{i} + a} = s i = 0 \sum L + 1 α_{i} f (p_{i}) = i = 0 \sum L + 1 α_{i} f (p_{i}) = E \geq c_{ϵ} .$ These nodes and weights give the required finite moment dual certificate on $[a / κ, B]$ .

∎

Lemma 17 turns polynomial indistinguishability into target separation. The moment cancellations control likelihood differences after Poissonization, while the final display gives a fixed gap for the inverse-weight expression that enters the treatment-effect contrast.

The rare-cell prior packages this dual certificate into a common-marginal experiment. For the transfer arguments, it is useful to state the construction as a recipe separate from any particular choice of sample intensities. The next definition records the raw rare-cell masses, the two branch laws, and the raw target centers that the lower-bound reductions manipulate; the later uniform-intensity lemma supplies the exact Poisson witness actually used by the converse proof.

Definition 28 [synth_20] (Common-marginal rare-cell recipe).

A common-marginal rare-cell recipe with $k$ rare cells at scale $(a, B, L)$ is a finite random construction $R$ consisting of a generator law $generator_{R}$ for a latent draw $w$ , together with two branch laws indexed by $b \in {0, 1}$ . For each generated $w$ , the recipe specifies rare-cell raw masses $p_{i} (w) \in [0, B]$ , $i = 1, \dots, k$ , branch-specific outcome regressions, and any reservoir mass needed to normalize the construction. The two branches have the same treatment–covariate raw masses and treatment propensities, so after normalization they induce the same $P_{X A}$ table, while their outcome regressions may differ.

For each branch $b \in {0, 1}$ , let $P_{b} (w)$ be the normalized law on $[d] \times {0, 1}^{2}$ induced by the recipe, and let $rawTarget_{R} (b, w)$ be the unnormalized target contribution assigned by the rare cells in branch $b$ . The branch center is $θ_{b} = \int rawTarget_{R} (b, w) d generator_{R} (w),$ and the raw-center separation is $Δ = ∣ θ_{1} - θ_{0} ∣$ . The recipe induces, for each branch $b$ , the mixture experiment obtained by first drawing $w \sim generator_{R}$ , then sampling from $P_{b} (w)$ for labeled observations and from the common marginal $P_{b, X A} (w)$ for auxiliary observations. Its raw Poisson version uses labeled intensity $u$ and auxiliary intensity $v$ with the same generated law $P_{b} (w)$ , so the treatment–covariate component is common across the two branches and the likelihood discrepancy is carried by the labeled outcome-bearing component.

Definition 28 fixes the bookkeeping used by the lower-bound priors. The random raw masses determine a common treatment–covariate table, and the branch-specific outcome regressions create a raw-center separation before normalization. This separation is the quantity later converted into minimax risk.

The raw Poisson comparison enters several concentration and transfer statements, so the notation for the branch predictive laws is named explicitly before those estimates.

Lemma 18 [lem:common-marginal-recipe-target-concentration] (Prior target concentration).

Fix $ϵ \in R$ , a common-marginal calibration at overlap level $ϵ$ with dual-gap constant $c_{ϵ}$ , and parameters $n, m, d, L, k \in N$ and $B, a, u, v, C, ρ \in R$ . Let $R$ be a common-marginal rare-cell recipe with atom-generator measure $Γ$ , branch laws $P_{b} (w)$ , raw targets $r_{b} (w)$ , raw total mass $S (w)$ , and branch priors $Π_{b} = Γ \circ P_{b}^{- 1}, θ_{b} = \int r_{b} (w) d Γ (w), b \in {0, 1} .$ Assume the recipe bounds hold at intensities $(u, v)$ with constants $(C, ρ)$ and dual gap $c_{ϵ}$ :

(Finite-atom construction.) The recipe uses the finite-atom construction.

(Model support.) Each prior is supported on $M_{d, ϵ}$ of Definition 6.

(Common table.) The two priors induce the same law of the random treatment-covariate table.

(Separation.) The raw centers satisfy $c_{ϵ} k a \leq ∣ θ_{1} - θ_{0} ∣.$

(Mass and variance bounds.) Under the usual measurability and integrability conditions, $\int S (w) d Γ (w) = 1, Var_{Γ} (S) \leq C k B a, b \in {0, 1} max Var_{Γ} (r_{b}) \leq C k B a .$

(Mixture proximity.) The two raw mixture laws at labeled intensity $u$ and auxiliary intensity $v$ have total variation distance at most $C u k a ρ^{L} .$

Then for every branch $b \in {0, 1}$ and every $Δ$ with $0 < Δ \leq 2$ , the branch prior concentrates its average treatment effect $τ (P)$ , defined in Definition 8, around its raw center: $Π_{b} {P : \frac{Δ}{4} < ∣ τ (P) - θ_{b} ∣} \leq \frac{512 C k B a}{Δ ^{2}} .$

⊢ Lean

Proof of Lemma 18.

Fix $b \in {0, 1}$ . Define the two exceptional events in the atom space by $A_{tar} := {w : \frac{Δ}{16} < ∣ r_{b} (w) - θ_{b} ∣}, A_{mass} := {w : \frac{Δ}{16} < ∣ S (w) - 1∣} .$ We first record the normalization implication used below. Outside a $Γ$ -null set the recipe gives $r_{b} (w) = S (w) τ (P_{b} (w)), S (w) > 0.$ Suppose $\frac{Δ}{4} < ∣ τ (P_{b} (w)) - θ_{b} ∣, w \in / A_{tar} \cup A_{mass} .$ The second condition means $∣ r_{b} (w) - θ_{b} ∣ \leq \frac{Δ}{16}, ∣ S (w) - 1∣ \leq \frac{Δ}{16} .$ The raw-center bound supplied by the recipe normalization is $∣ θ_{b} ∣ \leq 1.$ Hence $∣ S (w) θ_{b} - θ_{b} ∣ = ∣ (S (w) - 1) θ_{b} ∣ \leq \frac{Δ}{16} .$ Using $r_{b} (w) = S (w) τ (P_{b} (w))$ , the triangle inequality gives $S (w) ∣ τ (P_{b} (w)) - θ_{b} ∣ = ∣ S (w) τ (P_{b} (w)) - S (w) θ_{b} ∣ \leq \frac{Δ}{8} .$ Also $∣ S (w) - 1∣ \leq Δ/16$ and $Δ \leq 2$ imply $S (w) \geq 1 - \frac{Δ}{16} \geq \frac{7}{8} .$ Therefore $\frac{7}{8} ∣ τ (P_{b} (w)) - θ_{b} ∣ \leq S (w) ∣ τ (P_{b} (w)) - θ_{b} ∣ \leq \frac{Δ}{8},$ which contradicts $Δ/4 < ∣ τ (P_{b} (w)) - θ_{b} ∣$ . Thus, for $Γ$ -almost every $w$ , $\frac{Δ}{4} < ∣ τ (P_{b} (w)) - θ_{b} ∣ ⟹ w \in A_{tar} \cup A_{mass} .$

By the definition of the branch prior as the pushforward of $Γ$ under $P_{b}$ , with the two Boolean branches handled separately, $Π_{b} {P : \frac{Δ}{4} < ∣ τ (P) - θ_{b} ∣} = Γ {w : \frac{Δ}{4} < ∣ τ (P_{b} (w)) - θ_{b} ∣} .$ The implication above yields $Π_{b} {P : \frac{Δ}{4} < ∣ τ (P) - θ_{b} ∣} \leq Γ (A_{tar} \cup A_{mass}) .$

We next bound the two exceptional probabilities. Chebyshev’s inequality, applied under $Γ$ with mean $θ_{b} = \int r_{b} (w) d Γ (w),$ and the assumed variance bound $Var_{Γ} (r_{b}) \leq C k B a,$ gives $Γ (A_{tar}) \leq \frac{C k B a}{( Δ/16 ) ^{2}} .$ Similarly, since $\int S (w) d Γ (w) = 1, Var_{Γ} (S) \leq C k B a,$ Chebyshev’s inequality gives $Γ (A_{mass}) \leq \frac{C k B a}{( Δ/16 ) ^{2}} .$

Subadditivity of measure gives $Γ (A_{tar} \cup A_{mass}) \leq Γ (A_{tar}) + Γ (A_{mass}) .$ Combining the preceding displays, $Π_{b} {P : \frac{Δ}{4} < ∣ τ (P) - θ_{b} ∣} \leq \frac{C k B a}{( Δ/16 ) ^{2}} + \frac{C k B a}{( Δ/16 ) ^{2}} = \frac{512 C k B a}{Δ ^{2}},$ where the final equality uses $Δ > 0$ . This is the desired bound for the fixed branch $b$ , and hence for both branches.

∎

Definition 29 separates the prior randomness in the raw masses from the Poisson sampling randomness. With this notation, total variation between the two branch mixtures measures the experiment-level testing difficulty after the common treatment–covariate table has been integrated over.

The next two lemmas isolate the normalization estimates used by the transfer. The first converts raw target control into concentration of the average treatment effect, and the second records the corresponding lower-tail control for the raw scale.

Definition 29 [synth_21] (Raw Poisson prior-predictive mixture laws).

Fix a common-marginal rare-cell recipe $R$ with atom generator $Γ$ , branch laws $P_{b} (w)$ for $b \in {0, 1}$ , and branch priors $Π_{b} = Γ \circ P_{b}^{- 1}$ . At labeled and auxiliary Poisson intensities $u$ and $v$ , define $Q_{b}$ to be the branch- $b$ raw prior-predictive law obtained by first drawing $w \sim Γ$ , forming the raw branch law $P_{b} (w)$ , and then sampling the independent raw Poisson experiment with labeled intensity $u$ and auxiliary intensity $v$ from that raw law and its treatment–covariate marginal. Equivalently, $Q_{b} = \int {PoiExp_{u, v}^{raw} (P_{b} (w))} d Γ (w) = \int PoiExp_{u, v}^{raw} (P) d Π_{b} (P), b \in {0, 1},$ where $PoiExp_{u, v}^{raw} (P)$ denotes the product Poisson law for the outcome-labeled raw cell counts at intensity $u$ and the auxiliary treatment–covariate raw cell counts at intensity $v$ .

Lemma 19 [lem:common-marginal-recipe-scale-tail] (Small-scale prior tail).

Fix $ϵ \in R$ , a common-marginal calibration with positive constants including dual-gap constant $c_{ϵ} \leq 1/4$ , integers $n, m, d, L, k \geq 0$ , and reals $B, a, u, v, C, ρ$ . Let $R$ be a common-marginal rare-cell recipe with atom-generator $Γ$ , branch laws $P_{b} (w)$ , branch priors $Π_{b} = Γ \circ P_{b}^{- 1}$ , raw total mass $S (w)$ , raw targets $r_{b} (w)$ , and measurable raw scale $s_{R}$ satisfying $s_{R} (P_{b} (w)) = S (w)$ for $Γ$ -almost every $w$ .

Assume that the recipe bounds hold for $R$ at $(u, v, c_{ϵ}, C, ρ)$ :

(Finite-atom construction.) The recipe uses the finite-atom construction.
(Model support and common table.) The two priors are supported on $M_{d, ϵ}$ from Definition 6, and they induce the same distribution of the random treatment-covariate table $P_{X A}$ .
(Center separation.) If $θ_{b}$ denotes the raw prior center for branch $b$ , then $c_{ϵ} k a \leq ∣ θ_{1} - θ_{0} ∣.$
(Mass and variance bounds.) Under the usual measurability and integrability conditions, $\int S d Γ = 1, Var_{Γ} (S) \leq C k B a, max {Var_{Γ} (r_{0}), Var_{Γ} (r_{1})} \leq C k B a .$
(Mixture distance.) The two raw Poisson mixture laws generated by the recipe satisfy $TV (Q_{0}, Q_{1}) \leq C u k a ρ^{L} .$

Then, for each branch $b \in {0, 1}$ , $Π_{b} {P : max {s_{R} (P), 0} < \frac{1}{2}} \leq 4 C k B a .$

⊢ Lean

Proof of Lemma 19.

Define the mass-tail event on the atom space by $MassTail := {w : \frac{1}{2} < ∣ S (w) - 1∣} .$ The mass assumptions give $\int S d Γ = 1, Var_{Γ} (S) \leq C k B a .$ Since $Γ$ is a probability law, Chebyshev’s inequality at radius $1/2$ yields $Γ (MassTail) = Γ {w : \frac{1}{2} < ∣ S (w) - 1∣} \leq \frac{Var _{Γ} ( S )}{( 1/2 ) ^{2}} \leq 4 C k B a .$
For a fixed branch $b \in {0, 1}$ , set $SmallScale_{b} := {P : max {s_{R} (P), 0} < \frac{1}{2}} .$ Because $Π_{b} = Γ \circ P_{b}^{- 1}$ , $Π_{b} (SmallScale_{b}) = Γ {w : P_{b} (w) \in SmallScale_{b}} .$ For $Γ$ -almost every $w$ , the recipe supplies both identities $s_{R} (P_{b} (w)) = S (w) and 0 < S (w) .$ On this full-measure set, $P_{b} (w) \in SmallScale_{b}$ implies $S (w) = max {S (w), 0} < \frac{1}{2} .$ Hence $S (w) - 1 < - 1/2$ , and therefore $∣ S (w) - 1∣ = 1 - S (w) > \frac{1}{2} .$ Thus ${w : P_{b} (w) \in SmallScale_{b}} \subseteq MassTail up to a Γ-null set,$ and consequently $Π_{b} (SmallScale_{b}) \leq Γ (MassTail) .$
The preceding containment is applied to the two branch cases separately: for $b = 0$ it uses $Π_{0} = Γ \circ P_{0}^{- 1},$ and for $b = 1$ it uses $Π_{1} = Γ \circ P_{1}^{- 1} .$ Combining the appropriate branch identity with the bound from the first step gives, for each $b \in {0, 1}$ , $Π_{b} {P : max {s_{R} (P), 0} < \frac{1}{2}} \leq Γ (MassTail) \leq 4 C k B a .$

∎

Lemmas 18 and 19 make the raw construction usable after normalization. The target concentration bound relates the branch centers $θ_{b}$ to the actual functional values under the induced priors, while the scale-tail bound keeps the normalizing mass away from the region where raw-to-normalized conversion would amplify the rare-cell perturbation.

The next transfer lemma applies the recipe at fixed Poisson intensities large enough to dominate the fixed sample sizes. Its assumptions combine a total-variation bound with a scale condition that controls the random normalization.

Lemma 20 [lem:common-marginal-recipe-transfer] (Recipe transfer bound).

Let $0 < ϵ < 1/2$ , where $ϵ$ is the overlap level. Fix a common-marginal calibration at level $ϵ$ , consisting of positive constants including $γ_{ϵ}$ and dual gap $δ_{ϵ} \leq 1/4$ , a constant $C_{ϵ} > 0$ , and a real number $ρ_{ϵ}$ . Let $n, m, d, L, k$ be nonnegative integers, let $B, a \in R$ , and let $R$ be a common-marginal rare-cell recipe with $k$ rare cells at scale $(a, B, L)$ . For each branch $b \in {0, 1}$ , define the raw prior center $θ_{b} = \int rawTarget_{R} (b, w) d G_{R} (w),$ where $G_{R}$ is the atom-generator measure of the recipe, and put $Δ = ∣ θ_{1} - θ_{0} ∣.$ Assume:

(Sample and dimension.) $n \geq 1$ and $d \geq 2$ .

(Poisson-prior data.) With $u = 65536 n and v = 65536 (n + m),$ the recipe has bandwidth $B > 0$ , shift $a = \frac{γ _{ϵ} B}{L ^{2}},$ rare-shift size $k a \leq δ_{ϵ},$ and the recipe intensities agree with $u$ and $v$ .

(Bandwidth and likelihood bounds.) The realized scale satisfies $(u + v) B \leq b_{ϵ} L,$ where $b_{ϵ}$ is the calibration’s uniform-bandwidth constant, and the recipe satisfies the common-marginal Poisson-prior bounds at gap $δ_{ϵ}$ , intensity pair $(u, v)$ , variance constant $C_{ϵ}$ , and decay constant $ρ_{ϵ}$ .

(Mixture affinity.) The two prior-predictive raw Poisson-count laws generated by the two recipe priors and the recipe’s experiment kernel have total variation distance at most $\frac{1}{8} .$

(Variance scale.) The recipe scale obeys $k B a \leq \frac{Δ ^{2}}{65536 C _{ϵ}} .$

Then the minimax risk of Definition 11 satisfies $R_{ϵ} (n, m, d) \geq \frac{Δ ^{2}}{65536} .$

⊢ Lean

Proof of Lemma 20.

Let $Π_{b} := the branch- b prior induced by R, θ_{b} := \int r_{b} (w) d Γ (w), b \in {0, 1},$ where $Γ$ is the atom-generator measure and $r_{b}$ is the raw target of the recipe. Thus $Δ = ∣ θ_{1} - θ_{0} ∣.$ Also put $u := 65536 n, v := 65536 (n + m),$ and, for every law $P$ , define the totalized nonnegative recipe scale $s_{+} (P) := max {s_{R} (P), 0} .$

The witness bounds imply $∣ θ_{0} ∣ \leq 1, ∣ θ_{1} ∣ \leq 1.$ Indeed, each $θ_{b}$ is the $Γ$ -mean of the raw target, the raw target is $S (w) τ (P_{b} (w))$ , $∣ τ (P_{b} (w)) ∣ \leq 1$ on $M_{d, ϵ}$ , and the recipe bounds give $\int S d Γ = 1$ . Hence $0 \leq Δ \leq 2.$ The intensities satisfy $u \geq 0$ and $v \geq 0$ . The branch priors are probability measures, and the support part of the witness gives the required nonempty model class.
Let $K_{raw}$ be the ordered raw Poisson-count experiment which, at law $P$ , draws independent ordered Poisson samples from the labeled mark law and the auxiliary mark law with intensities $u s_{+} (P) and v s_{+} (P) .$ Let $Q_{b}$ be the prior-predictive law obtained from $Π_{b}$ through $K_{raw}$ . We now identify this law as a Markov image of the raw count law in the statement. Let $K_{cnt}$ denote the recipe’s raw Poisson-count kernel before ordering. Let $reindex$ be the deterministic map that rewrites a raw count table as labeled-mark and auxiliary-mark count functions, and let $order$ be the paired-histogram kernel that, conditional on those two count functions, orders the corresponding multiset of labeled marks and the corresponding multiset of auxiliary marks uniformly and independently. Put $Rec := order \circ reindex .$ For every law $P$ , the raw count kernel factors through this reconstruction as $Rec \circ K_{cnt} (P) = K_{raw} (P) .$ Indeed, after the deterministic reindexing, the labeled and auxiliary count functions have independent Poisson coordinates with means $u s_{+} (P)$ times the labeled mark law and $v s_{+} (P)$ times the auxiliary mark law; ordering each histogram gives exactly the ordered independent Poisson samples with those two intensities. Therefore, for each branch, $R_{raw, b} := \int K_{cnt} (P) d Π_{b} (P) satisfies Q_{b} = \int K_{raw} (P) d Π_{b} (P) = (Rec)_{#} R_{raw, b} .$ Total variation contracts under a Markov kernel, and the assumed raw mixture affinity gives $TV (Q_{0}, Q_{1}) \leq TV (R_{raw, 0}, R_{raw, 1}) \leq \frac{1}{8} .$
Set $α_{floor} := \frac{1}{100} .$ The fixed experiment has the parametric two-point floor $\frac{α _{floor}}{n} \leq R_{ϵ} (n, m, d) .$ To see this, take $δ_{n} := \frac{2}{5 n} .$ Choose a cell $x_{⋆} \in [d]$ . Under both laws below, let $X = x_{⋆}$ almost surely and let the treatment propensity be $1/2$ . Let the control outcome mean be $1/2$ under both laws; let the treated outcome mean be $1/2$ for $P_{par, 0}$ and $1/2 + δ_{n}$ for $P_{par, 1}$ . Since $n \geq 1$ , $0 \leq δ_{n} \leq \frac{2}{5},$ so these are valid binary outcome laws, and since $0 < ϵ < 1/2$ , both belong to $M_{d, ϵ}$ . Their treatment-covariate marginals agree, their average treatment effects are $τ (P_{par, 0}) = 0, τ (P_{par, 1}) = δ_{n},$ and hence the auxiliary samples have identical laws. For the labeled samples, the standard Bernoulli product testing bound applies because $n \cdot \frac{( 1/2 ) δ _{n}^{2}}{( 1/2 ) ( 1 - 1/2 )} = \frac{8}{25} \leq lo g 2,$ and therefore $TV (P_{par, 0}^{\otimes n}, P_{par, 1}^{\otimes n}) \leq \frac{1}{2} .$ Le Cam’s two-point bound with separation $δ_{n}$ yields $R_{ϵ} (n, m, d) \geq \frac{( δ _{n} /2 ) ^{2}}{4} = \frac{1}{100 n} .$ For the fixed-sample decision problem with the class-gated clipped squared loss, let $R_{dec} := ψ in f P sup \int ℓ_{ϵ} (P, ψ) d K_{fix} (P),$ where the infimum is over measurable fixed-sample rules, $K_{fix}$ is the fixed labeled-and-auxiliary experiment, and $ℓ_{ϵ} (P, z) := {(clip_{[- 1, 1]} (z) - τ (P))^{2}, 0, P \in M_{d, ϵ}, P \in / M_{d, ϵ} .$ We use the following elementary comparison between Definition 11 and the class-gated decision problem: $R_{ϵ} (n, m, d) \leq R_{dec} .$ Indeed, every measurable fixed-sample rule $ψ$ induces the measurable estimator $T_{ψ} := clip_{[- 1, 1]} \circ ψ .$ For every $P \in M_{d, ϵ}$ , the class-gated loss evaluates the same clipped rule, so $\int ℓ_{ϵ} (P, ψ (y)) d K_{fix} (P) (y) = E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [(T_{ψ} - τ (P))^{2}] .$ Thus the worst-case on-class decision risk of $ψ$ is exactly the worst-case mean-squared risk of $T_{ψ}$ . Since $T_{ψ}$ is one admissible estimator in $T_{n, m, d}$ , Definition 11 is no larger than that worst-case risk; taking the infimum over $ψ$ gives the displayed comparison. Combining it with the parametric floor gives $\frac{α _{floor}}{n} \leq R_{dec} .$
If $Δ = 0$ , then the desired lower bound is $0 \leq R_{ϵ} (n, m, d)$ , which follows from the preceding floor since $n \geq 1$ . Hence assume for the rest of the proof that $0 < Δ.$ The variance-scale hypothesis and $C_{ϵ} > 0$ give $C_{ϵ} k B a \leq \frac{Δ ^{2}}{65536} .$ By Lemma 18, for each $b \in {0, 1}$ , $Π_{b} {P : \frac{Δ}{4} < ∣ τ (P) - θ_{b} ∣} \leq \frac{512 C _{ϵ} k B a}{Δ ^{2}} \leq \frac{1}{128} \leq \frac{1}{16} .$ The assumed common-marginal Poisson-prior bounds also include the raw-center separation $δ_{ϵ} k a \leq ∣ θ_{1} - θ_{0} ∣ = Δ,$ which is the center-separation hypothesis required below. By Lemma 19, again for each $b \in {0, 1}$ , $Π_{b} {P : s_{+} (P) < 1/2} \leq 4 C_{ϵ} k B a \leq \frac{Δ ^{2}}{16384} .$
The two fuzzy hypotheses with centers $θ_{0}, θ_{1}$ , radius $Δ/4$ , prior tail bounds $1/16, 1/16$ , and predictive total variation bound $1/8$ give the raw lower bound as follows. For a measurable raw rule $φ$ , define $B_{b} (φ) := \int^{+} \int^{+} ℓ_{ϵ} (P, φ (y)) d K_{raw} (P) (y) d Π_{b} (P) \in \overline{R}_{+}, b \in {0, 1} .$ These are the branch Bayes decision risks for the class-gated clipped loss. Since the two branch priors are supported on $M_{d, ϵ}$ , the integrand on the support is $ℓ_{ϵ} (P, φ (y)) = (clip_{[- 1, 1]} (φ (y)) - τ (P))^{2} .$ Let $Q_{b}$ be the branch predictive laws from step 2, and first suppose $θ_{0} \leq θ_{1}$ . Put $θ_{mid} := \frac{θ _{0} + θ _{1}}{2}, A_{φ} := {y : clip_{[- 1, 1]} (φ (y)) \geq θ_{mid}},$ and define the good prior events $G_{0} := {P : ∣ τ (P) - θ_{0} ∣ \leq Δ/4}, G_{1} := {P : ∣ τ (P) - θ_{1} ∣ \leq Δ/4} .$ Because $θ_{1} - θ_{0} = ∣ θ_{1} - θ_{0} ∣ \geq Δ$ , on $G_{0} \cap A_{φ}$ the clipped decision is at least $Δ/4$ above $τ (P)$ , while on $G_{1} \cap A_{φ}^{c}$ it is at least $Δ/4$ below $τ (P)$ . Hence $B_{0} (φ) \geq \frac{Δ ^{2}}{16} Π_{0} \otimes K_{raw} Pr (G_{0} \cap A_{φ}), B_{1} (φ) \geq \frac{Δ ^{2}}{16} Π_{1} \otimes K_{raw} Pr (G_{1} \cap A_{φ}^{c}) .$ The concentration bounds from step 4 give $Π_{0} \otimes K_{raw} Pr (G_{0}^{c}) \leq \frac{1}{16}, Π_{1} \otimes K_{raw} Pr (G_{1}^{c}) \leq \frac{1}{16},$ and therefore $B_{0} (φ) \geq \frac{Δ ^{2}}{16} (Q_{0} (A_{φ}) - \frac{1}{16}), B_{1} (φ) \geq \frac{Δ ^{2}}{16} (1 - Q_{1} (A_{φ}) - \frac{1}{16}) .$ Using $max {x_{0}, x_{1}} \geq (x_{0} + x_{1}) /2$ and the total-variation bound, $max {B_{0} (φ), B_{1} (φ)} \geq \frac{Δ ^{2}}{32} (1 - (Q_{1} (A_{φ}) - Q_{0} (A_{φ})) - \frac{1}{16} - \frac{1}{16}) \geq \frac{Δ ^{2}}{32} (1 - \frac{1}{8} - \frac{1}{16} - \frac{1}{16}) = \frac{3 Δ ^{2}}{128} .$ If $θ_{1} \leq θ_{0}$ , the same calculation is applied with the two branches interchanged. Thus, for every measurable raw rule $φ$ , $\frac{3 Δ ^{2}}{128} \leq max {B_{0} (φ), B_{1} (φ)}$ in $\overline{R}_{+}$ .
The loss is bounded by $0 \leq ℓ_{ϵ} (P, z) \leq 4$ because both $clip_{[- 1, 1]} (z)$ and $τ (P)$ lie in $[- 1, 1]$ on the model class, and the loss is $0$ off the model class. The Poisson retention tails used to transfer raw rules to fixed-sample rules are as follows. For any integer $n \geq 1$ and any Poisson mean $λ \geq 32768 n$ , $Pr {Poi (λ) < n} \leq exp (- λ /8) \leq \frac{1}{4096 n} .$ Indeed, with $z_{λ} = λ /8$ , $2 λ z_{λ} = λ /2,$ and ${Poi (λ) < n} \subseteq {λ - Poi (λ) > λ /2}$ ; the Poisson Bernstein lower-tail bound gives the exponential term, and $exp (- z_{λ}) \leq 1/ z_{λ} \leq 1/ (4096 n)$ .

On the event $s_{+} (P) \geq 1/2$ , the labeled Poisson mean satisfies $u s_{+} (P) \geq 32768 n,$ so the labeled retention failure probability is at most $1/ (4096 n)$ . Similarly, $v s_{+} (P) \geq 32768 (n + m),$ and the event of retaining fewer than $m$ auxiliary observations is contained in the event of retaining fewer than $n + m$ observations; hence the auxiliary retention failure probability is at most $\frac{1}{4096 ( n + m )} \leq \frac{1}{4096 n} .$
We next transfer the raw fuzzy lower bound to the fixed experiment by the prefix coupling. Fix any measurable fixed-sample decision rule $ψ$ . Define its transferred raw rule $φ_{ψ}$ as follows. Given an ordered raw output $y_{raw}$ , set $φ_{ψ} (y_{raw}) = {ψ (the first n labeled marks and the first m auxiliary marks in y_{raw}), ζ_{⋆}, if those marks are present, otherwise .$ The rule uses only the first $m$ auxiliary marks, while the raw auxiliary Poisson process has intensity $v s_{+} (P) = 65536 (n + m) s_{+} (P)$ ; the auxiliary tail estimate below therefore bounds the event of having fewer than $m$ auxiliary marks by the larger-threshold event of having fewer than $n + m$ . Here $ζ_{⋆} \in R$ is any fixed real-valued action. For a law $P$ , let $F_{P} := {the ordered raw draw contains fewer than n labeled marks or fewer than m auxiliary marks} .$ On $F_{P}^{c}$ , the first $n$ labeled marks and first $m$ auxiliary marks have exactly the fixed-sample law $K_{fix} (P)$ , because the ordered raw experiment is generated by independent Poisson samples from the same labeled and auxiliary mark kernels and the fixed experiment keeps the first prescribed counts.

For branch $b$ , the exceptional probability under the branch prior predictive is bounded by integrating the conditional failure probability. Splitting according to the scale event gives $Π_{b} \otimes K_{raw} Pr (F_{P}) \leq Π_{b} {P : s_{+} (P) < 1/2} + \int_{{P : s_{+} (P) \geq 1/2}} [Pr {Poi (u s_{+} (P)) < n} + Pr {Poi (v s_{+} (P)) < m}] d Π_{b} (P) .$ The scale-tail bound from step 4 and the Poisson retention bounds from step 6 therefore yield, for each $b \in {0, 1}$ , $Π_{b} \otimes K_{raw} Pr (F_{P}) \leq \frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n} .$ Since $0 \leq ℓ_{ϵ} \leq 4$ , the Bayes risk of the transferred raw rule satisfies $B_{b} (φ_{ψ}) \leq \int\int ℓ_{ϵ} (P, ψ (y)) d K_{fix} (P) (y) d Π_{b} (P) + 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) \leq P sup \int ℓ_{ϵ} (P, ψ (y)) d K_{fix} (P) (y) + 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) .$ Combining this with the raw fuzzy bound above gives, for every measurable $ψ$ , $\frac{3 Δ ^{2}}{128} \leq P sup \int ℓ_{ϵ} (P, ψ (y)) d K_{fix} (P) (y) + 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) .$ Equivalently, $\frac{3 Δ ^{2}}{128} - 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) \leq P sup \int ℓ_{ϵ} (P, ψ (y)) d K_{fix} (P) (y) .$ Taking the infimum over $ψ$ gives $\frac{3 Δ ^{2}}{128} - 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) \leq R_{dec} .$ The floor inequality from step 3 also gives $α_{floor} / n \leq R_{dec}$ , so $max {\frac{3 Δ ^{2}}{128} - 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}), \frac{α _{floor}}{n}} \leq R_{dec} .$ For the reverse comparison to step 3, start with any measurable estimator $T$ in Definition 11 and use the same function as the fixed-sample decision rule $ψ_{T}$ . On $M_{d, ϵ}$ , clipping cannot increase squared error because $τ (P) \in [- 1, 1]$ , while off the model class the class-gated loss is zero. Hence the worst-case decision risk of $ψ_{T}$ is no larger than the worst-case mean-squared risk of $T$ . Taking the infimum over $T$ gives $R_{dec} \leq R_{ϵ} (n, m, d),$ and consequently $max {\frac{3 Δ ^{2}}{128} - 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}), \frac{α _{floor}}{n}} \leq R_{ϵ} (n, m, d) .$
It remains only to read off the deterministic arithmetic. From the floor bound, $R_{ϵ} (n, m, d) \geq \frac{α _{floor}}{n} > 0.$

First suppose $\frac{1}{n} \leq Δ^{2} .$ Then $4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) \leq 4 (\frac{Δ ^{2}}{16384} + \frac{2 Δ ^{2}}{4096}) = \frac{9 Δ ^{2}}{4096} .$ Therefore $\frac{3 Δ ^{2}}{128} - 4 (\frac{Δ ^{2}}{16384} + \frac{1}{4096 n} + \frac{1}{4096 n}) \geq \frac{87 Δ ^{2}}{4096} \geq \frac{Δ ^{2}}{65536} .$ The first branch of the maximum in the preceding display gives $R_{ϵ} (n, m, d) \geq \frac{Δ ^{2}}{65536} .$

Now suppose $Δ^{2} < \frac{1}{n} .$ Since $α_{floor} = 1/100$ , $\frac{Δ ^{2}}{65536} < \frac{1}{65536 n} \leq \frac{1}{100 n} = \frac{α _{floor}}{n} .$ The second branch of the maximum gives the same conclusion. Combining the two cases, $R_{ϵ} (n, m, d) \geq \frac{Δ ^{2}}{65536} .$

∎

Lemma 20 converts the raw recipe separation into a lower bound for $R_{ϵ} (n, m, d)$ under the fixed annotation sample sizes. The total-variation requirement supplies the testing affinity, and the variance-scale requirement keeps the normalization fluctuations small relative to the squared raw gap.

The Poisson construction is then converted to a general fixed-size annotation experiment. The transfer statement tracks the raw normalization explicitly, so the resulting lower bound applies to the minimax risk in Definition 11.

Lemma 21 [lem:fixed-sample-common-marginal-transfer] (Fixed-sample transfer).

There exist numerical constants $C_{fixed} > 0$ and $c_{risk} > 0$ such that the following holds. For every $0 < ϵ < 1/2$ and every $C_{ϵ} > 0$ , there is a constant $c_{ϵ, C_{ϵ}} > 0$ such that, for every common-marginal calibration at overlap level $ϵ$ with positive calibration constants and dual gap $δ_{ϵ} \leq 1/4$ , every $ρ \in R$ , every $n, m, d, L, k \in N$ , every $B, a \in R$ , and every common-marginal recipe $R$ at these indices, the implication below holds.

Write $G_{R}$ for the recipe’s rare-cell generator measure and $θ_{R, b} (w)$ for its raw target on branch $b$ . Let $Δ = \int θ_{R, 1} (w) d G_{R} (w) - \int θ_{R, 0} (w) d G_{R} (w) .$ Assume:

(Sample and dimension.) $n \geq 1$ and $d \geq 2$ .

(Witness data.) The recipe carries a common-marginal Poisson-prior witness at labeled and auxiliary intensities $u = C_{fixed} n, v = C_{fixed} (n + m) .$ Equivalently, the witness supplies $(u + v) B \leq b_{ϵ} L, B > 0, a = γ_{ϵ} B / L^{2}, k a \leq δ_{ϵ},$ identifies the recipe’s labeled and auxiliary intensities with $u$ and $v$ , and satisfies the corresponding common-marginal bounds at dual gap $δ_{ϵ}$ , variance-bound constant $C_{ϵ}$ , and likelihood-decay parameter $ρ$ .

(Testing distance.) The raw prior-predictive mixture laws $M_{0}$ and $M_{1}$ generated by the two branches of the recipe satisfy $TV (M_{0}, M_{1}) \leq \frac{1}{8} .$

(Concentration scale.) The realized scale obeys $k B a \leq c_{ϵ, C_{ϵ}} Δ^{2} .$

Then the minimax risk $R_{ϵ} (n, m, d)$ from Definition 11 satisfies $c_{risk} Δ^{2} \leq R_{ϵ} (n, m, d) .$

⊢ Lean

Proof of Lemma 21.

Take $C_{fixed} = 65536, c_{risk} = \frac{1}{65536} .$ Fix $0 < ϵ < 1/2$ and a positive variance-bound constant $C_{ϵ}$ , and set $c_{ϵ, C_{ϵ}} := \frac{1}{65536 C _{ϵ}} > 0.$ Let a calibration, a recipe, and all parameters satisfy the hypotheses of the statement. Put $\overset{ˉ}{θ}_{b} := \int θ_{b} (w) d G (w), b \in {0, 1},$ so that the raw-center separation in the statement is $Δ = \overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0} .$ The recipe’s raw prior centers are exactly these two numbers: $\overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0} = \int θ_{1} (w) d G (w) - \int θ_{0} (w) d G (w) .$ The Poisson-prior witness in the hypotheses identifies the recipe’s labeled and auxiliary intensities as $u = 65536 n, v = 65536 (n + m) .$ Separately, the concentration-scale hypothesis, the definition of $c_{ϵ, C_{ϵ}}$ , and $Δ = \overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0}$ give $k B a \leq c_{ϵ, C_{ϵ}} Δ^{2} = \frac{Δ ^{2}}{65536 C _{ϵ}} = \frac{1}{65536 C _{ϵ}} \overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0}^{2} .$ Together with $n \geq 1$ , $d \geq 2$ , and $TV (M_{0}, M_{1}) \leq 1/8$ , these are precisely the hypotheses of Lemma 20, applied with the present calibration, recipe, variance-bound constant $C_{ϵ}$ , likelihood-decay parameter $ρ$ , and intensities $u = 65536 n$ , $v = 65536 (n + m)$ . That result yields, for the minimax risk $R_{ϵ} (n, m, d)$ of Definition 11, $R_{ϵ} (n, m, d) \geq \frac{1}{65536} \overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0}^{2} .$ Since $∣ \overset{ˉ}{θ}_{1} - \overset{ˉ}{θ}_{0} ∣^{2} = Δ^{2}$ and $c_{risk} = 1/65536$ , this is exactly $c_{risk} Δ^{2} \leq R_{ϵ} (n, m, d) .$

∎

Lemma 21 is the bridge from a Poisson fuzzy-hypothesis calculation to the fixed- $n, m$ risk. The testing-distance condition supplies indistinguishability, the concentration-scale condition controls the random raw normalization, and the conclusion converts the raw-center separation $Δ$ into a lower bound for $R_{ϵ} (n, m, d)$ .

A separate one-dimensional ingredient records the unavoidable parametric contribution of the labeled sample. It is used together with the rare-cell construction to cover regimes in which the lower-bound rate is governed by outcome labels.

Lemma 22 [lem:parametric-label-floor] (Parametric label floor).

Let $R_{ϵ} (n, m, d)$ be the minimax risk in Definition 11. There exists a universal constant $c > 0$ such that, whenever

(Overlap level.) $0 < ϵ < 1/2$ ,

(Labeled sample size.) $n \geq 1$ ,

(Auxiliary sample size.) $m$ is a nonnegative integer,

(Alphabet size.) $d \geq 2$ ,

the minimax risk satisfies $R_{ϵ} (n, m, d) \geq \frac{c}{n} .$

⊢ Lean

Proof of Lemma 22.

Set $c = 1/100$ . Fix $0 < ϵ < 1/2$ , $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ . Define $η_{n} := \frac{2}{5 n} .$ Then $η_{n} \geq 0$ , and $η_{n}^{2} = \frac{4}{25 n} .$ Consider two laws $P_{null}$ and $P_{pert}$ in $M_{d, ϵ}$ with uniform covariate distribution on $[d]$ , propensity $e_{x} = 1/2$ in every cell, control regression $1/2$ in every cell, and treated regression respectively $1/2$ and $1/2 + η_{n}$ . Since $n \geq 1$ , $η_{n} \leq 2/5$ , hence $1/2 + η_{n} \leq 1$ . The overlap condition follows from $0 < ϵ < 1/2$ , so both laws belong to $M_{d, ϵ}$ .

For the labeled samples, the parametric total-variation bound applies once $n \cdot \frac{( 1/2 ) η _{n}^{2}}{( 1/2 ) ( 1 - 1/2 )} \leq lo g 2.$ Using the displayed value of $η_{n}^{2}$ , the left side is $8/25$ , and $8/25 \leq lo g 2$ . Therefore $TV (P_{null}^{\otimes n}, P_{pert}^{\otimes n}) \leq \frac{1}{2} .$

The auxiliary treatment-covariate marginals of the two laws are equal: $(P_{null})_{X A} = (P_{pert})_{X A} .$ Indeed, both laws have the same uniform covariate distribution and the same propensity $e_{x} = 1/2$ in every cell; summing over the binary outcome coordinate gives the same mass for each $(x, a)$ . Hence their $m$ -fold auxiliary product laws are identical. The product total-variation inequality gives $TV (P_{null}^{\otimes n} \otimes (P_{null})_{X A}^{\otimes m}, P_{pert}^{\otimes n} \otimes (P_{pert})_{X A}^{\otimes m}) \leq TV (P_{null}^{\otimes n}, P_{pert}^{\otimes n}) + 0 \leq \frac{1}{2} .$

The two average treatment effects are $τ (P_{null}) = 0, τ (P_{pert}) = η_{n} .$ The first identity follows because the two outcome regressions coincide in every cell, and the second because every cell has treated-minus-control contrast $η_{n}$ and the cell masses sum to one. Thus, with $s_{n} := η_{n} /2$ , $2 s_{n} = η_{n} \leq τ (P_{null}) - τ (P_{pert}) .$

We now record the two-point risk consequence used for Definition 11. Let $T \in T_{n, m, d}$ be any estimator, and let $A_{0} := {z : s_{n} \leq ∣ T (z) - τ (P_{null}) ∣}, A_{1} := {z : s_{n} \leq ∣ T (z) - τ (P_{pert}) ∣} .$ The testing bound at separation $2 s_{n}$ , together with the preceding total-variation bound, yields $max {P_{null}^{\otimes n} \otimes (P_{null})_{X A}^{\otimes m} Pr (A_{0}), P_{pert}^{\otimes n} \otimes (P_{pert})_{X A}^{\otimes m} Pr (A_{1})} \geq \frac{1}{4} .$ Moreover, by the elementary inequality $s_{n}^{2} Pr {s_{n} \leq ∣ T - θ ∣} \leq E [(T - θ)^{2}],$ applied under each of the two laws, the larger of the two mean-squared risks is at least $s_{n}^{2} /4$ . Taking the supremum over $P \in M_{d, ϵ}$ and then the infimum over $T \in T_{n, m, d}$ gives $\frac{s _{n}^{2}}{4} \leq R_{ϵ} (n, m, d) .$

Finally, $\frac{s _{n}^{2}}{4} = \frac{( η _{n} /2 ) ^{2}}{4} = \frac{1}{100 n} .$ Consequently $R_{ϵ} (n, m, d) \geq 1/ (100 n)$ , which proves the claim with the universal constant $c = 1/100$ .

∎

Lemma 22 fixes the labeled-outcome scale uniformly over auxiliary sample sizes and alphabet sizes. This lower bound matches the $1/ n$ component in Definition 12, so the remaining common-marginal construction can focus on the alphabet-dependent contribution.

One final finite-product identity records how independent coordinates can be embedded into a larger product experiment while adding a distinguished coordinate and deterministic padding. This bookkeeping is useful when the rare-cell construction is placed inside a larger finite alphabet with reservoir coordinates.

Lemma 23 [lem:finite-product-padding-identity] (Finite product padding identity).

Let $I$ and $J$ be finite index sets, and let $Y$ be a measurable space whose singletons are measurable. Suppose the following conditions hold:

(Embedding.) $f : I \to J$ is injective.

(Distinguished coordinate.) $r \in J$ satisfies $r \neq = f (i)$ for every $i \in I$ .

(Coordinate laws.) $μ_{i}$ , $i \in I$ , and $ν$ are probability measures on $Y$ , and $y_{0} \in Y$ .

Define $A : Y^{I} \times Y \to Y^{J}$ by $A (z, y) (j) = ⎩ ⎨ ⎧ y, z_{i}, y_{0}, j = r, j = f (i) for the unique i \in I, j \in J ∖ ({r} \cup f (I)) .$ Define the coordinate laws $H_{j}$ , $j \in J$ , by $H_{j} = ⎩ ⎨ ⎧ ν, μ_{i}, δ_{y_{0}}, j = r, j = f (i) for the unique i \in I, j \in J ∖ ({r} \cup f (I)) .$ Then $A_{#} ((i \in I ⨂ μ_{i}) \otimes ν) = j \in J ⨂ H_{j} .$

⊢ Lean

Proof of Lemma 23.

Write $Λ_{in} := (i \in I ⨂ μ_{i}) \otimes ν on Y^{I} \times Y, Λ_{out} := j \in J ⨂ H_{j} on Y^{J} .$ The map $A$ is measurable. Indeed, for each $j \in J$ , the $j$ -th coordinate of $A (z, y)$ is the second projection if $j = r$ , is the $i$ -th coordinate projection from $z$ if $j = f (i)$ , and is the constant $y_{0}$ on the remaining coordinates. The condition $r \neq = f (i)$ separates the distinguished coordinate from the embedded ones, and injectivity of $f$ makes the embedded coordinate $i$ unique. Since $I$ and $J$ are finite, coordinatewise measurability gives measurability of $A$ . It is therefore enough, by uniqueness of finite product measures on measurable rectangles, to prove that for every family of measurable sets $(G_{j})_{j \in J}, G_{j} \in Y,$ one has $Λ_{in} (A^{- 1} {w \in Y^{J} : \forall j \in J, w_{j} \in G_{j}}) = j \in J \prod H_{j} (G_{j}) .$
First suppose that there is an unused coordinate $j_{\circ} \in J$ such that $j_{\circ} \neq = r, j_{\circ} \neq = f (i) for every i \in I, y_{0} \in / G_{j_{\circ}} .$ For every $(z, y) \in Y^{I} \times Y$ , the definition of $A$ gives $A (z, y)_{j_{\circ}} = y_{0} .$ Thus $A^{- 1} {w : \forall j, w_{j} \in G_{j}} = \emptyset,$ so the left-hand side is $0$ . On the right-hand side, the same coordinate has $H_{j_{\circ}} = δ_{y_{0}}, H_{j_{\circ}} (G_{j_{\circ}}) = δ_{y_{0}} (G_{j_{\circ}}) = 0,$ because $y_{0} \in / G_{j_{\circ}}$ . Hence $j \in J \prod H_{j} (G_{j}) = 0,$ and the required rectangle identity holds in this case.
It remains to consider the complementary case: $\forall j \in J, j \neq = r and (\forall i \in I, j \neq = f (i)) ⟹ y_{0} \in G_{j} .$ Then the preimage of the rectangle is exactly $A^{- 1} {w : \forall j, w_{j} \in G_{j}} = {z \in Y^{I} : \forall i \in I, z_{i} \in G_{f (i)}} \times G_{r} .$ For the forward inclusion, membership in the rectangle gives the $r$ -coordinate condition $y \in G_{r}$ , and for each $i \in I$ it gives $z_{i} = A (z, y)_{f (i)} \in G_{f (i)},$ using $r \neq = f (i)$ and injectivity of $f$ . Conversely, if $z_{i} \in G_{f (i)}$ for all $i$ and $y \in G_{r}$ , then each coordinate $j \in J$ is checked by the three alternatives in the definition of $A$ : the coordinate $r$ gives $y$ , a coordinate $f (i)$ gives $z_{i}$ , and an unused coordinate gives $y_{0} \in G_{j}$ by the displayed complementary condition.
Using the preceding preimage identity and the finite product formula, $Λ_{in} (A^{- 1} {w : \forall j, w_{j} \in G_{j}}) = (i \in I ⨂ μ_{i}) {z : \forall i, z_{i} \in G_{f (i)}} ν (G_{r}) = (i \in I \prod μ_{i} (G_{f (i)})) ν (G_{r}) .$ The product over $J$ has the same value. The distinguished coordinate contributes $H_{r} (G_{r}) = ν (G_{r}),$ each embedded coordinate contributes, for $i \in I$ , $H_{f (i)} (G_{f (i)}) = μ_{i} (G_{f (i)}),$ again by $r \neq = f (i)$ and injectivity of $f$ , and every unused coordinate $j$ contributes $H_{j} (G_{j}) = δ_{y_{0}} (G_{j}) = 1$ by the complementary condition. Since $f$ is injective, the embedded coordinates are indexed without repetition, and therefore $j \in J \prod H_{j} (G_{j}) = ν (G_{r}) i \in I \prod μ_{i} (G_{f (i)}) .$ This agrees with the preceding expression for the pushforward on every measurable rectangle, so $A_{#} Λ_{in} = Λ_{out},$ which is the desired identity.

∎

Lemma 23 formalizes the product-measure preservation behind this padding step. Injectivity keeps the rare coordinates distinct, the distinguished coordinate carries the added law, and the remaining coordinates are fixed at point masses, so the assembled law is the intended product over the larger index set.

The final calibration statements choose constants so that the rare-cell prior works uniformly across the intensity range needed in Theorem 3. The first gives a ready-to-use recipe with explicit counting, variance, and total-variation controls; the second packages these ingredients into the common-marginal certificate family used by the main converse.

Lemma 24 [lem:common-marginal-uniform-intensity] (Uniform intensity common marginal).

For every overlap level $0 < ϵ < 1/2$ , there are positive constants $r_{ϵ}, D_{ϵ}, b_{0, ϵ}, b_{ϵ}, γ_{ϵ}, q_{ϵ}, c_{ϵ}, δ_{ϵ}, C_{ϵ}$ and a number $0 < ρ_{ϵ} < 1$ , with $c_{ϵ} \leq 1/4$ , such that the following calibration facts hold:

(Bandwidth slack.) $131072 b_{0, ϵ} \leq b_{ϵ}$ .

(Degree floor.) $2 \leq D_{ϵ}$ .

(Rare-cell scale.) $q_{ϵ} γ_{ϵ} b_{0, ϵ} \leq c_{ϵ}, q_{ϵ} = \frac{c _{ϵ}}{2 γ _{ϵ} b _{0, ϵ}}, 4 \leq q_{ϵ} .$

(Dimension scale.) $δ_{ϵ} r_{ϵ} \leq \frac{1}{65536}, δ_{ϵ} \leq \frac{c _{ϵ}^{4}}{16}, δ_{ϵ} \leq (\frac{c _{ϵ} γ _{ϵ} b _{0, ϵ}}{2 ( D _{ϵ} + 1 )})^{2} .$

(Variance and counting calibration.) $8 (D_{ϵ} + 1) b_{0, ϵ} \leq \frac{c _{ϵ}^{3}}{65536 C _{ϵ}},$ $\frac{4 ( D _{ϵ} + 1 )}{( 1/ ( 65536 C _{ϵ} )) c _{ϵ}^{2} γ _{ϵ}} \leq q_{ϵ},$ and ${\frac{4 ( D _{ϵ} + 1 ) ^{2}}{( 1/ ( 65536 C _{ϵ} )) c _{ϵ}^{2} γ _{ϵ}}}^{2} \leq r_{ϵ} .$

(Mixture decay.) For every integer $n \geq 1$ , $C_{ϵ} (65536 n) c_{ϵ} ρ_{ϵ}^{⌈ D_{ϵ} l o g (e n)⌉} \leq \frac{1}{8} .$

For every $n, m, L, d, k$ and every $u, v, B, a$ , if

$n \geq 1$ , $L \geq 2$ , and $d \geq 2$ ;

$u \geq 0$ , $v \geq 0$ , and $u + v > 0$ ;

$B > 0$ , $(u + v) B \leq b_{ϵ} L$ , and $a = \frac{γ _{ϵ} B}{L ^{2}};$

$k a \leq c_{ϵ}$ and $k + 1 \leq d$ ,

then there is a common-marginal rare-cell recipe with $k$ rare cells at scale $(a, B, L)$ whose labeled and auxiliary Poisson intensities are $u$ and $v$ . Its null and alternative priors are the two branch priors of the recipe, and they satisfy:

the recipe is in the finite-atom construction regime, and both priors are supported on $M_{d, ϵ}$ ;

the induced random treatment-covariate tables $P_{X A}$ have exactly the same law under the two priors;

the raw total mass has mean one;

the raw target centers obey $c_{ϵ} k a \leq ∣ θ_{1} - θ_{0} ∣,$ where $θ_{1}$ and $θ_{0}$ are the alternative and null raw prior centers;

the raw total-mass variance is at most $C_{ϵ} k B a,$ and the larger of the two raw target variances is at most $C_{ϵ} k B a;$

in the raw Poisson experiment with labeled intensity $u$ and auxiliary intensity $v$ , the two prior-predictive mixture laws have total variation distance at most $C_{ϵ} u k a ρ_{ϵ}^{L} .$

⊢ Lean

Proof of Lemma 24.

Fix $0 < ϵ < 1/2$ .

By Lemma 17, with $κ_{ϵ} := \frac{1 - 2 ϵ}{ϵ} > 0,$ there are $γ_{ϵ} > 0$ and $\overset{c}{ˉ}_{ϵ} > 0$ such that every $L \geq 2$ and $B > 0$ , with $a = γ_{ϵ} B / L^{2}$ , admit a finite signed certificate on $[a / κ_{ϵ}, B]$ , annihilating monomials through degree $L$ , whose signed integral of $p / (p + a)$ is at least $\overset{c}{ˉ}_{ϵ}$ . The one-coordinate marked-Poisson moment-matching estimate supplies constants $\overset{ˉ}{b}_{ϵ} > 0$ , $\overset{ˉ}{C}_{ϵ} > 0$ , and $ρ_{ϵ} \in (0, 1)$ with the following uniform property for the complete normalized certificate object that generates the coordinate mixtures. For every finite index set $I$ and every normalized finite signed moment certificate $C = {(p_{ι}, ω_{ι}) : ι \in I}$ of order $L$ , meaning that its signed measure $\sum_{ι \in I} ω_{ι} δ_{p_{ι}}$ has normalized variation $\sum_{ι \in I} ∣ ω_{ι} ∣ = 1$ and the certificate data include the moment equations through degree $L$ , if its variation measure is supported on $[a / κ_{ϵ}, B]$ and the support scale satisfies $(u + v) B \leq \overset{ˉ}{b}_{ϵ} L$ , then the two branch marked-Poisson predictive laws $Q_{0}, Q_{1}$ associated with that same certificate obey $TV (Q_{0}, Q_{1}) \leq \overset{ˉ}{C}_{ϵ} u a ρ_{ϵ}^{L} .$ Set $C_{ϵ} := max {\overset{ˉ}{C}_{ϵ}, 1}, c_{ϵ} := min {\overset{c}{ˉ}_{ϵ}, 1/4}, b_{ϵ} := \overset{ˉ}{b}_{ϵ} .$ Let $D_{ϵ} := max {2, \frac{max { 1 , lo g ( 8 C _{ϵ} 65536 c _{ϵ} ) + 2 }}{- lo g ρ _{ϵ}}}, s_{ϵ} := \frac{1}{65536 C _{ϵ}} .$ Define $b_{0, ϵ} := min {\frac{b _{ϵ}}{131072}, \frac{c _{ϵ}}{8 γ _{ϵ}}, \frac{s _{ϵ} c _{ϵ}^{3}}{8 ( D _{ϵ} + 1 )}},$ $q_{ϵ} := \frac{c _{ϵ}}{2 γ _{ϵ} b _{0, ϵ}}, r_{ϵ} := (max {1, \frac{4 ( D _{ϵ} + 1 ) ^{2}}{s _{ϵ} c _{ϵ}^{2} γ _{ϵ}}})^{2},$ and $δ_{ϵ} := min {\frac{c _{ϵ}^{4}}{16}, (\frac{c _{ϵ} γ _{ϵ} b _{0, ϵ}}{2 ( D _{ϵ} + 1 )})^{2}, \frac{1}{65536 r _{ϵ}}} .$ All these constants are positive, $0 < ρ_{ϵ} < 1$ , and $c_{ϵ} \leq 1/4$ . The first five displayed calibrations follow immediately from the three minima and the definitions: $131072 b_{0, ϵ} \leq b_{ϵ}$ , $D_{ϵ} \geq 2$ , $q_{ϵ} γ_{ϵ} b_{0, ϵ} = \frac{c _{ϵ}}{2} \leq c_{ϵ}, q_{ϵ} = \frac{c _{ϵ}}{2 γ _{ϵ} b _{0, ϵ}}, q_{ϵ} \geq 4,$ and $δ_{ϵ} r_{ϵ} \leq \frac{1}{65536}, δ_{ϵ} \leq \frac{c _{ϵ}^{4}}{16}, δ_{ϵ} \leq (\frac{c _{ϵ} γ _{ϵ} b _{0, ϵ}}{2 ( D _{ϵ} + 1 )})^{2} .$ Moreover, $8 (D_{ϵ} + 1) b_{0, ϵ} \leq \frac{c _{ϵ}^{3}}{65536 C _{ϵ}},$ and the same bound on $b_{0, ϵ}$ gives $\frac{4 ( D _{ϵ} + 1 )}{( 1/ ( 65536 C _{ϵ} )) c _{ϵ}^{2} γ _{ϵ}} \leq q_{ϵ} .$ The definition of $r_{ϵ}$ gives ${\frac{4 ( D _{ϵ} + 1 ) ^{2}}{( 1/ ( 65536 C _{ϵ} )) c _{ϵ}^{2} γ _{ϵ}}}^{2} \leq r_{ϵ} .$ Finally, put $λ_{ϵ} := - lo g ρ_{ϵ} > 0$ and $M_{ϵ} := 8 C_{ϵ} 65536 c_{ϵ}$ . Since $λ_{ϵ} D_{ϵ} \geq max {1, lo g M_{ϵ} + 2},$ for every $n \geq 1$ , $ρ_{ϵ}^{⌈ D_{ϵ} l o g (e n)⌉} \leq \frac{1}{M _{ϵ} n} .$ Thus $C_{ϵ} (65536 n) c_{ϵ} ρ_{ϵ}^{⌈ D_{ϵ} l o g (e n)⌉} \leq \frac{C _{ϵ} 65536 n c _{ϵ}}{8 C _{ϵ} 65536 c _{ϵ} n} = \frac{1}{8} .$
Now fix $n, m, L, d, k, u, v, B, a$ satisfying the hypotheses in the statement. By the bandwidth slack, $(u + v) B \leq b_{ϵ} L = \overset{ˉ}{b}_{ϵ} L,$ and $a = γ_{ϵ} B / L^{2} > 0$ . Applying Lemma 17 at this $L, B$ gives distinct points $p_{j} \in [a / κ_{ϵ}, B]$ and weights $ω_{j}$ with $j \sum ∣ ω_{j} ∣ = 1, j \sum ω_{j} p_{j}^{ℓ} = 0 (0 \leq ℓ \leq L), c_{ϵ} \leq j \sum ω_{j} \frac{p _{j}}{p _{j} + a} .$ Let $σ := j \sum ω_{j} δ_{p_{j}}, ∣ σ ∣ := j \sum ∣ ω_{j} ∣ δ_{p_{j}},$ and define the polar sign on all $p \in R$ by $h (p) := j \sum 1_{{p = p_{j}}} sgn (ω_{j}) .$ Then $∣ σ ∣$ is a probability measure, $∣ h (p) ∣ \leq 1$ , and $σ = h ∣ σ ∣, c_{ϵ} \leq \int \frac{p}{p + a} d σ (p) .$ Also, every certificate node satisfies $p_{j} \geq a / κ_{ϵ} > 0$ , so $0$ is not one of the nodes. Hence the displayed definition gives $h (0) = 0$ . Define the zero-inflated scalar law $ν (E) := \int_{E} \frac{a}{p + a} d ∣ σ ∣ (p) + (1 - \int \frac{a}{p + a} d ∣ σ ∣ (p)) 1_{E} (0) .$ Since $p \geq a / κ_{ϵ} > 0$ on the support of $∣ σ ∣$ , $0 \leq \frac{a}{p + a} \leq 1 ∣ σ ∣ -a.e.$ and $∣ σ ∣$ has total mass one. Hence $0 \leq \int \frac{a}{p + a} d ∣ σ ∣ (p) \leq 1,$ so the coefficient of $δ_{0}$ is nonnegative and $ν$ is a probability measure supported on ${0} \cup [a / κ_{ϵ}, B]$ . Its mean $\overset{p}{ˉ} := \int p d ν (p)$ is computed directly from the definition of $ν$ : the atom at zero contributes nothing, and therefore $\overset{p}{ˉ} = \int p \frac{a}{p + a} d ∣ σ ∣ (p) = a \int \frac{p}{p + a} d ∣ σ ∣ (p) .$ Because $0 \leq p / (p + a) \leq 1$ on the support, this gives $0 \leq \overset{p}{ˉ} \leq a .$ Let $G := ν^{\otimes k}$ . For every vector $ξ \in R^{k}$ , set $ξ_{i} := {ξ_{i}, 0, ξ_{i} \in {p_{j}}_{j}, ξ_{i} \in / {p_{j}}_{j} .$ Under $G$ , $ξ = ξ$ almost surely, and the range of $ξ$ is finite. Choose one reservoir cell $x_{⋆} \in [d]$ and distinct rare cells $x_{1}, \dots, x_{k} \in [d] ∖ {x_{⋆}}$ , possible because $k + 1 \leq d$ . Put $p_{⋆} := 1 - k \overset{p}{ˉ} .$ Since $k a \leq c_{ϵ} \leq 1/4$ , one has $p_{⋆} \geq 3/4 > 0$ . Define the raw cell masses and raw total mass by $Ω_{ξ} (x) := p_{⋆} 1_{{x = x_{⋆}}} + i = 1 \sum k ξ_{i} 1_{{x = x_{i}}}, S (ξ) := p_{⋆} + i = 1 \sum k ξ_{i} .$ Then $S (ξ) > 0$ and $S (ξ) = x = 1 \sum d Ω_{ξ} (x) .$
Define the propensity field on all cells by $e_{ξ} (x) := ⎩ ⎨ ⎧ 1/2, 1/2, ϵ \frac{\sum _{i} ξ _{i} 1 _{{x = x_{i}}} + a}{\sum _{i} ξ _{i} 1 _{{x = x_{i}}}}, x = x_{⋆}, x \neq = x_{⋆} and \sum_{i} ξ_{i} 1_{{x = x_{i}}} = 0, otherwise .$ If the last line applies, the denominator is some nonzero $ξ_{i} \in [a / κ_{ϵ}, B]$ , so $ϵ \leq ϵ (1 + \frac{a}{ξ _{i}}) \leq ϵ (1 + κ_{ϵ}) = 1 - ϵ .$ The other cases give $e_{ξ} (x) = 1/2 \in [ϵ, 1 - ϵ]$ . For branches $b \in {0, 1}$ , define control outcome means to be zero and treated outcome means by $μ_{1, ξ}^{(0)} (x_{i}) := \frac{1 - h ( ξ _{i} )}{2}, μ_{1, ξ}^{(1)} (x_{i}) := \frac{1 + h ( ξ _{i} )}{2},$ with $μ_{1, ξ}^{(b)} (x_{⋆}) = 0$ and $μ_{1, ξ}^{(b)} (x) = 0$ on unused cells. Since $∣ h ∣ \leq 1$ , all outcome means lie in $[0, 1]$ . Let $P_{b} (ξ)$ be the law on $[d] \times {0, 1}^{2}$ with cell mass $p_{ξ} (x) := \frac{Ω _{ξ} ( x )}{S ( ξ )},$ propensity $e_{ξ} (x)$ , zero control outcome mean, and treated outcome mean $μ_{1, ξ}^{(b)} (x)$ . This is a probability law because each raw mass is nonnegative and $x = 1 \sum d p_{ξ} (x) = \frac{\sum _{x} Ω _{ξ} ( x )}{S ( ξ )} = 1.$ For the reservoir cell, the raw mass is $p_{⋆} > 0$ and the propensity is $1/2$ , which lies in $[ϵ, 1 - ϵ]$ . For an unused cell, the raw mass is zero, so the occupied-cell overlap condition has no constraint; the totalized propensity there is $1/2$ . For a rare cell $x_{i}$ , if $ξ_{i} = 0$ then its raw mass is zero. If $ξ_{i} \neq = 0$ , then $ξ_{i} \in [a / κ_{ϵ}, B]$ , and the calculation above gives $ϵ \leq e_{ξ} (x_{i}) = ϵ (1 + \frac{a}{ξ _{i}}) \leq 1 - ϵ .$ The control outcome means are zero, and the treated outcome means are either zero or of the form $(1 \pm h (ξ_{i})) /2$ , hence lie in $[0, 1]$ . Thus every occupied cell satisfies the overlap bounds and all outcome means are legal binary means, so $P_{b} (ξ) \in M_{d, ϵ}$ by Definition 6. Define $Π_{b} := (P_{b})_{#} G, b \in {0, 1} .$ These are the two branch priors of the recipe, and the finite range of $ξ$ places the recipe in the finite-atom construction regime. The labeled and auxiliary intensities stored in the recipe are, by construction, $u$ and $v$ . Since $p_{ξ} (x)$ and $e_{ξ} (x)$ are independent of $b$ , $(P \mapsto P_{X A})_{#} Π_{0} = (P \mapsto P_{X A})_{#} Π_{1} .$
The raw total mass has mean one: $\int S (ξ) d G (ξ) = p_{⋆} + i = 1 \sum k \int ξ_{i} d G (ξ) = 1 - k \overset{p}{ˉ} + k \overset{p}{ˉ} = 1.$ For branch $b$ , the raw target is $T_{b} (ξ) := i = 1 \sum k ξ_{i} \frac{1 + ( - 1 ) ^{1 - b} h ( ξ _{i} )}{2} .$ It satisfies the raw version of Definition 8. Indeed, the control outcome means are zero, so $τ (P_{b} (ξ)) = x = 1 \sum d \frac{Ω _{ξ} ( x )}{S ( ξ )} μ_{1, ξ}^{(b)} (x) .$ Multiplying by $S (ξ)$ cancels the normalization. The reservoir and unused cells contribute zero, and the rare-cell terms give $S (ξ) τ (P_{b} (ξ)) = i = 1 \sum k ξ_{i} \frac{1 + ( - 1 ) ^{1 - b} h ( ξ _{i} )}{2} = T_{b} (ξ) .$ Moreover, $\int [p \frac{1 + h ( p )}{2} - p \frac{1 - h ( p )}{2}] d ν (p) = \int p h (p) d ν (p) .$ The atom of $ν$ at zero contributes zero. On the nonzero support, $d ν = (a / (p + a)) d ∣ σ ∣$ and $d σ = h d ∣ σ ∣$ , so $\int p h (p) d ν (p) = a \int \frac{p}{p + a} d σ (p) .$ Therefore the raw prior centers $θ_{b} := \int T_{b} d G$ obey $θ_{1} - θ_{0} = k a \int \frac{p}{p + a} d σ (p) .$ The alternation gap gives $c_{ϵ} \leq \int \frac{p}{p + a} d σ (p),$ so the displayed difference is nonnegative and at least $c_{ϵ} k a$ . Hence $c_{ϵ} k a \leq ∣ θ_{1} - θ_{0} ∣.$
The support and mean bounds for $ν$ give $Var_{G} (S) = Var_{G} (i = 1 \sum k ξ_{i}) \leq i = 1 \sum k E_{ν} [p^{2}] \leq k B E_{ν} [p] \leq k B a \leq C_{ϵ} k B a .$ For each branch define $g_{b} (p) := p \frac{1 + ( - 1 ) ^{1 - b} h ( p )}{2} .$ Then $0 \leq g_{b} (p) \leq p \leq B$ on the support of $ν$ , and $T_{b} (ξ) = i = 1 \sum k g_{b} (ξ_{i}) almost surely under G .$ Consequently, $Var_{G} (T_{b}) \leq i = 1 \sum k E_{ν} [g_{b} (p)^{2}] \leq k B E_{ν} [g_{b} (p)] \leq k B E_{ν} [p] \leq k B a \leq C_{ϵ} k B a .$ Taking the maximum over $b = 0, 1$ gives the asserted raw target-variance bound. The same bounded-support estimates also give the required square integrability of $S$ and of both $T_{b}$ .
It remains to bound the prior-predictive distance in the raw Poisson experiment. For a single rare cell, let $Z_{rare}$ be the finite product space consisting of the two labeled treated-outcome counts, the auxiliary treated count, and the labeled and auxiliary aggregate control counts, and let $z_{\emptyset}$ be its all-zero element. For $b \in {0, 1}$ and $p \in R$ , define the raw rare-cell kernel by $Q_{b, p}^{raw} := {δ_{z_{\emptyset}}, S_{α_{u, v}} M_{b, p}, p = 0, p \neq = 0, α_{u, v} := \frac{u}{u + v} \in [0, 1],$ where $M_{b, p}$ is the branch- $b$ marked-Poisson law associated with the certificate at latent mass $p$ , and $S_{α_{u, v}}$ is the fixed splitting kernel that separates the marked treated counts into labeled and auxiliary pieces and carries along the corresponding control-count components. Let $Q_{b} := \int Q_{b, p}^{raw} d ν (p), Q_{b} := \int S_{α_{u, v}} M_{b, p} d ν (p) .$ For every measurable event $E \subseteq Z_{rare}$ and every $p \in R$ , $Q_{1, p}^{raw} (E) - Q_{0, p}^{raw} (E) = (S_{α_{u, v}} M_{1, p}) (E) - (S_{α_{u, v}} M_{0, p}) (E) .$ Indeed, this is the definition when $p \neq = 0$ . When $p = 0$ , the left side is zero because both raw kernels are $δ_{z_{\emptyset}}$ . The right side is also zero: as shown above, $0$ is not a certificate node, so $h (0) = 0$ , and the two branch marked-Poisson laws agree at the zero fiber before the fixed splitting is applied. Integrating the displayed identity with respect to $ν$ gives, for every measurable $E$ , $(Q_{1} - Q_{0}) (E) = (Q_{1} - Q_{0}) (E) .$ Thus the signed finite measures $Q_{1} - Q_{0}$ and $Q_{1} - Q_{0}$ are equal, and their total variations are equal. Hence $TV (Q_{0}, Q_{1}) = TV (Q_{0}, Q_{1}) .$ Data processing under the splitting kernel and the one-coordinate marked-Poisson estimate give $TV (Q_{0}, Q_{1}) \leq \overset{ˉ}{C}_{ϵ} u a ρ_{ϵ}^{L} .$ The certificate used here is precisely the finite-node certificate constructed above: its node set is finite, its signed weights have total variation one, all nodes lie in $[a / κ_{ϵ}, B]$ , and its moments through degree $L$ vanish. Thus the normalized finite-certificate and support hypotheses of the one-coordinate estimate are all discharged by the displayed construction, rather than by an additional regularity convention. Independence of the $k$ rare coordinates yields $TV (Q_{0}^{\otimes k}, Q_{1}^{\otimes k}) \leq k TV (Q_{0}, Q_{1}) \leq \overset{ˉ}{C}_{ϵ} u k a ρ_{ϵ}^{L} .$ It remains to identify this product experiment with the raw Poisson experiment generated by the constructed laws. Let $R_{b} (ξ)$ be the conditional raw Poisson law generated by $P_{b} (ξ)$ at raw scale $S (ξ)$ . Let $Y_{cell} := {y : {0, 1} \to (N \times N) \times N}$ be the single-cell raw-count space; $y (α) = ((y_{α}^{s}, y_{α}^{f}), y_{α}^{aux})$ records labeled successes, labeled failures, and auxiliary counts in arm $α$ . Let $z_{cell, 0} \in Y_{cell}$ be the all-zero cell observation, and let $r : Z_{rare} \to Y_{cell}$ send the five rare-cell counts to the two arm-indexed cell observations: in the treated arm it records the treated success count, treated failure count, and auxiliary treated count, while in the control arm it records zero successes, the labeled control count, and the auxiliary control count. The common reservoir law is $Q_{⋆} := α \in {0, 1} ⨂ {Poi (0) \otimes Poi (\frac{u p _{⋆}}{2}) \otimes Poi (\frac{v p _{⋆}}{2})} .$ For fixed $ξ$ , define rare-coordinate cell laws by $H_{b, ξ, i} := (Q_{b, ξ_{i}}^{raw})_{#} r, i = 1, \dots, k,$ and define a zero law $H_{0} := δ_{z_{cell, 0}}$ . For $y = (y_{1}, \dots, y_{k}) \in Y_{cell}^{k}$ and $y_{⋆} \in Y_{cell}$ , define the deterministic assembly map by $[A (y, y_{⋆})] (x, α) := ⎩ ⎨ ⎧ y_{⋆} (α), y_{i} (α), z_{cell, 0} (α), x = x_{⋆}, x = x_{i} for the unique i, x \in / {x_{⋆}, x_{1}, \dots, x_{k}} .$ The middle line is unambiguous because the rare cells are distinct and none is the reservoir. By Lemma 23, applied with the embedding $i \mapsto x_{i}$ , distinguished coordinate $x_{⋆}$ , coordinate laws $H_{b, ξ, i}$ , reservoir law $Q_{⋆}$ , and filler $z_{cell, 0}$ , $A_{#} ((i = 1 ⨂ k H_{b, ξ, i}) \otimes Q_{⋆}) = x = 1 ⨂ d J_{b, ξ, x},$ where $J_{b, ξ, x} := ⎩ ⎨ ⎧ Q_{⋆}, H_{b, ξ, i}, H_{0}, x = x_{⋆}, x = x_{i} for the unique i, x \in / {x_{⋆}, x_{1}, \dots, x_{k}} .$

Equivalently, if $A_{rare}$ denotes the map $A_{rare} (z, z_{⋆}) := A ((r (z_{1}), \dots, r (z_{k})), z_{⋆}),$ then $(A_{rare})_{#} ((i = 1 ⨂ k Q_{b, ξ_{i}}^{raw}) \otimes Q_{⋆}) = x = 1 ⨂ d J_{b, ξ, x},$ because pushforward by $r$ in each rare coordinate turns $⨂_{i} Q_{b, ξ_{i}}^{raw}$ into $⨂_{i} H_{b, ξ, i}$ . More explicitly, coordinatewise pushforward gives $(i = 1 ⨂ k Q_{b, ξ_{i}}^{raw})_{#} [(z_{i})_{i} \mapsto (r (z_{i}))_{i}] = i = 1 ⨂ k H_{b, ξ, i} .$ Taking the product with $Q_{⋆}$ , then composing this coordinate map with $A$ , gives $A_{rare}$ . Functoriality of pushforward therefore turns the left side of the padding identity into the asserted assembled rare-coordinate law.

We now compare these coordinate laws with the conditional raw Poisson law cell by cell. At the reservoir, raw scale multiplication cancels the normalization: $S (ξ) p_{ξ} (x_{⋆}) = p_{⋆}, e_{ξ} (x_{⋆}) = \frac{1}{2}, μ_{1, ξ}^{(b)} (x_{⋆}) = μ_{0, ξ}^{(b)} (x_{⋆}) = 0.$ Thus each arm has raw treatment-covariate mass $p_{⋆} /2$ , zero labeled-success rate, labeled-failure rate $u p_{⋆} /2$ , and auxiliary rate $v p_{⋆} /2$ . Therefore the reservoir coordinate law of $R_{b} (ξ)$ is $J_{b, ξ, x_{⋆}} = Q_{⋆} .$ At a rare cell $x_{i}$ , put $p = ξ_{i}$ . If $p = 0$ , then $S (ξ) p_{ξ} (x_{i}) = 0,$ so all labeled and auxiliary Poisson rates at this cell are zero. The conditional raw Poisson coordinate law is $δ_{z_{cell, 0}}$ . The raw rare-cell kernel is also $δ_{z_{\emptyset}}$ , and $r (z_{\emptyset}) = z_{cell, 0}$ , hence $(Q_{b, 0}^{raw})_{#} r = δ_{z_{cell, 0}} = J_{b, ξ, x_{i}} .$ If $p > 0$ , define $s_{tr} (p) := ϵ (p + a), s_{ct} (p) := p - s_{tr} (p), η_{1} := 1, η_{0} := - 1.$ The construction gives $S (ξ) p_{ξ} (x_{i}) = p, e_{ξ} (x_{i}) = \frac{s _{tr} ( p )}{p}, μ_{1, ξ}^{(b)} (x_{i}) = \frac{1 + η _{b} h ( p )}{2}, μ_{0, ξ}^{(b)} (x_{i}) = 0.$ Consequently the raw treatment-covariate masses in the treated and control arms are $p e_{ξ} (x_{i}) = s_{tr} (p), p {1 - e_{ξ} (x_{i})} = s_{ct} (p) .$ The conditional raw Poisson coordinate law at $x_{i}$ is therefore the product law on $Y_{cell}$ with treated-arm rates $\frac{u s _{tr} ( p ) ( 1 + η _{b} h ( p ))}{2}, \frac{u s _{tr} ( p ) ( 1 - η _{b} h ( p ))}{2}, v s_{tr} (p)$ for labeled successes, labeled failures, and auxiliary counts, and with control-arm rates $0, u s_{ct} (p), v s_{ct} (p)$ for labeled successes, labeled failures, and auxiliary counts. The split marked rare-cell law $Q_{b, p}^{raw}$ is the fivefold product law whose coordinates have exactly the nonzero rates just displayed: treated success, treated failure, auxiliary treated, labeled control, and auxiliary control. Applying $r$ places the first three counts in the treated arm and the last two in the control arm, with zero control successes. Product Poisson laws are determined by their coordinate rates, so $(Q_{b, p}^{raw})_{#} r = J_{b, ξ, x_{i}} (p = ξ_{i} > 0) .$ Combining the two rare-cell cases gives $(Q_{b, ξ_{i}}^{raw})_{#} r = J_{b, ξ, x_{i}} (1 \leq i \leq k) .$ At an unused cell the raw mass is zero, so all rates are zero and its coordinate law is $J_{b, ξ, x} = H_{0} .$ Thus the coordinate laws of the conditional raw Poisson experiment are precisely the padded product coordinate laws: reservoir law at $x_{⋆}$ , the pushed-forward rare laws at the rare cells, and the zero law elsewhere. Since conditional raw Poisson counts are independent across cells, $R_{b} (ξ) = x = 1 ⨂ d J_{b, ξ, x} .$ The preceding coordinate identification, together with the padding identity $A_{#} ((i = 1 ⨂ k H_{b, ξ, i}) \otimes Q_{⋆}) = x = 1 ⨂ d J_{b, ξ, x},$ which follows from Lemma 23, yields the pointwise fiber identity $R_{b} (ξ) = (A_{rare})_{#} ((i = 1 ⨂ k Q_{b, ξ_{i}}^{raw}) \otimes Q_{⋆}) .$ This equality uses more than the abstract padding identity: the reservoir calculation fixes the distinguished coordinate, the two rare-cell cases identify every pushed-forward rare law, and the unused-cell calculation supplies the Dirac fillers. Independence of the raw Poisson counts then identifies the conditional experiment with the resulting finite product. Hence every coordinate law on the two sides is the same before finite-product uniqueness is invoked. The identity holds for every $ξ$ . Integrating it against $G$ and using $ξ = ξ$ almost surely gives $\int R_{b} (ξ) d G (ξ) = (A_{rare})_{#} ([\int i = 1 ⨂ k Q_{b, ξ_{i}}^{raw} d ν^{\otimes k} (ξ)] \otimes Q_{⋆}) .$ For cylinder events $E_{1} \times \dots \times E_{k} \subseteq Z_{rare}^{k}$ , independence under $ν^{\otimes k}$ gives $\int i = 1 \prod k Q_{b, ξ_{i}}^{raw} (E_{i}) d ν^{\otimes k} (ξ) = i = 1 \prod k \int Q_{b, p}^{raw} (E_{i}) d ν (p) = i = 1 \prod k Q_{b} (E_{i}),$ because each map $p \mapsto Q_{b, p}^{raw} (E_{i})$ is measurable and takes values in $[0, 1]$ . Finite Fubini therefore factors the integral of their product under $ν^{\otimes k}$ into the product of the coordinate integrals. Both sides define probability measures on the finite product space, and measurable rectangles form a generating $π$ -system; the $π$ - $λ$ uniqueness theorem extends the displayed equality from rectangles to the whole product $σ$ -algebra. Thus $\int i = 1 ⨂ k Q_{b, ξ_{i}}^{raw} d ν^{\otimes k} (ξ) = Q_{b}^{\otimes k} .$ Consequently the branch- $b$ raw prior-predictive law is $R_{b} := \int R_{b} (ξ) d G (ξ) = (A_{rare})_{#} (Q_{b}^{\otimes k} \otimes Q_{⋆}) .$ Appending the common reservoir law and applying the deterministic assembly map cannot increase total variation, so $TV (R_{0}, R_{1}) \leq TV (Q_{0}^{\otimes k}, Q_{1}^{\otimes k}) \leq \overset{ˉ}{C}_{ϵ} u k a ρ_{ϵ}^{L} \leq C_{ϵ} u k a ρ_{ϵ}^{L} .$ The constructed recipe with priors $Π_{0}, Π_{1}$ therefore satisfies all displayed calibration, support, common-table, mean, separation, variance, and mixture-distance conclusions.

∎

Lemma 25 [lem:common-marginal-canonical-certificate-family] (Canonical certificate family).

There exist real constants $C_{fixed}$ and $c_{risk}$ with $0 < C_{fixed}, 0 < c_{risk},$ such that, for every overlap level $ϵ$ with $0 < ϵ < 1/2$ , there are a common-marginal calibration in the sense of Definition 19 and real constants $C_{ϵ}, ρ_{ϵ}, c_{small}, c_{floor}$ with the following properties. Write $b_{0}$ for the calibration bandwidth constant, $b_{ϵ}$ for the uniform bandwidth constant, $γ_{ϵ}$ for the shift constant, $c_{ϵ}$ for the dual-gap constant, and $C_{d}$ for the dimension constant supplied by the calibration. These calibration constants are positive, and $c_{ϵ} \leq \frac{1}{4} .$ The auxiliary constants satisfy $0 < C_{ϵ}, ρ_{ϵ} \in (0, 1), 0 < c_{small}, 0 < c_{floor}, c_{floor} \leq c_{risk} .$ Moreover:

(Fixed-sample transfer.) For every $n, m, d$ with $n \geq 1$ and $d \geq 2$ , and every common-marginal prior handle $H$ at $(n, m, d)$ in the sense of Definition 19, let $L, B, a, k$ denote the degree, bandwidth, shift, and rare-cell count components of $H$ , and let $θ_{0} (H)$ and $θ_{1} (H)$ denote the two raw prior centers of its recipe. If the recipe of $H$ carries the exact common-marginal Poisson-prior construction at bandwidth constant $b_{0}$ , information size $n + m$ , labeled intensity $u = C_{fixed} n,$ auxiliary intensity $v = C_{fixed} (n + m),$ and constants $(C_{ϵ}, ρ_{ϵ})$ , including the corresponding finite-support prior data, exact common treatment-covariate marginal table, moment-matching identities through degree $L$ , raw-center separation data, and Poisson likelihood-ratio bound at rate $C_{ϵ} u k a ρ_{ϵ}^{L}$ , if the two raw prior-predictive Poisson experiment laws supplied by that recipe have total variation distance at most $1/8$ , and if $k B a \leq c_{small} ∣ θ_{1} (H) - θ_{0} (H) ∣^{2},$ then $c_{risk} ∣ θ_{1} (H) - θ_{0} (H) ∣^{2} \leq R_{ϵ} (n, m, d),$ where $R_{ϵ} (n, m, d)$ is the minimax risk from Definition 11.

(Canonical handles.) For every $n, m, d$ with $n \geq 1$ and $d \geq 2$ , there exists a common-marginal prior handle $H$ at $(n, m, d)$ . Let $L, B, a, k$ denote its degree, bandwidth, shift, and rare-cell count components, set $Δ_{H} = c_{ϵ} k a, D (n, m, d) = min {1, \frac{d ^{2}}{( n + m ) ^{2} lo g ( e n ) ^{2}}},$ and let $θ_{0} (H)$ and $θ_{1} (H)$ denote the two raw prior centers of its recipe. The same constants satisfy $L \geq 2, 2 C_{fixed} b_{0} \leq b_{ϵ},$ $(C_{fixed} n + C_{fixed} (n + m)) B \leq b_{ϵ} L,$ $L (P_{X A} ∣ Π_{0} (H)) = L (P_{X A} ∣ Π_{1} (H)), \frac{c _{floor}}{n} \leq R_{ϵ} (n, m, d) .$ On the finite-atom branch, the handle also satisfies $k a \leq c_{ϵ},$ and its recipe carries the exact common-marginal Poisson-prior construction at bandwidth constant $b_{0}$ , information size $n + m$ , labeled intensity $u = C_{fixed} n$ , auxiliary intensity $v = C_{fixed} (n + m)$ , and constants $(C_{ϵ}, ρ_{ϵ})$ , including the finite-support prior data, exact common treatment-covariate marginal table, moment-matching identities through degree $L$ , raw-center separation data, and Poisson likelihood-ratio bound at rate $C_{ϵ} u k a ρ_{ϵ}^{L}$ . The recipe bounds hold at these intensities: both priors are supported on $M_{d, ϵ}$ from Definition 6, they induce the same random $P_{X A}$ law, their raw-center separation is at least $c_{ϵ} k a$ , the raw total mass of the generator has mean one, the raw total mass and raw targets are square-integrable with generator variances at most $C_{ϵ} k B a$ , and the raw mixture total variation is at most $C_{ϵ} u k a ρ_{ϵ}^{L}$ . Moreover, the two raw prior-predictive Poisson experiment laws supplied by the recipe have total variation distance at most $1/8$ , and $Δ_{H} \leq ∣ θ_{1} (H) - θ_{0} (H) ∣, k B a \leq c_{small} Δ_{H}^{2},$ $c_{floor} Δ_{H}^{2} \leq R_{ϵ} (n, m, d), C_{d} D (n, m, d) \leq Δ_{H}^{2} .$ On the parametric branch, $C_{d} D (n, m, d) \leq \frac{c _{floor}}{n} .$

⊢ Lean

Proof of Lemma 25.

Set $C_{fixed} = 65536, c_{risk} = \frac{1}{65536} .$ These constants are positive. Fix $ϵ \in (0, 1/2)$ . By Lemma 24, choose a common-marginal calibration and constants $C_{ϵ}, ρ_{ϵ}$ with $C_{ϵ} > 0, ρ_{ϵ} \in (0, 1),$ together with the uniform construction and closure bounds supplied there. Define $c_{small} = \frac{1}{65536 C _{ϵ}}, c_{floor} = \frac{1}{65536} .$ Then $c_{small} > 0, c_{floor} > 0, c_{floor} \leq c_{risk},$ and Lemma 24 supplies positive $b_{0}, b_{ϵ}, γ_{ϵ}, q_{ϵ}, c_{ϵ}, C_{d}$ , with $c_{ϵ} \leq 1/4$ . We retain that statement’s notation: $q_{ϵ}$ is the rare-count coefficient, $c_{ϵ}$ is the dual-gap constant, and $C_{d}$ is the dimension constant.

First consider the fixed-sample transfer assertion. Let $n, m, d, H$ satisfy the hypotheses in that assertion, and write $Δ = ∣ θ_{1} (H) - θ_{0} (H) ∣ .$ By the hypotheses of the fixed-sample assertion, the recipe carried by $H$ comes with an exact Poisson-prior witness at $u = C_{fixed} n, v = C_{fixed} (n + m),$ with constants $C_{ϵ}, ρ_{ϵ}$ . The same hypotheses give total variation at most $1/8$ for its prior-predictive laws, and the assumed variance absorption is $k B a \leq c_{small} Δ^{2} = \frac{1}{65536 C _{ϵ}} Δ^{2} .$ Applying the fixed-sample recipe transfer bound Lemma 20 at these intensities gives $\frac{1}{65536} Δ^{2} \leq R_{ϵ} (n, m, d) .$ Since $c_{risk} = 1/65536$ , this is exactly the fixed-sample transfer conclusion.

It remains to construct the canonical handle for arbitrary $n, m, d$ with $n \geq 1$ and $d \geq 2$ . Write $N = n + m, ℓ_{n} = lo g (e n), D (n, m, d) = min {1, \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}}} .$ Let $r_{ϵ}$ denote the calibration regime constant, $D_{ϵ}$ the calibration degree constant, and $q_{ϵ}$ the calibration rare-count constant. The proof splits according to whether the parametric regime holds: $D (n, m, d) \leq \frac{r _{ϵ}}{n} .$ In both regimes set $L = ⌈ D_{ϵ} ℓ_{n} ⌉, B = \frac{b _{0} L}{N}, a = \frac{γ _{ϵ} B}{L ^{2}},$ and $k = min {d - 1, ⌊ q_{ϵ} N L ⌋} .$ The calibration closure in Lemma 24 gives $L \geq 2$ and $2 C_{fixed} b_{0} = 131072 b_{0} \leq b_{ϵ} .$ Moreover, since $n \leq N$ , $(C_{fixed} n + C_{fixed} N) B \leq (131072 N) \frac{b _{0} L}{N} = 131072 b_{0} L \leq b_{ϵ} L .$ Thus the degree and bandwidth conclusions hold for either branch.

Suppose first that $D (n, m, d) \leq \frac{r _{ϵ}}{n} .$ The local testing gap used by the parametric handle is admissible because $\frac{c _{ϵ}}{n} \leq c_{ϵ} \leq \frac{1}{4},$ where $n \geq 1$ . Let $H$ be the parametric common-marginal prior handle with the above values of $L, B, a, k$ . Its branch condition makes the finite-atom predicate false in this case. Consequently all conclusions whose hypotheses include the finite-atom predicate are vacuous. The common treatment-covariate marginal identity is the handle identity $L (P_{X A} ∣ Π_{0} (H)) = L (P_{X A} ∣ Π_{1} (H)) .$ The risk floor follows from the parametric two-point floor: $\frac{c _{floor}}{n} = \frac{1/65536}{n} \leq \frac{1/100}{n} \leq R_{ϵ} (n, m, d) .$ Finally, the calibration closure gives $C_{d} r_{ϵ} \leq \frac{1}{65536} = c_{floor},$ and hence, using the parametric-regime inequality, $C_{d} D (n, m, d) \leq C_{d} \frac{r _{ϵ}}{n} \leq \frac{c _{floor}}{n} .$ This proves all required canonical-handle properties on the parametric branch.

Now suppose that the parametric regime fails: $D (n, m, d) > \frac{r _{ϵ}}{n} .$ With $L, B, a, k$ as above, the quantities $B$ and $a$ are positive. From $k = min {d - 1, ⌊ q_{ϵ} N L ⌋}$ and the elementary inequality $⌊ q_{ϵ} N L ⌋ \leq q_{ϵ} N L$ , valid since $q_{ϵ} N L \geq 0$ , we have $k \leq q_{ϵ} N L .$ Therefore $k a \leq q_{ϵ} N L \cdot \frac{γ _{ϵ} b _{0}}{N L} = q_{ϵ} γ_{ϵ} b_{0} \leq c_{ϵ},$ where the last inequality is the rare-scale calibration inequality. Hence the rare shift is small.

Apply the finite-atom part of Lemma 24 with $u = 65536 n, v = 65536 N .$ The preceding bandwidth bound, positivity of $B$ , the identity $a = γ_{ϵ} B / L^{2}$ , the rare-shift bound $k a \leq c_{ϵ}$ , and $k + 1 \leq d$ produce a recipe $R$ at these intensities. The bounds supplied for this recipe include the finite-atom construction flag, support in $M_{d, ϵ}$ , equality of the induced treatment-covariate marginal table, the raw total-mass mean-one identity, the raw-center separation, the raw total-mass and target variance bounds, and the raw mixture total-variation bound $TV (P_{0, R}^{Pois}, P_{1, R}^{Pois}) \leq C_{ϵ} u k a ρ_{ϵ}^{L} .$ Together with $n \geq 1$ , $L \geq 2$ , $d \geq 2$ , the nonnegativity of $u, v$ , the positivity of $u + v$ , the bandwidth and shift identities, and the rare-count fit $k + 1 \leq d$ , these same bounds form the exact Poisson-prior witness used below. Let $H$ be the corresponding handle. Its branch condition identifies this recipe with the finite-atom branch. The common marginal identity again gives $L (P_{X A} ∣ Π_{0} (H)) = L (P_{X A} ∣ Π_{1} (H)) .$ The recipe bounds also give $∣ θ_{1} (H) - θ_{0} (H) ∣ \geq c_{ϵ} k a = Δ_{H},$ the raw total mass has mean one, the raw total mass and raw targets are square-integrable with variances at most $C_{ϵ} k B a$ , and the raw mixture total variation is bounded by $C_{ϵ} u k a ρ_{ϵ}^{L} .$ The calibration decay inequality and the bound $k a \leq c_{ϵ}$ imply $TV (P_{0, H}^{Pois}, P_{1, H}^{Pois}) \leq C_{ϵ} (65536 n) k a ρ_{ϵ}^{L} \leq C_{ϵ} (65536 n) c_{ϵ} ρ_{ϵ}^{L} \leq \frac{1}{8} .$ Thus the prior-predictive total variation condition required by the transfer rule holds.

It remains to record the two absorptions used by the handle. The counting estimate used in the finite-atom branch is $\frac{L ^{2}}{c _{small} c _{ϵ}^{2} γ _{ϵ}} \leq k .$ Indeed, write $N = n + m$ and $ℓ_{n} = lo g (e n)$ . Since the branch condition is $D (n, m, d) > \frac{r _{ϵ}}{n},$ the minimum in $D (n, m, d)$ forces $\frac{r _{ϵ}}{n} < \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}} .$ Put $K = c_{small} c_{ϵ}^{2} γ_{ϵ} > 0.$ The calibration inequality $(\frac{4 ( D _{ϵ} + 1 ) ^{2}}{K})^{2} \leq r_{ϵ}$ implies $\frac{4 ( D _{ϵ} + 1 ) ^{2}}{K} \leq r_{ϵ} .$ Meanwhile, the failed-regime inequality above gives $\frac{r _{ϵ} N ℓ _{n}}{n} < d .$ Using the ceiling bound $L \leq (D_{ϵ} + 1) ℓ_{n}$ and $ℓ_{n} n \leq 2 n \leq 2 N,$ we obtain the complete cross-multiplication $\frac{2 L ^{2}}{K} \leq \frac{2 ( D _{ϵ} + 1 ) ^{2} ℓ _{n}^{2}}{K} = (\frac{4 ( D _{ϵ} + 1 ) ^{2}}{K}) \frac{N ℓ _{n}}{n} \frac{ℓ _{n} n}{2 N} \leq \frac{r _{ϵ} N ℓ _{n}}{n} < d .$ Equivalently, the proof has compared the nonnegative quantities through the strict square inequality $(\frac{4 ( D _{ϵ} + 1 ) ^{2}}{K} N ℓ_{n})^{2} < (d n)^{2},$ and then divided by the positive factor $n$ . Weakening the strict bound gives $\frac{2 L ^{2}}{K} \leq d .$ Now split according to the actual comparison defining the minimum in $k$ . If $d - 1 \leq ⌊ q_{ϵ} N L ⌋,$ then $k = d - 1.$ Since $d \geq 2$ , $\frac{d}{2} \leq d - 1,$ and hence $\frac{L ^{2}}{K} \leq \frac{d}{2} \leq d - 1 = k .$ If instead $⌊ q_{ϵ} N L ⌋ < d - 1,$ then $k = ⌊ q_{ϵ} N L ⌋ .$ The calibration gives $\frac{4 ( D _{ϵ} + 1 )}{K} \leq q_{ϵ},$ and, since $q_{ϵ} N L \geq 1$ , the elementary floor bound $⌊ q_{ϵ} N L ⌋ \geq q_{ϵ} N L /2$ gives $k = ⌊ q_{ϵ} N L ⌋ \geq \frac{q _{ϵ} N L}{2} \geq \frac{2 ( D _{ϵ} + 1 ) N L}{K} .$ Using $L \leq (D_{ϵ} + 1) ℓ_{n}$ and $ℓ_{n} \leq N$ , this is at least $\frac{L ^{2}}{K} .$ Thus the displayed counting estimate holds in both subcases. Recall here that $c_{ϵ}$ denotes the calibration dual-gap constant; the rare-count coefficient is $q_{ϵ}$ . Using $a = γ_{ϵ} B / L^{2}$ , the counting estimate yields $k B a = (\frac{k a ^{2}}{γ _{ϵ}}) L^{2} \leq \frac{k a ^{2}}{γ _{ϵ}} (k c_{small} c_{ϵ}^{2} γ_{ϵ}) = c_{small} (c_{ϵ} k a)^{2} = c_{small} Δ_{H}^{2} .$ The recipe separation bound gives $Δ_{H} = c_{ϵ} k a \leq ∣ θ_{1} (H) - θ_{0} (H) ∣ .$ Consequently $k B a \leq c_{small} ∣ θ_{1} (H) - θ_{0} (H) ∣^{2} .$ The fixed-sample transfer already proved above therefore gives $c_{floor} ∣ θ_{1} (H) - θ_{0} (H) ∣^{2} \leq R_{ϵ} (n, m, d),$ and hence $c_{floor} Δ_{H}^{2} \leq R_{ϵ} (n, m, d) .$ As before, the parametric floor gives $\frac{c _{floor}}{n} \leq R_{ϵ} (n, m, d) .$ Finally we verify the dimension absorption for the above $L, k, a$ . Since $a = \frac{γ _{ϵ} B}{L ^{2}} = \frac{γ _{ϵ} b _{0}}{N L},$ it is enough to compare $C_{d} D (n, m, d)$ with $(c_{ϵ} k \frac{γ _{ϵ} b _{0}}{N L})^{2} .$ Split according to which term realizes $k = min {d - 1, ⌊ q_{ϵ} N L ⌋} .$ On the cap-active subcase, $k = d - 1$ . Because $d \geq 2$ , $\frac{d}{2} \leq d - 1 = k .$ The definition of $D (n, m, d)$ and the dimension calibration give $C_{d} D (n, m, d) \leq C_{d} \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}} \leq (\frac{c _{ϵ} γ _{ϵ} b _{0}}{2 ( D _{ϵ} + 1 )})^{2} \frac{d ^{2}}{N ^{2} ℓ _{n}^{2}} .$ All factors here are nonnegative. The two bounds $d \leq 2 k$ and $L \leq (D_{ϵ} + 1) ℓ_{n}$ therefore give, after cross-multiplication by the positive denominators, $0 \leq \frac{c _{ϵ} γ _{ϵ} b _{0} d}{2 ( D _{ϵ} + 1 ) N ℓ _{n}} \leq c_{ϵ} k \frac{γ _{ϵ} b _{0}}{N L} = c_{ϵ} k a .$ Squaring this nonnegative comparison proves $C_{d} D (n, m, d) \leq (c_{ϵ} k a)^{2}$ on the cap-active subcase. On the floor-active subcase, $k = ⌊ q_{ϵ} N L ⌋$ . The calibration includes the rare-count identity $q_{ϵ} = \frac{c _{ϵ}}{2 γ _{ϵ} b _{0}} .$ The calibration has $q_{ϵ} \geq 4$ , while $N \geq 1$ and $L \geq 2$ , so $q_{ϵ} N L \geq 1$ . The elementary floor bound therefore gives $k \geq \frac{q _{ϵ} N L}{2} .$ Thus, with every factor nonnegative, $k a \geq \frac{q _{ϵ} N L}{2} \cdot \frac{γ _{ϵ} b _{0}}{N L} = \frac{c _{ϵ}}{4} .$ Since $D (n, m, d) \leq 1$ , the calibration gives $C_{d} \leq c_{ϵ}^{4} /16$ , and squaring preserves the preceding comparison, $C_{d} D (n, m, d) \leq C_{d} \leq \frac{c _{ϵ}^{4}}{16} = (c_{ϵ} \frac{c _{ϵ}}{4})^{2} \leq (c_{ϵ} k a)^{2} .$ Thus in both subcases $C_{d} D (n, m, d) \leq (c_{ϵ} k a)^{2} = Δ_{H}^{2} .$ This proves every required canonical-handle property on the finite-atom branch, and completes the construction for all $n, m, d$ .

∎

The constants in Lemma 24 coordinate the degree, bandwidth, rare-cell count, and mixture decay. The shared-law conclusion for the random treatment–covariate tables is the structural feature that makes the prior compatible with auxiliary annotation data, and the final total-variation display supplies the testing bound used by Lemmas 20 and 21. Lemma 25 then assembles the transfer constants, the canonical handle, the parametric label floor, and the finite-atom branch into the certificate invoked by Theorem 3.

One-coordinate marked-Poisson input.

The load-bearing analytic estimate is a one-coordinate marked-Poisson total-variation bound for normalized signed certificates. For a normalized signed certificate supported on $[a / κ_{ϵ}, B]$ , with moments through degree $L$ equal to zero and $(u + v) B \leq b_{ϵ} L$ , its two marked-Poisson predictive laws satisfy $TV (Q_{0}, Q_{1}) \leq C_{ϵ} u a ρ_{ϵ}^{L}, 0 < ρ_{ϵ} < 1.$ Expanding each marked-Poisson likelihood in the latent mass gives the intuition behind the bound: moment matching cancels the polynomial terms through degree $L$ , and the bandwidth condition leaves a geometrically controlled exponential-series tail. Tensorization over rare cells contributes the factor $k$ , while the common reservoir and deterministic splitting and assembly maps preserve or contract total variation.

Known-marginal experiment and verification note

The benchmark in Definition 15, Theorem 1, and Theorem 5 uses the same labeled experiment as the annotation problem, but supplies the treatment–covariate marginal as exact side information. This appendix records the corresponding experiment explicitly so that the comparison with the auxiliary-sample experiment in Definition 9 has a fixed statistical object.

Definition 30 [def:known-marginal-experiment] (Known-marginal experiment

E_{n, d, ϵ}^{known}

For $n \geq 1$ and $d \geq 2$ , the known-marginal experiment is $E_{n, d, ϵ}^{known} := {(P^{\otimes n}, P_{X A}) : P \in M_{d, ϵ}} .$ The first coordinate generates $L_{n}$ , and the second coordinate is the exact $2 d$ -cell table observed as deterministic side information.

⊢ Lean

The deterministic table $P_{X A}$ contains the full joint distribution of treatment and covariates under the same observed-data law $P$ used for the labeled sample. Thus the benchmark isolates the contribution of outcome labels after the treatment–covariate frequencies are known exactly. This is the population analogue of having unlimited annotation data for $(X, A)$ , and it is the object compared with finite- $m$ annotation in Theorem 5.

The risk $R_{ϵ}^{known} (n, d)$ in Definition 15 takes the infimum over $T_{n, d}^{known}$ , the estimators measurable with respect to the labeled sample and this exact table. The boundary result in Theorem 1 therefore describes the benchmark rate for the known-marginal problem, while Theorem 5 identifies the annotation regimes in which the finite auxiliary sample attains the same limiting benchmark. The construction is aligned with the finite-alphabet causal model in Definition 6 and the average treatment effect functional in Definition 8; the overlap and sampling conditions remain those stated in Assumptions 1 and 2.

The next statements record the finite side-information comparison underlying that benchmark. The finite-cover statement is the compactness input that turns exact side information into finitely many stable decision rules; the causal specialization then yields convergence of empirical side samples to the exact-side benchmark; and the measurable comparison formalizes exact side information as at least as informative as an empirical side sample.

Lemma 26 [lem:finite-side-cover-approximate-decision] (Finite side cover).

Let $Θ$ be a nonempty compact topological space, let $Z$ be a finite set, let $C$ be a finite nonempty set, and let $l \leq u$ . Suppose the following conditions hold.

(Laws.) For each $θ \in Θ$ , $Λ_{θ}$ is a probability vector on $Z$ and $Ξ_{θ}$ is a probability vector on $C$ .

(Target.) For each $θ \in Θ$ , $υ_{θ} \in [l, u]$ .

(Continuity.) For every $z \in Z$ and $c \in C$ , the maps $θ \mapsto Λ_{θ} (z)$ , $θ \mapsto Ξ_{θ} (c)$ , and $θ \mapsto υ_{θ}$ are continuous.

For probability vectors $ν, ν^{'}$ on $C$ , set $D_{\infty} (ν, ν^{'}) = c \in C max ∣ ν (c) - ν^{'} (c) ∣.$ Define $R_{⋆} = ψ in f θ \in Θ sup z \in Z \sum Λ_{θ} (z) (ψ (z, Ξ_{θ}) - υ_{θ})^{2},$ where $ψ$ ranges over all maps $ψ : Z \times Prob (C) \to [l, u] .$ Then for every $δ > 0$ , there exist an integer $k \geq 1$ , centers $θ_{1}^{\circ}, \dots, θ_{k}^{\circ} \in Θ$ , decisions $β_{1}, \dots, β_{k} : Z \to [l, u]$ , radii $rad_{1}, \dots, rad_{k}$ , and a margin $η > 0$ such that $η < rad_{j}$ for every $j$ , every $θ \in Θ$ admits some $j \in {1, \dots, k}$ with $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} - η,$ and, for every $j \in {1, \dots, k}$ and every $θ \in Θ$ , $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} ⟹ z \in Z \sum Λ_{θ} (z) (β_{j} (z) - υ_{θ})^{2} \leq R_{⋆} + δ .$

⊢ Lean

Proof of Lemma 26.

For a decision $b : Z \to [l, u]$ , write $R_{θ} (b) = z \in Z \sum Λ_{θ} (z) (b (z) - υ_{θ})^{2} .$ Since $b (z), υ_{θ} \in [l, u]$ , each squared term is at most $(u - l)^{2}$ , and since $Λ_{θ}$ is a probability vector, $0 \leq R_{θ} (b) \leq (u - l)^{2} for every θ \in Θ.$ In particular the constant decision $b_{0} (z) = l$ is admissible.
Fix $θ_{0} \in Θ$ . Let $F (θ_{0}) = {θ \in Θ : Ξ_{θ} = Ξ_{θ_{0}}}$ and let $V_{fib} (θ_{0}) = β : Z \to [l, u] in f θ \in F (θ_{0}) sup R_{θ} (β) .$ The fiber is nonempty because it contains $θ_{0}$ . By the previous bound, the worst-case risk of the constant decision $b_{0}$ over this fiber is at most $(u - l)^{2}$ , hence $V_{fib} (θ_{0}) \leq (u - l)^{2}$ . Thus the set of all fiber values is bounded above.

For an exact-side rule $ψ : Z \times Prob (C) \to [l, u]$ , write $R_{θ}^{ex} (ψ) = z \in Z \sum Λ_{θ} (z) (ψ (z, Ξ_{θ}) - υ_{θ})^{2}, R_{⋆} = ψ in f θ \in Θ sup R_{θ}^{ex} (ψ) .$ For any such $ψ$ , define the fiber decision $β_{ψ, θ_{0}} (z) = ψ (z, Ξ_{θ_{0}}) .$ If $θ \in F (θ_{0})$ , then $Ξ_{θ} = Ξ_{θ_{0}}$ , so $R_{θ} (β_{ψ, θ_{0}}) = R_{θ}^{ex} (ψ) .$ Consequently $V_{fib} (θ_{0}) \leq θ \in F (θ_{0}) sup R_{θ} (β_{ψ, θ_{0}}) \leq θ \in Θ sup R_{θ}^{ex} (ψ) .$ Taking the infimum over $ψ$ gives $V_{fib} (θ_{0}) \leq R_{⋆}, ϑ \in Θ sup V_{fib} (ϑ) \leq R_{⋆} .$

For the reverse inequality, set $s = ϑ \in Θ sup V_{fib} (ϑ) .$ Let $a$ be any real number with $s < a$ . Then every fiber value is strictly below $a$ . Since the fiber risks are nonnegative, the strict infimum property gives, for each $θ_{0}$ , a decision $β_{θ_{0}} : Z \to [l, u]$ such that $θ \in F (θ_{0}) sup R_{θ} (β_{θ_{0}}) < a .$ Define a representative map on the whole simplex by $rep (ν) = {one θ \in Θ with Ξ_{θ} = ν, an arbitrary point of Θ, if such a θ exists, otherwise .$ This is defined for every $ν \in Prob (C)$ because $Θ$ is nonempty. Now assemble the exact-side rule $ψ_{a} (z, ν) = β_{rep (ν)} (z) .$ For each $θ \in Θ$ , the representative $rep (Ξ_{θ})$ has side law $Ξ_{θ}$ , so $θ \in F (rep (Ξ_{θ}))$ . Hence $R_{θ}^{ex} (ψ_{a}) = R_{θ} (β_{rep (Ξ_{θ})}) < a .$ Therefore $R_{⋆} \leq θ \in Θ sup R_{θ}^{ex} (ψ_{a}) \leq a .$ Since this holds for every $a > s$ , $R_{⋆} \leq s .$ Combining the two inequalities yields $R_{⋆} = ϑ \in Θ sup V_{fib} (ϑ),$ and in particular $V_{fib} (θ_{0}) \leq R_{⋆} .$
For $α > 0$ , define the closed side-law neighborhood value $V_{α} (θ_{0}) = β : Z \to [l, u] in f θ : D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) \leq α sup R_{θ} (β) .$ The required shrinking-neighborhood fact is the reciprocal-radius convergence $V_{1/ (N + 1)} (θ_{0}) ⟶ V_{fib} (θ_{0}) as N \to \infty.$ Indeed, for every $α \geq 0$ , the exact fiber $F (θ_{0})$ is contained in the closed $α$ -neighborhood, and therefore $V_{fib} (θ_{0}) \leq V_{α} (θ_{0}) .$ This gives the lower side of the limit. For the upper side, fix a real number $b > V_{fib} (θ_{0})$ and put $c = \frac{V _{fib} ( θ _{0} ) + b}{2}, V_{fib} (θ_{0}) < c < b .$ Since the fiber risks are nonnegative, the strict infimum property gives a decision $β^{♯} : Z \to [l, u]$ with $θ \in F (θ_{0}) sup R_{θ} (β^{♯}) < c,$ so $θ \in F (θ_{0}) ⟹ R_{θ} (β^{♯}) < c .$ For this fixed decision the map $θ \mapsto R_{θ} (β^{♯})$ is continuous, because $Z$ is finite and the coordinates $Λ_{θ} (z)$ and $υ_{θ}$ are continuous. Hence the set $B = {θ \in Θ : c \leq R_{θ} (β^{♯})}$ is closed, and compactness of $Θ$ makes $B$ compact. The side-law map $θ \mapsto Ξ_{θ}$ is continuous, so its image $Ξ (B) = {Ξ_{θ} : θ \in B}$ is compact and therefore closed in the finite simplex. The preceding fiber inequality implies $Ξ_{θ_{0}} \in / Ξ (B),$ for otherwise some $θ \in B$ would satisfy $Ξ_{θ} = Ξ_{θ_{0}}$ and hence both $c \leq R_{θ} (β^{♯})$ and $R_{θ} (β^{♯}) < c$ . Since a closed set misses $Ξ_{θ_{0}}$ , there is $α_{0} > 0$ such that $D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) < α_{0} ⟹ R_{θ} (β^{♯}) < c .$ For all sufficiently large $N$ , $1/ (N + 1) < α_{0}$ . For such $N$ , every $θ$ with $D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) \leq \frac{1}{N + 1}$ satisfies $R_{θ} (β^{♯}) < c$ , and therefore $V_{1/ (N + 1)} (θ_{0}) \leq θ : D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) \leq 1/ (N + 1) sup R_{θ} (β^{♯}) \leq c < b .$ Together with $V_{fib} (θ_{0}) \leq V_{1/ (N + 1)} (θ_{0})$ , this proves the displayed convergence.

Apply the convergence with $b = V_{fib} (θ_{0}) + δ /2$ . Choose an integer $N_{θ_{0}} \geq 0$ with $V_{1/ (N_{θ_{0}} + 1)} (θ_{0}) < V_{fib} (θ_{0}) + δ /2.$ Set $r (θ_{0}) = \frac{1}{N _{θ_{0}} + 1} > 0.$ Together with $V_{fib} (θ_{0}) \leq R_{⋆}$ , this gives $V_{r (θ_{0})} (θ_{0}) < R_{⋆} + δ .$ Since all risks are nonnegative, the range of worst-case risks over admissible decisions is bounded below by $0$ . Hence the strict inequality for the infimum yields a decision $β_{θ_{0}} : Z \to [l, u]$ such that $θ : D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) \leq r (θ_{0}) sup R_{θ} (β_{θ_{0}}) < R_{⋆} + δ .$ Consequently, whenever $D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) < r (θ_{0}),$ the same $θ$ belongs to the closed neighborhood, and therefore $R_{θ} (β_{θ_{0}}) \leq R_{⋆} + δ .$
The map $θ \mapsto Ξ_{θ}$ is continuous because each coordinate $θ \mapsto Ξ_{θ} (c)$ is continuous and $C$ is finite. For each $θ_{0}$ , define $U_{θ_{0}} = {θ \in Θ : D_{\infty} (Ξ_{θ}, Ξ_{θ_{0}}) < \frac{r ( θ _{0} )}{2}} .$ Each $U_{θ_{0}}$ is open, and $θ_{0} \in U_{θ_{0}}$ . Thus ${U_{θ_{0}} : θ_{0} \in Θ}$ is an open cover of $Θ$ . Compactness gives finitely many centers $θ_{1}^{\circ}, \dots, θ_{k}^{\circ}$ such that these sets still cover $Θ$ . Since $Θ$ is nonempty, this finite subcover has $k \geq 1$ .
For each selected center set $rad_{j} = r (θ_{j}^{\circ}), β_{j} = β_{θ_{j}^{\circ}} .$ Define $η = 1 \leq j \leq k min \frac{rad _{j}}{2} .$ Every $rad_{j}$ is positive, so $η > 0$ , and $η \leq \frac{rad _{j}}{2} < rad_{j} for every j .$ If $θ \in Θ$ , the finite subcover gives an index $j$ with $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < \frac{rad _{j}}{2} .$ Since $η \leq rad_{j} /2$ , this implies $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} - η .$ Finally, if for some $j$ and $θ$ , $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j},$ then the local guarantee from the construction at $θ_{j}^{\circ}$ gives $z \in Z \sum Λ_{θ} (z) (β_{j} (z) - υ_{θ})^{2} = R_{θ} (β_{j}) \leq R_{⋆} + δ .$ These are exactly the required centers, decisions, radii, and margin.

∎

Lemma 26 supplies the finite selection step used in the side-information limit. Compactness and continuity allow the exact-side decision problem to be represented, up to any tolerance $δ$ , by finitely many neighborhoods of side laws. The positive margin $η$ gives room for an empirical side table to select one of these neighborhoods uniformly once the auxiliary sample is large.

Lemma 27 [lem:finite-side-information-convergence] (Finite side-information convergence).

Let the overlap parameter $ϵ$ satisfy $0 < ϵ < 1/2$ , let $n \geq 0$ , and let $d \geq 2$ . Let $Θ_{ϵ, d} = {(π, ϱ, ω) : π \in Prob ([d]), ϱ_{x} \in [ϵ, 1 - ϵ], ω_{α x} \in [0, 1] for x \in [d], α \in {0, 1}} .$ For $θ = (π, ϱ, ω) \in Θ_{ϵ, d}$ , set $P_{θ} (x, α, y) = π_{x} ϱ_{x}^{α} (1 - ϱ_{x})^{1 - α} ω_{α x}^{y} (1 - ω_{α x})^{1 - y},$ let $λ_{θ} = P_{θ}^{\otimes n}$ be the labeled-sample law on $([d] \times {0, 1}^{2})^{n}$ , define $ξ_{θ} (x, α) = y \in {0, 1} \sum P_{θ} (x, α, y), υ_{θ} = x \in [d] \sum π_{x} (ω_{1 x} - ω_{0 x}) .$ For each $m \geq 0$ , define $R_{m} = φ : ([d] \times {0, 1}^{2})^{n} \times ([d] \times {0, 1})^{m} \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup \int (φ (Z, U) - υ_{θ})^{2} d (λ_{θ} \otimes ξ_{θ}^{\otimes m}) (Z, U),$ and define $R_{⋆} = ψ : ([d] \times {0, 1}^{2})^{n} \times Prob ([d] \times {0, 1}) \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup \int (ψ (Z, ξ_{θ}) - υ_{θ})^{2} d λ_{θ} (Z) .$ Then $m \to \infty lim R_{m} = R_{⋆} .$

⊢ Lean

Proof of Lemma 27.

Let $Z_{n} = ([d] \times {0, 1}^{2})^{n}, C_{d} = [d] \times {0, 1} .$ For $θ \in Θ_{ϵ, d}$ , write $Λ_{θ} (z) = λ_{θ} ({z}), Ξ_{θ} (c) = ξ_{θ} (c), c \in C_{d} .$ The assumptions $d \geq 2$ and $0 < ϵ < 1/2$ make $[d]$ nonempty and make $[ϵ, 1 - ϵ]$ nonempty. The coordinate map $(π, ϱ, ω) ⟼ (π, (ϱ_{x})_{x \in [d]}, (ω_{α x})_{(α, x) \in {0, 1} \times [d]})$ identifies $Θ_{ϵ, d}$ , with its coordinate topology, with the finite product $Prob ([d]) \times [ϵ, 1 - ϵ]^{[d]} \times [0, 1]^{{0, 1} \times [d]} .$ Here $Prob ([d])$ is the closed simplex $⎩ ⎨ ⎧ π \in [0, 1]^{[d]} : x \in [d] \sum π_{x} = 1 ⎭ ⎬ ⎫,$ which is compact as a closed subset of the finite cube $[0, 1]^{[d]}$ . The two interval powers are finite products of compact intervals. Hence $Θ_{ϵ, d}$ is nonempty and compact.

For each $θ$ , $Λ_{θ}$ is a probability law on $Z_{n}$ , because it is the $n$ -fold product of the one-record law $P_{θ}$ . Also $Ξ_{θ}$ is a probability law on $C_{d}$ . Indeed, every summand is nonnegative, and the normalization expands as $x \in [d] \sum α \in {0, 1} \sum Ξ_{θ} (x, α) = x \in [d] \sum α \in {0, 1} \sum y \in {0, 1} \sum π_{x} ϱ_{x}^{α} (1 - ϱ_{x})^{1 - α} ω_{α x}^{y} (1 - ω_{α x})^{1 - y} = x \in [d] \sum π_{x} α \in {0, 1} \sum ϱ_{x}^{α} (1 - ϱ_{x})^{1 - α} y \in {0, 1} \sum ω_{α x}^{y} (1 - ω_{α x})^{1 - y} = x \in [d] \sum π_{x} ((1 - ϱ_{x}) + ϱ_{x}) = 1.$ The maps $θ \mapsto Λ_{θ} (z)$ , $θ \mapsto Ξ_{θ} (c)$ , and $θ \mapsto υ_{θ}$ are continuous, since each is obtained from finitely many coordinate projections by finite sums and products. Finally, $- 1 \leq ω_{1 x} - ω_{0 x} \leq 1 for every x,$ and the weights $π_{x}$ are nonnegative with total mass one, so $- 1 = x \sum π_{x} (- 1) \leq x \sum π_{x} (ω_{1 x} - ω_{0 x}) = υ_{θ} \leq x \sum π_{x} (1) = 1.$

With this notation, the empirical-side minimax value at auxiliary size $m$ is $R_{m} = φ : Z_{n} \times C_{d}^{m} \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup z \in Z_{n} \sum Λ_{θ} (z) u \in C_{d}^{m} \sum Ξ_{θ}^{\otimes m} (u) (φ (z, u) - υ_{θ})^{2},$ and the exact-side minimax value is $R_{⋆} = ψ : Z_{n} \times Prob (C_{d}) \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup z \in Z_{n} \sum Λ_{θ} (z) (ψ (z, Ξ_{θ}) - υ_{θ})^{2} .$ We now use the finite side-information convergence argument for these data. First, for any empirical estimator $φ : Z_{n} \times C_{d}^{m} \to [- 1, 1]$ , define the corresponding exact-side estimator $\overline{φ} (z, ν) = u \in C_{d}^{m} \sum ν^{\otimes m} (u) φ (z, u), z \in Z_{n}, ν \in Prob (C_{d}) .$ Convexity of $v \mapsto (v - υ_{θ})^{2}$ gives, for every $θ$ , $\int (\overline{φ} (Z, Ξ_{θ}) - υ_{θ})^{2} d λ_{θ} (Z) \leq \int (φ (Z, U) - υ_{θ})^{2} d (λ_{θ} \otimes Ξ_{θ}^{\otimes m}) (Z, U) .$ Taking suprema over $θ$ and then infima over $φ$ yields $R_{⋆} \leq R_{m} for every m \geq 0.$

It remains to prove the reverse asymptotic inequality. Put $D_{\infty} (ν, ν^{'}) = c \in C_{d} max ∣ ν (c) - ν^{'} (c) ∣$ for probability laws on $C_{d}$ . Fix $δ > 0$ . With $Θ = Θ_{ϵ, d}$ , $Z = Z_{n}$ , $C = C_{d}$ , $Λ_{θ} = λ_{θ}$ , $Ξ_{θ} = ξ_{θ}$ , $l = - 1$ , and $u = 1$ , the compactness, simplex, continuity, and target-range checks above give the analytic hypotheses of Lemma 26. Its remaining structural hypotheses are immediate but worth recording: $Z_{n}$ is a finite power of a finite set, $C_{d}$ is finite and nonempty because $d \geq 2$ , and $- 1 \leq 1$ . Thus every hypothesis of Lemma 26 holds. Hence, for the present $δ$ , there are an integer $k \geq 1$ , centers $θ_{1}^{\circ}, \dots, θ_{k}^{\circ} \in Θ_{ϵ, d}$ , bounded decisions $β_{1}, \dots, β_{k} : Z_{n} \to [- 1, 1]$ , radii $rad_{1}, \dots, rad_{k}$ , and a margin $η > 0$ such that $η < rad_{j} (1 \leq j \leq k),$ $\forall θ \in Θ_{ϵ, d} \exists j \in {1, \dots, k}, D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} - η,$ and, for every $j \in {1, \dots, k}$ and every $θ \in Θ_{ϵ, d}$ , $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} ⟹ \int (β_{j} (Z) - υ_{θ})^{2} d λ_{θ} (Z) \leq R_{⋆} + δ .$

For $m \geq 1$ and $u \in C_{d}^{m}$ , let $Ξ_{u} \in Prob (C_{d})$ be the empirical law of $u$ , and put $J := {1, \dots, k}$ . Since $k \geq 1$ , this set is nonempty. Choose a selector $σ : Prob (C_{d}) \to J$ as follows: if at least one $ \in J$ satisfies $D_{\infty} (ν, Ξ_{θ_{}^{\circ}}) < rad_{} - η /2,$ let $σ (ν)$ be the least such index; if there is no such index, set $σ (ν) = 1$ . Define $Φ_{δ, m} (z, u) = β_{σ (Ξ_{u})} (z) .$ On the event $G_{θ, m} = {u \in C_{d}^{m} : D_{\infty} (Ξ_{u}, Ξ_{θ}) < η /2},$ choose a cover index $j \in J$ with $D_{\infty} (Ξ_{θ}, Ξ_{θ_{j}^{\circ}}) < rad_{j} - η .$ The good-event bound and the triangle inequality give $D_{\infty} (Ξ_{u}, Ξ_{θ_{j}^{\circ}}) < η /2 + rad_{j} - η = rad_{j} - η /2,$ so $j$ is eligible for the selector. Hence $D_{\infty} (Ξ_{u}, Ξ_{θ_{σ (Ξ_{u})}^{\circ}}) < rad_{σ (Ξ_{u})} - η /2.$ Applying the triangle inequality once more, now through $Ξ_{u}$ , yields $D_{\infty} (Ξ_{θ}, Ξ_{θ_{σ (Ξ_{u})}^{\circ}}) < η /2 + rad_{σ (Ξ_{u})} - η /2 = rad_{σ (Ξ_{u})} .$ Therefore $\int (Φ_{δ, m} (Z, U) - υ_{θ})^{2} d (λ_{θ} \otimes Ξ_{θ}^{\otimes m}) (Z, U) \leq R_{⋆} + δ + 4 Ξ_{θ}^{\otimes m} (G_{θ, m}^{c}),$ because all estimates and all targets lie in $[- 1, 1]$ . For the finite-alphabet concentration step, fix $θ$ . For $c \in C_{d}$ and $u = (u_{1}, \dots, u_{m}) \in C_{d}^{m}$ , write $N_{c} (u) = i = 1 \sum m 1 {u_{i} = c}, Ξ_{u} (c) = \frac{N _{c} ( u )}{m} .$ Under $Ξ_{θ}^{\otimes m}$ , the variables $1 {U_{i} = c}$ are independent Bernoulli variables with mean $Ξ_{θ} (c)$ . Thus $E_{θ} Ξ_{U} (c) = Ξ_{θ} (c)$ and, since a Bernoulli variance is at most $1/4$ , $Var_{θ} (Ξ_{U} (c)) = \frac{1}{m ^{2}} i = 1 \sum m Var_{θ} (1 {U_{i} = c}) \leq \frac{1}{4 m} .$ Chebyshev’s inequality gives, for every $t > 0$ , $Ξ_{θ}^{\otimes m} (Ξ_{U} (c) - Ξ_{θ} (c) \geq t) \leq \frac{1}{4 m t ^{2}} .$ If $D_{\infty} (Ξ_{U}, Ξ_{θ}) \geq η /2$ , then the defining identity $D_{\infty} (Ξ_{U}, Ξ_{θ}) = c \in C_{d} max Ξ_{U} (c) - Ξ_{θ} (c)$ implies that some $c \in C_{d}$ satisfies $Ξ_{U} (c) - Ξ_{θ} (c) \geq η /2.$ Hence the bad event satisfies $G_{θ, m}^{c} = {D_{\infty} (Ξ_{U}, Ξ_{θ}) \geq η /2} \subseteq c \in C_{d} ⋃ {Ξ_{U} (c) - Ξ_{θ} (c) \geq η /2} .$ The union bound over the finite alphabet gives $Ξ_{θ}^{\otimes m} (G_{θ, m}^{c}) \leq c \in C_{d} \sum Ξ_{θ}^{\otimes m} (Ξ_{U} (c) - Ξ_{θ} (c) \geq η /2) \leq \frac{∣ C _{d} ∣}{m η ^{2}} .$ The right-hand side is independent of $θ$ and tends to zero, hence $θ \in Θ_{ϵ, d} sup Ξ_{θ}^{\otimes m} (G_{θ, m}^{c}) \leq \frac{∣ C _{d} ∣}{m η ^{2}} ⟶ 0.$

Taking the supremum over $θ$ and then using the displayed estimator $Φ_{δ, m}$ in the minimax infimum yields $m \to \infty lim sup R_{m} \leq R_{⋆} + δ .$ Since $δ > 0$ was arbitrary, $m \to \infty lim sup R_{m} \leq R_{⋆} .$

Combining $R_{⋆} \leq R_{m}$ for every $m$ with the last display gives $m \to \infty lim inf R_{m} \geq R_{⋆} and m \to \infty lim sup R_{m} \leq R_{⋆} .$ Thus $lim_{m \to \infty} R_{m} = R_{⋆}$ .

∎

Lemma 27 states the population-side-information approximation directly: as the number of auxiliary treatment–covariate records grows, the finite-sample minimax risk converges to the risk obtained when the treatment–covariate marginal is observed exactly. This supplies the finite-dimensional continuity step behind the known-marginal limit in Theorem 5 and keeps the benchmark comparison on the same labeled-sample scale as the annotation experiment.

Finite-cover selection input.

The finite-cover mechanism in Lemma 26 applies to the discrete causal parametrization because the parameter set is compact and its labeled probabilities, side-law probabilities, and target are continuous functions of the cell probabilities. Concentration of the empirical side table then selects a valid neighborhood with probability tending uniformly to one, while Lemma 28 gives the complementary comparison from an empirical side sample to its exact probability vector.

Lemma 28 [lem:finite-side-measurable-comparison] (Empirical side comparison).

Let $Θ$ be a nonempty parameter set, let $Z$ and $C$ be nonempty finite sets, and let $β_{-} \leq β_{+}$ be real numbers. Suppose that, for each $θ \in Θ$ , the following objects are given:

(Observed side law.) A probability vector $λ_{θ}$ on $Z$ .

(Empirical side law.) A probability vector $ξ_{θ}$ on $C$ .

(Target.) A number $υ_{θ} \in [β_{-}, β_{+}]$ .

For each integer $m \geq 0$ , define $R_{m} = φ : Z \times C^{m} \to [β_{-}, β_{+}] in f θ \in Θ sup z \in Z \sum λ_{θ} (z) v \in C^{m} \sum (j = 1 \prod m ξ_{θ} (v_{j})) (φ (z, v) - υ_{θ})^{2} .$ Define also $R_{⋆} = ψ in f θ \in Θ sup z \in Z \sum λ_{θ} (z) (ψ (z, ξ_{θ}) - υ_{θ})^{2},$ where the infimum ranges over all maps $ψ : Z \times R^{C} \to [β_{-}, β_{+}]$ such that, for each $z \in Z$ , the section $w \mapsto ψ (z, w)$ is Borel measurable. Then, for every integer $m \geq 0$ , $R_{⋆} \leq R_{m} .$

⊢ Lean

Lemma 28 gives the corresponding monotone comparison for finite spaces: replacing an empirical side sample by its exact probability vector weakly lowers the minimax risk. In the known-marginal application, this comparison expresses the benchmark as the risk associated with observing the treatment–covariate law itself, rather than a random finite sample from that law.

A complementary asymptotic comparison spells out when auxiliary annotation improves the rate in Definition 12 relative to the labeled-only case. It expresses the improvement entirely through the alphabet growth and the effective treatment–covariate sample size.

Lemma 29 [lem:annotation-frontier-strict-improvement] (Strict frontier improvement).

Let $(m_{n})_{n \geq 1}$ and $(d_{n})_{n \geq 1}$ be sequences of nonnegative integers such that $d_{n} \geq 2$ for every $n \geq 1$ . For each $n \geq 1$ , set $N_{n} = n + m_{n}$ and $ℓ_{n} = lo g (e n)$ , and let $r_{ϵ} (n, m, d)$ be the annotation-frontier rate in Definition 12. Then, as $n \to \infty$ , $r_{ϵ} (n, m_{n}, d_{n}) = o (r_{ϵ} (n, 0, d_{n}))$ if and only if $\frac{d _{n}}{n ℓ _{n}} \to \infty$ and $\frac{d _{n} / ( N _{n} ℓ _{n} )}{min { 1 , d _{n} / ( n ℓ _{n} )}} \to 0.$

⊢ Lean

Proof of Lemma 29.

All limits are taken as $n \to \infty$ , and changing finitely many initial values is immaterial. Put $ι_{n} := \frac{1}{n}, α_{n} := \frac{d _{n}}{n ℓ _{n}}, β_{n} := \frac{d _{n}}{N _{n} ℓ _{n}} .$ For every sufficiently large $n$ , $ℓ_{n} = lo g (e n) > 0$ , $0 < ι_{n} \leq 1$ , $α_{n} > 0$ , $β_{n} \geq 0$ , and $β_{n} \leq α_{n}$ . The last inequality uses $N_{n} = n + m_{n} \geq n$ , while positivity of $α_{n}$ uses $d_{n} \geq 2$ . By Definition 12, $r_{ϵ} (n, m_{n}, d_{n}) = min {1, ι_{n} + β_{n}^{2}}, r_{ϵ} (n, 0, d_{n}) = min {1, ι_{n} + α_{n}^{2}} .$
We use the following numerical reduction. Suppose sequences $ι_{n}, α_{n}, β_{n}$ satisfy, eventually, $0 < ι_{n} \leq 1, 0 < α_{n}, 0 \leq β_{n} \leq α_{n} .$ Define $R_{n} := min {1, ι_{n} + β_{n}^{2}}, R_{n}^{0} := min {1, ι_{n} + α_{n}^{2}}, H_{n} := min {1, α_{n}} .$ Then $R_{n} = o (R_{n}^{0}) ⟺ \frac{ι _{n}}{R _{n}^{0}} \to 0 and \frac{β _{n}}{H _{n}} \to 0.$ Indeed, if $R_{n} = o (R_{n}^{0})$ , then $R_{n} / R_{n}^{0} \to 0$ . Since $ι_{n} \leq R_{n}$ and $R_{n}^{0} > 0$ eventually, $0 \leq \frac{ι _{n}}{R _{n}^{0}} \leq \frac{R _{n}}{R _{n}^{0}},$ so $ι_{n} / R_{n}^{0} \to 0$ . Also $R_{n} / R_{n}^{0} < 1$ eventually; hence $R_{n} < 1$ eventually, and therefore $R_{n} = ι_{n} + β_{n}^{2}$ eventually. Thus $0 \leq \frac{β _{n}^{2}}{R _{n}^{0}} \leq \frac{R _{n}}{R _{n}^{0}},$ so $β_{n}^{2} / R_{n}^{0} \to 0$ . To replace $R_{n}^{0}$ by $H_{n}^{2}$ , observe that $H_{n}^{2} = min {1, α_{n}^{2}},$ and hence $H_{n}^{2} \leq R_{n}^{0} \leq ι_{n} + H_{n}^{2} .$ Writing $Q_{n} := ι_{n} / R_{n}^{0}$ , we have $Q_{n} \to 0$ , and whenever $Q_{n} < 1$ , $0 \leq \frac{ι _{n}}{H _{n}^{2}} \leq \frac{Q _{n}}{1 - Q _{n}} .$ Consequently $ι_{n} / H_{n}^{2} \to 0$ , and the displayed two-sided bound gives $\frac{R _{n}^{0}}{H _{n}^{2}} \to 1.$ Therefore $\frac{β _{n}^{2}}{H _{n}^{2}} = \frac{β _{n}^{2}}{R _{n}^{0}} \frac{R _{n}^{0}}{H _{n}^{2}} \to 0,$ and taking square roots, using $β_{n} \geq 0$ and $H_{n} > 0$ , yields $β_{n} / H_{n} \to 0$ .

Conversely, assume $\frac{ι _{n}}{R _{n}^{0}} \to 0, \frac{β _{n}}{H _{n}} \to 0.$ Then $β_{n}^{2} / H_{n}^{2} \to 0$ , and since $H_{n}^{2} \leq R_{n}^{0}$ , $0 \leq \frac{β _{n}^{2}}{R _{n}^{0}} \leq \frac{β _{n}^{2}}{H _{n}^{2}} \to 0.$ Finally, $0 \leq \frac{R _{n}}{R _{n}^{0}} \leq \frac{ι _{n} + β _{n}^{2}}{R _{n}^{0}} = \frac{ι _{n}}{R _{n}^{0}} + \frac{β _{n}^{2}}{R _{n}^{0}} \to 0,$ which is exactly $R_{n} = o (R_{n}^{0})$ .
We also need the supervised floor equivalence $\frac{( 1/ n )}{r _{ϵ} ( n , 0 , d _{n} )} \to 0 ⟺ \frac{d _{n}}{n ℓ _{n}} \to \infty.$ Set $σ_{n} := \frac{d _{n}}{n ℓ _{n}} .$ For $n \geq 1$ , $r_{ϵ} (n, 0, d_{n}) = min {1, ι_{n} + α_{n}^{2}},$ and direct algebra gives $\frac{ι _{n}}{ι _{n} + α _{n}^{2}} = \frac{1}{1 + σ _{n}^{2}} .$ Assume first that $ι_{n} / r_{ϵ} (n, 0, d_{n}) \to 0$ . Given a real threshold $Y$ , put $Y_{+} := max {Y, 0} + 1.$ Eventually $\frac{ι _{n}}{r _{ϵ} ( n , 0 , d _{n} )} < \frac{1}{1 + Y _{+}^{2}} .$ Since $r_{ϵ} (n, 0, d_{n}) \leq ι_{n} + α_{n}^{2}$ , $\frac{1}{1 + σ _{n}^{2}} = \frac{ι _{n}}{ι _{n} + α _{n}^{2}} \leq \frac{ι _{n}}{r _{ϵ} ( n , 0 , d _{n} )} < \frac{1}{1 + Y _{+}^{2}} .$ Because $σ_{n} \geq 0$ , this forces $σ_{n} \geq Y_{+}$ , and hence $σ_{n} \geq Y$ eventually. Thus $σ_{n} \to \infty$ .

Conversely, if $σ_{n} \to \infty$ , then $ι_{n} \to 0$ and $1/ σ_{n}^{2} \to 0$ . With $H_{n} := min {1, α_{n}},$ the same identity $H_{n}^{2} = min {1, α_{n}^{2}}$ gives $H_{n}^{2} \leq r_{ϵ} (n, 0, d_{n}) .$ Therefore $\frac{ι _{n}}{r _{ϵ} ( n , 0 , d _{n} )} \leq \frac{ι _{n}}{H _{n}^{2}} .$ If $α_{n} \leq 1$ , then $\frac{ι _{n}}{H _{n}^{2}} = \frac{ι _{n}}{α _{n}^{2}} = \frac{1}{σ _{n}^{2}};$ if $α_{n} > 1$ , then $H_{n} = 1$ and $\frac{ι _{n}}{H _{n}^{2}} = ι_{n} \leq ι_{n} + \frac{1}{σ _{n}^{2}} .$ Thus, in all cases, $0 \leq \frac{ι _{n}}{r _{ϵ} ( n , 0 , d _{n} )} \leq ι_{n} + \frac{1}{σ _{n}^{2}} \to 0,$ which proves the equivalence.
Applying the reduction from the second step to the sequences in the first step yields $r_{ϵ} (n, m_{n}, d_{n}) = o (r_{ϵ} (n, 0, d_{n}))$ if and only if $\frac{ι _{n}}{r _{ϵ} ( n , 0 , d _{n} )} \to 0 and \frac{β _{n}}{min { 1 , α _{n} }} \to 0.$ Substituting the definitions of $α_{n}$ and $β_{n}$ , the second condition is exactly $\frac{d _{n} / ( N _{n} ℓ _{n} )}{min { 1 , d _{n} / ( n ℓ _{n} )}} \to 0.$ By the supervised floor equivalence from the third step, the first condition is exactly $\frac{d _{n}}{n ℓ _{n}} \to \infty.$ Combining the two equivalences proves the claimed characterization.

∎

Lemma 29 identifies the regimes in which the annotation rate is asymptotically smaller than the labeled-only rate. The first condition places the alphabet beyond the labeled-only parametric scale, so treatment–covariate information can affect the leading rate. The second condition requires the combined treatment–covariate sample size to reduce the alphabet-dependent term relative to its labeled-only analogue. Together these conditions give an exact rate-level criterion for when auxiliary annotation changes the asymptotic order delivered by Theorem 4.

Verification note.

The displayed theorem, proposition, lemma, definition, and assumption statements, together with their mathematical proofs, are machine-checked in Lean 4. Displayed objects marked presentation-synthesized organize or restate the checked layer for exposition. Causal interpretation, bibliographic comparison, and surrounding econometric prose are reader-facing synthesis built around that mathematical layer.

The stable statement-to-declaration manifest is presentation_crosswalk.json in this presentation directory; it records the source file, fully qualified declaration, mapping status, and sorry-free status for each mapped object. The corresponding extracted statement layer is recorded in formal_layer.json, and proof-audit receipts are recorded in proof_audit_cache.json. Verification used repository commit db544e5fa3f974dbfb320a2ecc9407c400aae9d2, Lean $4.33.0$ (release commit d8b18978322de05a8f3dba51ef03cf5461676c17), and Lake $5.0.0$ .

Proofs of the main results

Proof of Theorem 1.

Fix $0 < ϵ < 1/2$ . We prove the result with $c_{ϵ} = \frac{1}{100}, C_{ϵ} = 2 ϵ^{- 1} .$ The first constant is positive, and $c_{ϵ} \leq C_{ϵ}$ because $0 < ϵ < 1/2$ gives $\frac{1}{100} \leq \frac{2}{ϵ} ⟺ \frac{ϵ}{100} \leq 2.$ It remains to verify the two risk inequalities uniformly over $n \geq 1$ and $d \geq 2$ .

For the lower bound, put $δ := \frac{2}{5 n} .$ Then $δ \geq 0$ , $δ \leq 2/5$ , and $δ^{2} = \frac{4}{25 n} .$ Consider two laws on $[d] \times {0, 1}^{2}$ . Under $P_{\circ}$ , the covariate is uniform on $[d]$ , the propensity is $1/2$ , and both outcome regressions equal $1/2$ . Under $P_{δ}$ , the covariate is again uniform, the propensity is $1/2$ , the control regression is $1/2$ , and the treated regression is $1/2 + δ$ . Since $δ \leq 2/5$ , the perturbed treated regression lies in $[0, 1]$ . Both laws belong to $M_{d, ϵ}$ from Definition 6, because their occupied-cell propensity is $1/2$ , which satisfies the overlap condition in Assumption 1.
The two laws have the same supplied treatment-covariate table: $(P_{\circ})_{X A} (x, a) = \frac{1}{2 d} = (P_{δ})_{X A} (x, a), x \in [d], a \in {0, 1} .$ Their ATE values, computed from Definition 8, are $τ (P_{\circ}) = 0, τ (P_{δ}) = x = 1 \sum d \frac{1}{d} (\frac{1}{2} + δ - \frac{1}{2}) = δ .$ Thus, with the testing half-separation $η_{test} := \frac{δ}{2},$ we have $2 η_{test} = δ = τ (P_{\circ}) - τ (P_{δ}) .$
The product laws are sufficiently close in total variation. For one labeled observation, with the perturbed law compared to the null law, the chi-squared increment is $χ_{1}^{2} := \frac{( 1/2 ) δ ^{2}}{( 1/2 ) ( 1 - 1/2 )} = 2 δ^{2} = \frac{8}{25 n} .$ Thus $n χ_{1}^{2} = \frac{8}{25} \leq lo g 2.$ The product chi-squared divergence tensorizes as $1 + χ^{2} (P_{δ}^{\otimes n} ∥ P_{\circ}^{\otimes n}) = (1 + χ_{1}^{2})^{n} .$ Since $1 + u \leq e^{u}$ for $u \geq 0$ , the preceding calibration gives $1 + χ^{2} (P_{δ}^{\otimes n} ∥ P_{\circ}^{\otimes n}) \leq exp (n χ_{1}^{2}) \leq exp (lo g 2) = 2,$ and hence $χ^{2} (P_{δ}^{\otimes n} ∥ P_{\circ}^{\otimes n}) \leq 1.$ The perturbed product law is absolutely continuous with respect to the null product law because every one-observation atom under the null has positive mass. Therefore the standard Cauchy–Schwarz comparison between total variation and chi-squared divergence yields $TV (P_{\circ}^{\otimes n}, P_{δ}^{\otimes n}) = TV (P_{δ}^{\otimes n}, P_{\circ}^{\otimes n}) \leq \frac{1}{2} χ^{2} (P_{δ}^{\otimes n} ∥ P_{\circ}^{\otimes n}) \leq \frac{1}{2} .$
Let $T \in T_{n, d}^{known}$ be any known-marginal estimator, and define $f (L_{n}) := T (L_{n}, (P_{\circ})_{X A}) = T (L_{n}, (P_{δ})_{X A}),$ where the equality uses the common supplied table. Since the two targets are separated by $2 η_{test}$ , for every sample point at least one of $f - τ (P_{\circ}) \geq η_{test}, f - τ (P_{δ}) \geq η_{test}$ holds. The total-variation bound therefore gives $max {P_{\circ}^{\otimes n} (f - τ (P_{\circ}) \geq η_{test}), P_{δ}^{\otimes n} (f - τ (P_{δ}) \geq η_{test})} \geq \frac{1 - TV ( P _{\circ}^{\otimes n} , P _{δ}^{\otimes n} )}{2} \geq \frac{1}{4} .$ For either law $P_{⋆} \in {P_{\circ}, P_{δ}}$ , $η_{test}^{2} P_{⋆}^{\otimes n} (f - τ (P_{⋆}) \geq η_{test}) \leq E_{P_{⋆}^{\otimes n}} [(T (L_{n}, (P_{⋆})_{X A}) - τ (P_{⋆}))^{2}] .$ Taking the larger of the two risks, then the supremum over $M_{d, ϵ}$ and the infimum in Definition 15, yields $R_{ϵ}^{known} (n, d) \geq \frac{η _{test}^{2}}{4} = \frac{δ ^{2}}{16} = \frac{1}{100 n} .$
For the upper bound, take any $P \in M_{d, ϵ}$ . For a supplied table $Λ = (Λ_{x, a})_{x, a}$ , define the known-marginal score on a record $(x, a, y)$ by $ψ_{Λ} (x, a, y) := ⎩ ⎨ ⎧ (Λ_{x, 0} + Λ_{x, 1}) / Λ_{x, 1}, - (Λ_{x, 0} + Λ_{x, 1}) / Λ_{x, 0}, 0, y = 1, a = 1, y = 1, a = 0, y = 0,$ with any ratio having denominator $0$ assigned value $0$ . The raw estimator is $\overline{ψ}_{Λ} (L_{n}) := \frac{1}{n} i = 1 \sum n ψ_{Λ} (X_{i}, A_{i}, Y_{i}),$ and the estimator used in the risk is its projection to $[- 1, 1]$ . This projection is measurable on the finite sample space and weakly decreases squared loss because $τ (P) \in [- 1, 1]$ .
Now specialize the table to the true table $P_{X A}$ . Write $p_{x} = P (X = x), s_{a x} = P (X = x, A = a), q_{a x} = P (X = x, A = a, Y = 1) .$ If $p_{x} > 0$ , overlap gives $ϵ p_{x} \leq s_{1 x}, ϵ p_{x} \leq s_{0 x};$ if $p_{x} = 0$ , then $s_{0 x} = s_{1 x} = q_{0 x} = q_{1 x} = 0$ . Hence $E_{P} [ψ_{P_{X A}} (X, A, Y)] = x \sum (q_{1 x} \frac{p _{x}}{s _{1 x}} - q_{0 x} \frac{p _{x}}{s _{0 x}}) = x \sum p_{x} (μ_{1 x} - μ_{0 x}) = τ (P),$ with zero-mass cells contributing $0$ . Similarly, $E_{P} [ψ_{P_{X A}} (X, A, Y)^{2}] = x \sum {q_{1 x} (\frac{p _{x}}{s _{1 x}})^{2} + q_{0 x} (\frac{p _{x}}{s _{0 x}})^{2}} \leq x \sum {\frac{p _{x}^{2}}{s _{1 x}} + \frac{p _{x}^{2}}{s _{0 x}}} \leq x \sum 2 ϵ^{- 1} p_{x} = 2 ϵ^{- 1} .$ Thus $Var_{P} (ψ_{P_{X A}} (X, A, Y)) \leq 2 ϵ^{- 1} .$
By the labeled product sampling in Assumption 2, $E_{P^{\otimes n}} [\overline{ψ}_{P_{X A}} (L_{n})] = τ (P), Var_{P^{\otimes n}} (\overline{ψ}_{P_{X A}} (L_{n})) \leq \frac{2 ϵ ^{- 1}}{n} .$ After clipping, the squared risk is bounded by the same quantity. Taking the supremum over $P \in M_{d, ϵ}$ and then the infimum over estimators in Definition 15 gives $R_{ϵ}^{known} (n, d) \leq \frac{2 ϵ ^{- 1}}{n} = \frac{C _{ϵ}}{n} .$ Together with the lower bound, this proves the claimed inequalities for all $n \geq 1$ and $d \geq 2$ .

∎

Proof of Theorem 2.

Fix $0 < ϵ < 1/2$ .

By Lemma 14, choose a universal constant $C_{tr} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , $d \geq 2$ , and $P \in M_{d, ϵ}$ , $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq 2 R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{C _{tr}}{n} .$ By Lemma 13, choose $C_{Pois, ϵ} > 0$ , depending only on $ϵ$ , such that, for the same range of parameters and laws, $R_{hyb}^{Pois} (P; n, m, ϵ) \leq C_{Pois, ϵ} r_{ϵ} (n, m, d) .$ Set $C_{ϵ} := 2 C_{Pois, ϵ} + C_{tr} .$ Then $C_{ϵ} > 0$ .
It remains to verify the displayed bound uniformly over $P$ . Fix $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ . For any $P \in M_{d, ϵ}$ , the two estimates above give $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq 2 R_{hyb}^{Pois} (P; n, m, ϵ) + \frac{C _{tr}}{n} \leq 2 C_{Pois, ϵ} r_{ϵ} (n, m, d) + C_{tr} r_{ϵ} (n, m, d) .$ The last inequality uses the elementary rate comparison $\frac{1}{n} \leq r_{ϵ} (n, m, d),$ which follows directly from Definition 12, since $\frac{1}{n} \leq min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2} ℓ _{n}^{2}}} = r_{ϵ} (n, m, d)$ for $n \geq 1$ . Therefore $E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq (2 C_{Pois, ϵ} + C_{tr}) r_{ϵ} (n, m, d) = C_{ϵ} r_{ϵ} (n, m, d) .$
Since the preceding bound holds for every $P \in M_{d, ϵ}$ , taking the supremum over the model class yields $P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq C_{ϵ} r_{ϵ} (n, m, d) .$ The rate is positive for $n \geq 1$ by Definition 12, so the right-hand side is a well-defined finite real number. The asserted uniform bound follows for all $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ .

∎

Proof of Theorem 3.

Fix $0 < ϵ < 1/2$ . By Lemma 25, there are positive numerical constants $C_{fix} > 0, c_{risk} > 0,$ and, at this $ϵ$ , calibration constants as in Definition 19, together with $ρ \in (0, 1), c_{floor} > 0, c_{floor} \leq c_{risk} .$ Write $C_{d} > 0$ for the dimension constant in this calibration, and write $c_{gap} > 0$ for the calibration’s dual-gap constant. For every $n, m, d$ with $n \geq 1$ and $d \geq 2$ , Lemma 25 supplies a handle $H$ satisfying $L_{Π_{0} (H)} (P_{X A}) = L_{Π_{1} (H)} (P_{X A})$ and the floor certificate $c_{floor} \frac{1}{n} \leq R_{ϵ} (n, m, d) .$ For this handle, put $D (n, m, d) := min {1, \frac{d ^{2}}{( n + m ) ^{2} lo g ( e n ) ^{2}}}$ and $Δ_{H} := c_{gap} k (H) a (H),$ where $k (H)$ and $a (H)$ are the rare-cell count and shift in Definition 19. In the finite-atom branch, the same cited certificate gives the nonparametric dimension comparison $C_{d} D (n, m, d) \leq Δ_{H}^{2}$ and the fixed-sample lower bound attached to the canonical handle, $c_{floor} Δ_{H}^{2} \leq R_{ϵ} (n, m, d) .$ The displayed lower bound follows once the handle’s common-marginal witness, total-variation bound, center separation, and variance absorption satisfy the hypotheses of Lemma 21. In the complementary branch, Lemma 25 gives directly $C_{d} D (n, m, d) \leq \frac{c _{floor}}{n} .$
Define $c_{lb} := min {\frac{c _{floor}}{2}, \frac{C _{d}}{2}, \frac{c _{floor} C _{d}}{2}} .$ Since $c_{floor} > 0$ and $C_{d} > 0$ , this constant is positive.
Fix $n, m, d$ with $n \geq 1$ and $d \geq 2$ , and take the handle $H$ from the certificate package. Put $Q (n, m, d) := \frac{d ^{2}}{( n + m ) ^{2} lo g ( e n ) ^{2}}, D (n, m, d) = min {1, Q (n, m, d)} .$ Both $Q (n, m, d)$ and $D (n, m, d)$ are nonnegative. The frontier rate in Definition 12 satisfies $r_{ϵ} (n, m, d) = min {1, \frac{1}{n} + Q (n, m, d)} \leq \frac{1}{n} + D (n, m, d) .$ Indeed, if $Q (n, m, d) \leq 1$ , the right side is $1/ n + Q (n, m, d)$ , which dominates the minimum. If $Q (n, m, d) > 1$ , then $D (n, m, d) = 1$ , and the left side is at most $1 \leq 1/ n + 1$ .
Suppose first that $H$ is in the finite-atom branch. Since $c_{floor} > 0$ , multiplying the finite-branch dimension comparison by $c_{floor}$ and then using the transferred fixed-sample lower bound gives $c_{floor} C_{d} D (n, m, d) \leq c_{floor} Δ_{H}^{2} \leq R_{ϵ} (n, m, d) .$ Together with the floor inequality, $c_{floor} \frac{1}{n} \leq R_{ϵ} (n, m, d),$ this supplies the two lower bounds used in the averaging step. Using the definition of $c_{lb}$ and the bound from the previous step, $c_{lb} r_{ϵ} (n, m, d) \leq c_{lb} (\frac{1}{n} + D (n, m, d)) = c_{lb} \frac{1}{n} + c_{lb} D (n, m, d) \leq \frac{c _{floor}}{2} \frac{1}{n} + \frac{c _{floor} C _{d}}{2} D (n, m, d) = \frac{1}{2} (c_{floor} \frac{1}{n} + c_{floor} C_{d} D (n, m, d)) \leq R_{ϵ} (n, m, d) .$ Thus the asserted lower bound holds in the finite-atom branch.
It remains to treat the complementary branch. In this branch, the certificate clause in Lemma 25 gives directly $C_{d} D (n, m, d) \leq \frac{c _{floor}}{n} = c_{floor} \frac{1}{n} .$ Together with the floor inequality $c_{floor} \frac{1}{n} \leq R_{ϵ} (n, m, d),$ we obtain $c_{lb} r_{ϵ} (n, m, d) \leq c_{lb} (\frac{1}{n} + D (n, m, d)) = c_{lb} \frac{1}{n} + c_{lb} D (n, m, d) \leq \frac{c _{floor}}{2} \frac{1}{n} + \frac{C _{d}}{2} D (n, m, d) \leq \frac{1}{2} (c_{floor} \frac{1}{n} + c_{floor} \frac{1}{n}) = c_{floor} \frac{1}{n} \leq R_{ϵ} (n, m, d) .$ This proves the risk lower bound in the complementary branch as well.
Combining the two branches, for every $n, m, d$ with $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , the handle $H$ supplies priors $Π_{0} (H)$ and $Π_{1} (H)$ with common random treatment-covariate marginal law and satisfies $c_{lb} r_{ϵ} (n, m, d) \leq R_{ϵ} (n, m, d) .$ Taking the theorem’s lower-bound constant $c_{ϵ}$ to be $c_{lb}$ , and using $c_{lb} > 0$ with the fixed calibration constants chosen above, gives the asserted conclusion.

∎

Proof of Theorem 4.

By Theorem 3, choose a constant $c_{ϵ} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , $c_{ϵ} r_{ϵ} (n, m, d) \leq R_{ϵ} (n, m, d) .$ By Theorem 2, choose $C_{ϵ}^{up} > 0$ such that $P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] \leq C_{ϵ}^{up} r_{ϵ} (n, m, d)$ for the same range of $n, m, d$ . Set $C_{ϵ} := max {c_{ϵ}, C_{ϵ}^{up}} .$ Then $0 < c_{ϵ} \leq C_{ϵ}$ . The estimator $τ_{n, m, d}^{mix}$ is measurable, so the definition of $R_{ϵ} (n, m, d)$ in Definition 11 gives $R_{ϵ} (n, m, d) \leq P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{mix} - τ (P)}^{2}] .$ Combining the last two displays, and using $r_{ϵ} (n, m, d) > 0$ , yields $R_{ϵ} (n, m, d) \leq C_{ϵ}^{up} r_{ϵ} (n, m, d) \leq C_{ϵ} r_{ϵ} (n, m, d) .$ Together with the lower bound from Theorem 3, this proves the finite-sample bracket.
The unannotated endpoint is the specialization $m = 0$ of Definition 12. Since $N = n$ in that case, $r_{ϵ} (n, 0, d) = min {1, \frac{1}{n} + \frac{d ^{2}}{n ^{2} lo g ^{2} ( e n )}} .$
The known-marginal limit is exactly the conclusion of Theorem 5: for fixed $n \geq 1$ and $d \geq 2$ , $m \to \infty lim R_{ϵ} (n, m, d) = R_{ϵ}^{known} (n, d) .$
It remains to record the sequence consequences. Fix sequences $(m_{n})$ and $(d_{n})$ with $d_{n} \geq 2$ , and set $ρ_{n} := R_{ϵ} (n, m_{n}, d_{n}), q_{n} := r_{ϵ} (n, m_{n}, d_{n}),$ $ρ_{n}^{0} := R_{ϵ} (n, 0, d_{n}), q_{n}^{0} := r_{ϵ} (n, 0, d_{n}) .$ The finite-sample bracket gives, for all $n \geq 1$ , $c_{ϵ} q_{n} \leq ρ_{n} \leq C_{ϵ} q_{n}, c_{ϵ} q_{n}^{0} \leq ρ_{n}^{0} \leq C_{ϵ} q_{n}^{0} .$ Since all four sequences are nonnegative, these two-sided bounds imply $(ρ_{n}) = Θ (q_{n}), (ρ_{n}^{0}) = Θ (q_{n}^{0}) .$ Indeed, the upper inequalities give the $O$ -bounds in one direction, and the lower inequalities give $q_{n} \leq c_{ϵ}^{- 1} ρ_{n}$ and $q_{n}^{0} \leq c_{ϵ}^{- 1} ρ_{n}^{0}$ .

With $N_{n} = n + m_{n}, b_{n} := \frac{d _{n}}{N _{n} ℓ _{n}},$ Definition 12 gives $q_{n} = min {1, \frac{1}{n} + b_{n}^{2}} .$ Because $n^{- 1} \to 0$ , this display satisfies $q_{n} \to 0 ⟺ b_{n} \to 0.$ The implication from right to left follows by continuity of $x \mapsto min {1, x}$ ; the implication from left to right follows because eventually $q_{n} < 1/2$ , hence $q_{n} = \frac{1}{n} + b_{n}^{2}$ eventually, so $b_{n}^{2} \to 0$ . Using $(ρ_{n}) = Θ (q_{n})$ , we obtain $R_{ϵ} (n, m_{n}, d_{n}) \to 0 ⟺ \frac{d _{n}}{N _{n} ℓ _{n}} \to 0.$

Next, $q_{n} = O (n^{- 1}) ⟺ b_{n}^{2} = O (n^{- 1}) ⟺ d_{n} = O (\frac{N _{n} ℓ _{n}}{n}) .$ For the first equivalence, the term $n^{- 1}$ is already present in $q_{n}$ , and the same eventual equality $q_{n} = n^{- 1} + b_{n}^{2}$ applies after the $O (n^{- 1})$ bound forces $q_{n} \to 0$ . The second equivalence is obtained by taking square roots of nonnegative sequences and multiplying by $N_{n} ℓ_{n}$ . Transferring through $(ρ_{n}) = Θ (q_{n})$ gives $R_{ϵ} (n, m_{n}, d_{n}) = O (n^{- 1}) ⟺ d_{n} = O (\frac{N _{n} ℓ _{n}}{n}) .$

Finally put $a_{n} := \frac{d _{n}}{n ℓ _{n}} .$ Then $q_{n} = min {1, \frac{1}{n} + b_{n}^{2}}, q_{n}^{0} = min {1, \frac{1}{n} + a_{n}^{2}}, 0 \leq b_{n} \leq a_{n} .$ By Lemma 29, the numerical strict-improvement criterion for these two rates is $q_{n} = o (q_{n}^{0})$ if and only if both $\frac{d _{n}}{n ℓ _{n}} \to \infty$ and $\frac{b _{n}}{min { 1 , a _{n} }} \to 0.$ To see this, observe first that $q_{n} \geq n^{- 1}$ , so $q_{n} = o (q_{n}^{0})$ forces $n^{- 1} = o (q_{n}^{0})$ , which is equivalent to $a_{n} n = d_{n} / (n ℓ_{n}) \to \infty$ . Under this condition the floor term $n^{- 1}$ is negligible in $q_{n}^{0}$ , and $q_{n}^{0} ≍ min {1, a_{n}^{2}} .$ Since $0 \leq b_{n} \leq a_{n}$ , the remaining requirement $min {1, n^{- 1} + b_{n}^{2}} = o (q_{n}^{0})$ is equivalent to $b_{n}^{2} = o (min {1, a_{n}^{2}}),$ which, for nonnegative sequences, is the same as $\frac{b _{n}}{min { 1 , a _{n} }} \to 0.$ Substituting the definitions of $a_{n}$ and $b_{n}$ , this last condition is exactly $\frac{d _{n} / ( N _{n} ℓ _{n} )}{min { 1 , d _{n} / ( n ℓ _{n} )}} \to 0.$ The two-sided bounds $(ρ_{n}) = Θ (q_{n})$ and $(ρ_{n}^{0}) = Θ (q_{n}^{0})$ transfer $q_{n} = o (q_{n}^{0})$ precisely to $R_{ϵ} (n, m_{n}, d_{n}) = o (R_{ϵ} (n, 0, d_{n})),$ which proves the stated strict-improvement equivalence.

∎

Proof of Theorem 5.

Let $Z_{n} := ([d] \times {0, 1}^{2})^{n}, C_{d} := [d] \times {0, 1} .$ We use the compact coordinate space $Θ_{ϵ, d} = {(π, ϱ, ω) : π \in Prob ([d]), ϱ_{x} \in [ϵ, 1 - ϵ], ω_{α x} \in [0, 1]} .$ The assumptions $d \geq 2$ and $0 < ϵ < 1/2$ make this space nonempty. For $θ = (π, ϱ, ω) \in Θ_{ϵ, d}$ , set $P_{θ} (x, α, y) = π_{x} ϱ_{x}^{α} (1 - ϱ_{x})^{1 - α} ω_{α x}^{y} (1 - ω_{α x})^{1 - y},$ and define $λ_{θ} (z) = i = 1 \prod n P_{θ} (z_{i}), ξ_{θ} (x, α) = y \in {0, 1} \sum P_{θ} (x, α, y), υ_{θ} = x \in [d] \sum π_{x} (ω_{1 x} - ω_{0 x}) .$ Also, $- 1 = x \sum π_{x} (- 1) \leq x \sum π_{x} (ω_{1 x} - ω_{0 x}) \leq x \sum π_{x} (1) = 1,$ so $υ_{θ} \in [- 1, 1]$ for every $θ$ .

Define the finite-side values $R_{m} = φ : Z_{n} \times C_{d}^{m} \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup z \in Z_{n} \sum λ_{θ} (z) u \in C_{d}^{m} \sum (j = 1 \prod m ξ_{θ} (u_{j})) (φ (z, u) - υ_{θ})^{2}$ and $R_{⋆} = ψ : Z_{n} \times Prob (C_{d}) \to [- 1, 1] in f θ \in Θ_{ϵ, d} sup z \in Z_{n} \sum λ_{θ} (z) (ψ (z, ξ_{θ}) - υ_{θ})^{2} .$ With the preceding coordinates, Lemma 27 gives $m \to \infty lim R_{m} = R_{⋆} .$
Let $R_{⋆}$ be the measurable ambient exact-table value $R_{⋆} = ψ in f θ \in Θ_{ϵ, d} sup z \in Z_{n} \sum λ_{θ} (z) (ψ (z, ξ_{θ}) - υ_{θ})^{2},$ where the infimum ranges over all rules $ψ : Z_{n} \times R^{C_{d}} \to [- 1, 1]$ whose section $w \mapsto ψ (z, w)$ is Borel measurable for each $z$ . The target range established above gives $υ_{θ} \in [- 1, 1]$ , and Lemma 27 supplies the convergence $m \to \infty lim R_{m} = R_{⋆} .$ The measurable ambient exact-table value is tied to this limiting sequence by two comparisons. The exact-side value infimizes over all rules on $Prob (C_{d})$ , while the ambient value infimizes over rules on $R^{C_{d}}$ with Borel-measurable sections. Restricting any ambient rule to $Prob (C_{d})$ gives an admissible exact-side rule, so the exact-side class is at least as large after restriction, and therefore $R_{⋆} \leq R_{⋆} .$ Conversely, Lemma 28 gives $R_{⋆} \leq R_{m}$ for every $m \geq 0$ . For completeness, its measurable conditional-average construction in the present coordinates is as follows. Fix an integer $m \geq 0$ and an empirical-side rule $φ : Z_{n} \times C_{d}^{m} \to [- 1, 1]$ . Define first the polynomial conditional average on the ambient table space by $ψ_{φ}^{\circ} (z, w) = u \in C_{d}^{m} \sum (j = 1 \prod m w (u_{j})) φ (z, u), w \in R^{C_{d}} .$ This function is Borel measurable in $w$ for each fixed $z$ . Use its clipped version as the ambient exact-table rule, $ψ_{φ} (z, w) = min {1, max {- 1, ψ_{φ}^{\circ} (z, w)}}, w \in R^{C_{d}} .$ Then $ψ_{φ} : Z_{n} \times R^{C_{d}} \to [- 1, 1]$ is admissible for $R_{⋆}$ . When $w = r \in Prob (C_{d})$ , the coefficients $\prod_{j = 1}^{m} r (u_{j})$ form a probability vector on $C_{d}^{m}$ , and since $φ$ takes values in $[- 1, 1]$ , the polynomial average already lies in $[- 1, 1]$ . Hence, on probability tables, $ψ_{φ} (z, r) = u \in C_{d}^{m} \sum (j = 1 \prod m r (u_{j})) φ (z, u) .$ For every $θ$ and $z$ , Jensen’s inequality for the product law $ξ_{θ}^{\otimes m}$ gives $(ψ_{φ} (z, ξ_{θ}) - υ_{θ})^{2} = u \in C_{d}^{m} \sum (j = 1 \prod m ξ_{θ} (u_{j})) (φ (z, u) - υ_{θ})^{2} \leq u \in C_{d}^{m} \sum (j = 1 \prod m ξ_{θ} (u_{j})) (φ (z, u) - υ_{θ})^{2} .$ Multiplying by $λ_{θ} (z)$ , summing over $z$ , and then taking the supremum over $θ$ gives $θ sup z \in Z_{n} \sum λ_{θ} (z) (ψ_{φ} (z, ξ_{θ}) - υ_{θ})^{2} \leq θ sup z \in Z_{n} \sum λ_{θ} (z) u \in C_{d}^{m} \sum (j = 1 \prod m ξ_{θ} (u_{j})) (φ (z, u) - υ_{θ})^{2} .$ Taking the infimum over $φ$ yields $R_{⋆} \leq R_{m} for every m \geq 0.$ Taking the limit along the displayed convergence yields $R_{⋆} \leq m \to \infty lim R_{m} = R_{⋆} .$ The two inequalities identify the ambient measurable exact-table value with the exact-side value: $R_{⋆} = R_{⋆} .$
We use here the standing theorem hypotheses $d \geq 2$ and $0 < ϵ < 1/2$ , which are also the parameter-range clauses in the definition of $M_{d, ϵ}$ . Under these hypotheses, the compact coordinates parametrize the laws in $M_{d, ϵ}$ from Definition 6. For every $θ \in Θ_{ϵ, d}$ , the law $P_{θ}$ satisfies occupied-cell overlap because, whenever $π_{x} > 0$ , its treatment propensity is $\frac{\sum _{y} P _{θ} ( x , 1 , y )}{\sum _{α, y} P _{θ} ( x , α , y )} = ϱ_{x} \in [ϵ, 1 - ϵ] .$ Conversely, given any $P \in M_{d, ϵ}$ , choose coordinates by $π_{x} = p_{x}, ϱ_{x} = {e_{x}, ϵ, p_{x} > 0, p_{x} = 0, ω_{α x} = μ_{α x} .$ The value of $ϱ_{x}$ in an empty cell is immaterial because every atom in that cell is multiplied by $π_{x} = 0$ . For occupied cells, the overlap condition places $ϱ_{x}$ in $[ϵ, 1 - ϵ]$ , and the binary outcome regressions place $ω_{α x}$ in $[0, 1]$ . These coordinates recover the atom probabilities, $P (x, α, y) = π_{x} ϱ_{x}^{α} (1 - ϱ_{x})^{1 - α} ω_{α x}^{y} (1 - ω_{α x})^{1 - y},$ with the empty-cell case giving zero on both sides. Thus the map $θ \mapsto P_{θ}$ is surjective onto the model class.

Under this parametrization, the three quantities entering the known-marginal risk are transported by $P_{θ}^{\otimes n} (z) = λ_{θ} (z), (P_{θ})_{X A} (x, α) = ξ_{θ} (x, α), τ (P_{θ}) = υ_{θ} .$ The last identity follows from Definition 8: on occupied cells the outcome regressions are $ω_{1 x}$ and $ω_{0 x}$ , while empty cells have coefficient $π_{x} = 0$ . Hence, for every bounded measurable known-table rule $ψ$ , $\int (ψ (Z, (P_{θ})_{X A}) - τ (P_{θ}))^{2} d P_{θ}^{\otimes n} (Z) = z \in Z_{n} \sum λ_{θ} (z) (ψ (z, ξ_{θ}) - υ_{θ})^{2} .$ The known-table sample space in Definition 15 is $Z_{n} \times R^{C_{d}}$ . Its bounded measurable procedures are in bijection with the ambient procedures used in $R_{⋆}$ . Explicitly, a paper procedure $f$ is sent to $ψ_{f} (z, w) := f (z, w),$ and an ambient procedure $ψ$ is sent back to $f_{ψ} (z, w) := ψ (z, w)$ . Because $Z_{n}$ is finite and discrete, measurability of every section $w \mapsto ψ (z, w)$ is equivalent to measurability of the uncurried map; the two assignments are pointwise inverse and preserve the range $[- 1, 1]$ .

For corresponding $f$ and $ψ_{f}$ , evaluation at the known table $w = (P_{θ})_{X A} = ξ_{θ}$ , followed by expansion of the finite labeled product law, gives $\int (f (Z, (P_{θ})_{X A}) - τ (P_{θ}))^{2} d P_{θ}^{\otimes n} (Z) = z \in Z_{n} \sum λ_{θ} (z) (ψ_{f} (z, ξ_{θ}) - υ_{θ})^{2} .$ Thus the estimator bijection preserves every coordinate risk exactly. Reindexing the supremum over $P \in M_{d, ϵ}$ by the surjective coordinate map, and then taking the same infimum over the bijective procedure classes, gives $R_{ϵ}^{known} (n, d) = R_{⋆} .$ Together with the equality $R_{⋆} = R_{⋆}$ above, this gives $R_{ϵ}^{known} (n, d) = R_{⋆} .$
For each $m$ , the minimax risk in Definition 11 equals the empirical finite-side value: $R_{ϵ} (n, m, d) = R_{m} .$ First reindex the supremum over $P \in M_{d, ϵ}$ by the surjective map $θ \mapsto P_{θ}$ . The annotation sample space is the finite product $Z_{n} \times C_{d}^{m}$ , so every real-valued map on it is measurable. For any measurable estimator $T$ in the annotation experiment, write its two-sample mean-squared risk at $P_{θ}$ as $V_{T} (θ) = \int (T (Z, U) - τ (P_{θ}))^{2} d (P_{θ}^{\otimes n} \otimes (P_{θ})_{X A}^{\otimes m}) (Z, U) .$ Expanding both finite product measures, and using $(P_{θ})_{X A} = ξ_{θ}$ , $P_{θ}^{\otimes n} = λ_{θ}$ , and $τ (P_{θ}) = υ_{θ}$ , gives the exact table identity $V_{T_{φ}} (θ) = z \in Z_{n} \sum λ_{θ} (z) u \in C_{d}^{m} \sum (j = 1 \prod m ξ_{θ} (u_{j})) (φ (z, u) - υ_{θ})^{2}$ for $T_{φ} (z, u) := φ (z, u)$ .

Thus every empirical-side procedure $φ : Z_{n} \times C_{d}^{m} \to [- 1, 1]$ induces the paper estimator $T_{φ} (z, u) = φ (z, u),$ viewed as an estimator in the annotation experiment. For this induced estimator, the preceding expansion gives equality of the coordinate risks for every $θ$ , hence equality of the corresponding worst-case risks after the reindexing by $θ$ . Since the paper infimum ranges over all measurable estimators, including these induced estimators, taking the infimum over $φ$ gives $R_{ϵ} (n, m, d) \leq R_{m} .$ For the reverse inequality, let $T$ be any measurable paper estimator and define the clipped empirical-side rule $φ_{T} (z, u) = min {1, max {- 1, T (z, u)}} .$ The transported identity $τ (P_{θ}) = υ_{θ}$ , together with the target-range bound, gives the pointwise clipping inequality $(φ_{T} (z, u) - τ (P_{θ}))^{2} \leq (T (z, u) - τ (P_{θ}))^{2} for all (z, u), θ .$ Integrating this inequality against $P_{θ}^{\otimes n} \otimes (P_{θ})_{X A}^{\otimes m}$ yields $\int (φ_{T} (Z, U) - τ (P_{θ}))^{2} d (P_{θ}^{\otimes n} \otimes (P_{θ})_{X A}^{\otimes m}) (Z, U) \leq V_{T} (θ) .$ By the same finite product expansion as above, the left side is exactly the empirical-side table risk of $φ_{T}$ at $θ$ . The comparison of worst-case risks also requires a finite bound for the unclipped estimator. Set $B_{T} := (z, u) \in Z_{n} \times C_{d}^{m} \sum (∣ T (z, u) ∣ + 1)^{2} .$ Since $∣ υ_{θ} ∣ \leq 1$ , $(T (z, u) - υ_{θ})^{2} \leq (∣ T (z, u) ∣ + 1)^{2} \leq B_{T}$ for every $(z, u)$ and $θ$ . Hence the worst-case risk of $T$ over the reindexed model is bounded, and the integrated clipping inequality implies $θ \in Θ_{ϵ, d} sup \int (φ_{T} (Z, U) - τ (P_{θ}))^{2} d (P_{θ}^{\otimes n} \otimes (P_{θ})_{X A}^{\otimes m}) (Z, U) \leq θ \in Θ_{ϵ, d} sup V_{T} (θ) .$ Taking infima over all measurable paper estimators $T$ gives $R_{m} \leq R_{ϵ} (n, m, d) .$ The two inequalities prove $R_{ϵ} (n, m, d) = R_{m}$ .
The preceding displayed identities give $m \to \infty lim R_{ϵ} (n, m, d) = m \to \infty lim R_{m} = R_{⋆} = R_{ϵ}^{known} (n, d) .$

∎

Proof of Theorem 6.

By Lemma 2, choose $C_{0} > 0$ , depending only on $ϵ$ , such that for every $d$ , every $u, t > 0$ , and every $P \in M_{d, ϵ}$ , $Var_{P} (Z^{H}) \leq C_{0} (u^{- 1} + t^{- 1}), E_{P} Z^{H} - τ (P) \leq C_{0} min {1, d / t} .$ Hence, in the independent Poisson count experiment, $E_{P} {Z^{H} - τ (P)}^{2} = Var_{P} (Z^{H}) + {E_{P} Z^{H} - τ (P)}^{2} \leq C_{0} (u^{- 1} + t^{- 1}) + C_{0}^{2} min {1, d / t}^{2} .$ Set $C_{ϵ} = 100000 (1 + C_{0} + C_{0}^{2}) > 0.$
Fix $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ . Let $M_{0} = ⌊ n /3 ⌋, n_{p} = ⌊ \frac{n - M _{0}}{2} ⌋, n_{f} = n - M_{0} - n_{p}, m_{f} = m - ⌊ m /2 ⌋,$ and put $u = \frac{M _{0}}{8}, t = \frac{n _{f} + m _{f}}{8}, q = \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}} .$ The estimator is a finite sum of measurable prefix statistics with deterministic weights, so it is measurable. The class $M_{d, ϵ}$ is nonempty: the law with uniform covariate distribution, propensity $1/2$ , and equal outcome regressions in the two arms satisfies Assumption 1 because $0 < ϵ < 1/2$ .
First suppose $24 \leq n$ . The block arithmetic gives $n \leq 4 M_{0}, n + m \leq 4 (n_{f} + m_{f}),$ and therefore $M_{0} > 0$ , $n_{f} + m_{f} > 0$ , $u > 0$ , and $t > 0$ . Consequently $\frac{n}{32} \leq u, \frac{n + m}{32} \leq t,$ so $u^{- 1} \leq \frac{32}{n}, t^{- 1} \leq \frac{32}{n + m} \leq \frac{32}{n} .$ Also, $\frac{d}{t} \leq \frac{32 d}{n + m},$ and hence $min {1, d / t}^{2} \leq (d / t)^{2} \leq 1024 \frac{d ^{2}}{( n + m ) ^{2}} .$
For the Poisson prefix overflow terms, the elementary bound $Pr {Poi (Λ_{tail} /8) > Λ_{tail}} \leq \frac{8}{Λ _{tail}}, Λ_{tail} \geq 1,$ is obtained as follows. Bernstein’s upper-tail inequality for a Poisson variable with mean $Λ_{tail} /8$ , applied with deviation parameter $Λ_{tail} /4$ , yields $Pr {Poi (Λ_{tail} /8) > Λ_{tail}} \leq exp (- Λ_{tail} /4) .$ Since $exp (x) \geq 1 + x$ , $exp (- Λ_{tail} /4) = \frac{1}{exp ( Λ _{tail} /4 )} \leq \frac{1}{Λ _{tail} /4} = \frac{4}{Λ _{tail}} \leq \frac{8}{Λ _{tail}} .$ Applying this with $Λ_{tail} = M_{0}$ and $Λ_{tail} = n_{f} + m_{f}$ gives $Pr {Poi (u) > M_{0}} \leq \frac{8}{M _{0}}, Pr {Poi (t) > n_{f} + m_{f}} \leq \frac{8}{n _{f} + m _{f}} .$ Together with $n \leq 4 M_{0}$ and $n + m \leq 4 (n_{f} + m_{f})$ , this implies $\frac{8}{M _{0}} \leq \frac{32}{n}, \frac{8}{n _{f} + m _{f}} \leq \frac{32}{n + m} \leq \frac{32}{n} .$
Fix $P \in M_{d, ϵ}$ . Under the probability law $P_{P, u, t}^{pref}$ , draw two independent finite prefixes: an outcome-marked prefix from $P$ with Poisson length of mean $u$ , and a treatment–covariate prefix from $P_{X A}$ with Poisson length of mean $t$ . If $S_{a x}$ is the number of outcome-marked prefix records with $(X, A, Y) = (x, a, 1)$ and $K_{a x}$ is the number of treatment–covariate prefix records with $(X, A) = (x, a)$ , set $τ_{u, t}^{IC, pref} = clip_{[- 1, 1]} [x = 1 \sum d {\frac{S _{1 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{1 x} + 1} - \frac{S _{0 x}}{u} \frac{K _{0 x} + K _{1 x} + 1}{K _{0 x} + 1}}],$ and define $R_{pref} (P; u, t) := E_{P_{P, u, t}^{pref}} [{τ_{u, t}^{IC, pref} - τ (P)}^{2}] .$ The prefix transfer in Lemma 4, applied with the present block sizes and intensities, gives $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq R_{pref} (P; u, t) + 4 [Pr {Poi (u) > M_{0}} + Pr {Poi (t) > n_{f} + m_{f}}] .$ Here the fixed-sample estimator is the finite Rao–Blackwell average over admissible prefix lengths in Definition 20; the displayed inequality accounts for the two overflow events, and the coefficient $4$ is the squared diameter of $[- 1, 1]$ for the clipped statistic and the target $τ (P)$ .
The Poisson-prefix counts have the Poisson count law by Lemma 3: for every $x \in [d]$ and $a \in {0, 1}$ , $S_{a x} \sim Poi (u q_{a x}), K_{a x} \sim Poi (t s_{a x}),$ and these coordinates are mutually independent. Thus the pushforward of $P_{P, u, t}^{pref}$ under the count map is exactly the product law used in Lemma 2. We first record the range of the target. The cell masses satisfy $p_{x} \geq 0$ and $\sum_{x = 1}^{d} p_{x} = 1$ , and the binary-outcome regressions satisfy $0 \leq μ_{a x} \leq 1$ . Hence $∣ τ (P) ∣ = x = 1 \sum d p_{x} (μ_{1 x} - μ_{0 x}) \leq x = 1 \sum d p_{x} ∣ μ_{1 x} - μ_{0 x} ∣ \leq x = 1 \sum d p_{x} = 1.$ Thus $τ (P) \in [- 1, 1]$ . Since projection onto $[- 1, 1]$ cannot increase distance from a point of $[- 1, 1]$ , clipping gives $R_{pref} (P; u, t) \leq E_{P} {Z^{H} - τ (P)}^{2} .$ Combining this with Step 1 gives $R_{pref} (P; u, t) \leq C_{0} (u^{- 1} + t^{- 1}) + C_{0}^{2} min {1, d / t}^{2} .$
The preceding displays imply $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq (64 C_{0} + 1024 C_{0}^{2} + 256) {\frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$ Indeed, the variance contribution is bounded by $C_{0} (u^{- 1} + t^{- 1}) \leq 64 C_{0} \frac{1}{n},$ the squared-bias contribution by $C_{0}^{2} min {1, d / t}^{2} \leq 1024 C_{0}^{2} \frac{d ^{2}}{( n + m ) ^{2}},$ and the overflow contribution by $4 (\frac{8}{M _{0}} + \frac{8}{n _{f} + m _{f}}) \leq 256 \frac{1}{n} .$ Since $64 C_{0} + 1024 C_{0}^{2} + 256 \leq C_{ϵ},$ the preceding display gives the large-sample estimate $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq C_{ϵ} q .$ If $q \leq 1$ , then $min {1, q} = q$ , and hence $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq C_{ϵ} min {1, q} .$ When $q > 1$ , both the estimator and $τ (P)$ lie in $[- 1, 1]$ , so $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq 4 \leq C_{ϵ} min {1, q} .$
It remains to cover $n < 24$ . Then $n \leq 23$ , and $\frac{1}{23} \leq \frac{1}{n} \leq q, \frac{1}{23} \leq min {1, q} .$ Again using the boundedness of the estimator and of $τ (P)$ , $E_{P} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq 4 \leq C_{ϵ} min {1, q},$ because $C_{ϵ} \geq 92$ .
Taking the supremum over $P \in M_{d, ϵ}$ in the preceding bound gives $P \in M_{d, ϵ} sup E_{P^{\otimes n} \otimes P_{X A}^{\otimes m}} [{τ_{n, m, d}^{IC} - τ (P)}^{2}] \leq C_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$ Finally, $τ_{n, m, d}^{IC}$ is an admissible measurable estimator in the class defining $R_{ϵ} (n, m, d)$ in Definition 11. Therefore $R_{ϵ} (n, m, d) \leq C_{ϵ} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$

∎

Proof of Theorem 7.

Fix $0 < ϵ < 1/2$ .

By Lemma 22, choose $c_{lab} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , $\frac{c _{lab}}{n} \leq R_{ϵ} (n, m, d) .$ By Theorem 6, choose $C_{IC} > 0$ such that, for every $n \geq 1$ , $m \geq 0$ , and $d \geq 2$ , $R_{ϵ} (n, m, d) \leq C_{IC} min {1, \frac{1}{n} + \frac{d ^{2}}{( n + m ) ^{2}}} .$ Set $c_{ϵ} := c_{lab}, C_{ϵ} := max {c_{lab}, 5 C_{IC}} .$ Then $0 < c_{ϵ} \leq C_{ϵ}$ .
For $d = 2$ , the lower bound supplied by Lemma 22 gives, for every $n \geq 1$ and $m \geq 0$ , $\frac{c _{ϵ}}{n} = \frac{c _{lab}}{n} \leq R_{ϵ} (n, m, 2) .$
It remains to bound the right-hand side. Fix $n \geq 1$ and $m \geq 0$ , and put $N := n + m .$ Then $N \geq n \geq 1$ , hence $N > 0$ , and $n \leq N \leq N^{2},$ because $N \geq 1$ . Therefore $\frac{4}{N ^{2}} \leq \frac{4}{n} .$ Using Theorem 6 with $d = 2$ , $R_{ϵ} (n, m, 2) \leq C_{IC} min {1, \frac{1}{n} + \frac{4}{N ^{2}}} \leq C_{IC} (\frac{1}{n} + \frac{4}{N ^{2}}) \leq C_{IC} \frac{5}{n} = \frac{5 C _{IC}}{n} \leq \frac{C _{ϵ}}{n} .$ This proves the asserted two-sided bound for all $n \geq 1$ and $m \geq 0$ .

∎

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Formal layer — Lean correspondence

Every object below is formalized in Lean 4. Click a verified row to view its statement; the rest link to the Lean source.

Setups (3)

Setup S-1 — A two-sample semi-supervised discrete causal experiment. A known covariate alphabet $[d]=\{1,\ldots,d\}$ with $d\ge2$ , binary treatment $A\in\{0,1\}$ and binary outcome $Y\in\{0,1\}$ give the complete record space $[d]\times\{0,1\}^2$ . A law $P\in\mathsf{Prob}([d]\times\{0,1\}^2)$ determines cell masses $p_x=P(X=x)$ , propensities $e_x=P(A=1\mid X=x)$ on occupied cells, outcome regressions $\mu_{ax}=E_P[Y\mid A=a,X=x]$ , treatment--covariate cell masses $s_{ax}=P(X=x,A=a)$ , outcome-marked masses $q_{ax}=s_{ax}\mu_{ax}$ , and the target $\tau(P)=\sum_{x=1}^d p_x(\mu_{1x}-\mu_{0x})\in[-1,1]$ . For a known overlap constant $0<\epsilon<1/2$ , the unrestricted class $\mathcal M_{d,\epsilon}$ collects the laws with $\epsilon\le e_x\le1-\epsilon$ on every occupied cell; no minimum cell mass is imposed and null cells are permitted. The observation experiment $\mathcal E_{n,m,d,\epsilon}$ supplies $n\ge1$ complete labeled records $\mathcal L_n\sim P^{\otimes n}$ together with $m\ge0$ outcome-unlabeled records $\mathcal U_m\sim P_{XA}^{\otimes m}$ from the same population, the two samples being independent, so that outcome-bearing information has size $n$ while treatment--covariate information has size $N=n+m$ . Estimators $\mathcal T_{n,m,d}$ are the measurable real functions of $(\mathcal L_n,\mathcal U_m)$ ; the minimax risk is $R_\epsilon(n,m,d)=\inf_{T\in\mathcal T_{n,m,d}}\sup_{P\in\mathcal M_{d,\epsilon}}E[(T-\tau(P))^2]$ , compared with the explicit annotation-frontier rate $r_\epsilon(n,m,d)=\min\{1,n^{-1}+d^2/(N^2\log^2(en))\}$ and attained by the mixed-information hybrid $\widehat\tau^{\mathrm{mix}}_{n,m,d}$ . Asymptotic statements run along arbitrary sequences $(m_n)_{n\ge1}$ and $(d_n)_{n\ge1}$ ; the two-sided constants $c_\epsilon,C_\epsilon\in(0,\infty)$ depend only on $\epsilon$ . Lower bounds are indexed by fuzzy hypotheses $\Pi_0,\Pi_1$ on $\mathcal M_{d,\epsilon}$ whose full random table $P_{XA}$ has identical distribution, together with the finite illustrative pair $\Pi_0^{\mathrm{small}},\Pi_1^{\mathrm{small}}$ and the finite calculation $\mathfrak C_{\mathrm{small}}$ at $d=2$ , $\epsilon=1/4$ . — Encoding: abbrev Obs (d : Nat) := Fin d x Bool x Bool, abbrev AuxObs (d : Nat) := Fin d x Bool, structure DiscreteLaw (d : Nat) where pmf : PMF (Obs d), Sample n m d := (Fin n -> Obs d) x (Fin m -> AuxObs d), with jointMass, cellMass, armMass, propensity, outcomeMean, markedMass, auxMarginal : DiscreteLaw d -> PMF (AuxObs d), labeledProductLaw, auxProductLaw, annotationLaw = (labeledProductLaw P n).prod (auxProductLaw P m). All in Basic.lean, namespace CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier. Substrate survey (bypass-justified, define-local): searched ../doc/API.md sections 10z Reusable minimax and nonparametric reduction substrate, 8h PO/, 18 Experimentation/ and the cluster roots Causalean/Stat/, Causalean/Estimation/, Causalean/Estimation/Efficiency/, Causalean/Estimation/MinimaxATE/, Causalean/PO/ for a two-sample labeled/unlabeled finite-alphabet discrete-ATE world; concept words grepped: semisupervised, unlabeled, two-sample, annotation, auxiliary sample, discrete ATE, overlap cone. Causalean carries the general measure-theoretic PO/ATE machinery and the experiment-level minimax tools but no finite-alphabet two-sample discrete-ATE world; every sibling Stat run (CausalSmith/Stat/STAT_DiscreteAteMinimaxLoggap_Research/Basic.lean, CausalSmith/Stat/STAT_DiscreteAteHeterogeneityFrontier_Research/Basic.lean) owns its own finite model layer over Causalean.Stat.Sample + Mathlib PMF, and doc/API.md Workspace context fixes the dependency direction CausalSmith -> Causalean only (no cross-run imports). BYPASS-JUSTIFIED: abstraction mismatch (measure-theoretic vs finite algebraic) plus the one-way package rule. Analogous module F2 should mirror line-for-line: CausalSmith/Stat/STAT_DiscreteAteMinimaxLoggap_Research/Basic.lean (DiscreteLaw, Obs, cellMass, propensity, outcomeMean, Overlap, ateFunctional, mse, worstCaseMSE, minimaxRisk) - the one-sample (m = 0) version of this exact world; the gap to bridge is the second, outcome-free sample and the n vs n+m information split.
Setup S-2 — A finite potential-outcome overlay on the same discrete world. A full-data law on $[d]\times\{0,1\}^4$ carries the record $(X,A,Y,Y(0),Y(1))$ with binary potential outcomes $Y(a)\in\{0,1\}$ ; consistency requires $Y=Y(A)$ almost surely and conditional exchangeability requires $(Y(0),Y(1))\perp\!\!\!\perp A\mid X$ . Together with occupied-cell overlap these identify the causal contrast $E[Y(1)-Y(0)]$ as the observed-data functional $\tau(P)$ of the induced observed marginal. — Encoding: abbrev FullObs (d) := Fin d x Bool x Bool x Bool x Bool, structure PotentialLaw (d) where pmf : PMF (FullObs d), fullMass, poAtom, observedMarginal : PotentialLaw d -> DiscreteLaw d; Consistency and ConditionalExchangeability are finite-sum Props on PotentialLaw. Search trace: grepped Causalean/PO/ and ../doc/API.md section 8h PO/ for Consistency, conditional exchangeability, ignorability, PODataFrame, POSystem (Causalean does carry PO systems, Backdoor, POManskiIVSystem); Mathlib-side check lean_leansearch "conditional independence of potential outcomes given covariates" returns only generic ProbabilityTheory.CondIndep. BYPASS-JUSTIFIED: the Causalean PO analogue is a measure-theoretic PO system with abstract outcome spaces, whereas this note needs a purely finite PMF overlay used only to certify identification of tau once; the sibling STAT_DiscreteAteMinimaxLoggap_Research/Basic.lean (PotentialLaw, Consistency, ConditionalExchangeability, observedMarginal) is the exact analogous encoding to copy.
Setup S-3 — The known-treatment--covariate-marginal benchmark experiment. An estimator observes $n\ge1$ complete labeled records $\mathcal L_n\sim P^{\otimes n}$ together with the exact $2d$ -cell table $P_{XA}$ supplied as deterministic side information, so that the decision class $\mathcal T^{\mathrm{known}}_{n,d}$ consists of the measurable functions of $(\mathcal L_n,P_{XA})$ and the benchmark risk is $R_\epsilon^{\mathrm{known}}(n,d)=\inf_{T\in\mathcal T^{\mathrm{known}}_{n,d}}\sup_{P\in\mathcal M_{d,\epsilon}}E_{P^{\otimes n}}[(T(\mathcal L_n,P_{XA})-\tau(P))^2]$ . This is the $m\to\infty$ limit of the annotation experiment: all treatment--covariate information is revealed and no outcome information is added. — Encoding: AuxTable d := AuxObs d -> Real (the exact 2d-cell table), knownSample n d := (Fin n -> Obs d) x AuxTable d, knownMarginalExperiment, knownMarginalRisk in Basic.lean. Search trace: grepped Causalean/Stat/Minimax/ + Causalean/Estimation/MinimaxATE/ for known marginal, side information, oracle nuisance, deterministic side information, ancillary - the only relevant hit is Causalean.Stat.chiSqDiv_prod_ancillary (Causalean/Stat/Minimax/ChiSquared.lean:451), which is the information-theoretic tool for the converse, not the experiment object; Mathlib-side lean_leansearch "minimax risk with known nuisance side information" returns only ProbabilityTheory.minimaxRisk (Mathlib/Probability/Decision/Risk/Defs.lean), an abstract Bayes-risk shell over a kernel, which does not carry the finite table. BYPASS-JUSTIFIED for the experiment object; the abstract Mathlib/Causalean risk shells stay unused because this note's risk must be the same inf-sup shape as minimaxRisk for thm:known-marginal-limit to compare them.

Definitions (10)

Definition P-1 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.ModelClass $\mathcal M_{d,\epsilon}:=\{P\in\mathsf{Prob}([d]\times\{0,1\}^2): d\ge2,\ 0<\epsilon<1/2,\text{ and }P\text{ satisfies }\mathrm{ass:overlap}\}$ .
Definition P-2 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.ateFunctional $\tau(P):=\sum_{x=1}^{d}p_x(\mu_{1x}-\mu_{0x})$ .
Definition P-3 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotationExperiment $\mathcal E_{n,m,d,\epsilon}:=\{P^{\otimes n}\otimes P_{XA}^{\otimes m}:P\in\mathcal M_{d,\epsilon},\ n\ge1,\ m\ge0,\ d\ge2\}$ , with the product coupling specified by $\mathrm{ass:labeled-iid}$ , $\mathrm{ass:auxiliary-iid}$ , and $\mathrm{ass:sample-independence}$ .
Definition P-4 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.minimaxRisk $R_\epsilon(n,m,d):=\inf_{T\in\mathcal T_{n,m,d}}\sup_{P\in\mathcal M_{d,\epsilon}}E_{P^{\otimes n}\otimes P_{XA}^{\otimes m}}[(T-\tau(P))^2]$ .
Definition P-5 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.frontierRate Put $N=n+m$ and $\ell_n=\log(en)$ . The unequal-information annotation-frontier rate is $r_\epsilon(n,m,d):=\min\!\left\{1,\frac1n+\frac{d^2}{N^2\ell_n^2}\right\}.$
Definition P-6 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.mixedEstimator Put $A_\star=7056$ and $c^\circ=1/256$ . Let $j_\epsilon$ be the least integer $j\ge1$ for which $2^{-j}\le\epsilon$ , and put $\bar\epsilon=2^{-j_\epsilon}$ ; thus $0<\bar\epsilon\le\epsilon<2\bar\epsilon$ . Define $H_\epsilon$ to be the least positive integer $H$ satisfying the finite list H\ge4,\qquad \frac H4\ge15+\frac{24}{c^\circ},\qquad Hc^\circ\bar\epsilon\ge672, \qquad Hc^\circ\bar\epsilon\ge24. \tag{C1} The set in this minimum is nonempty because $c^\circ\bar\epsilon>0$ , and all of its tests involve integers and dyadic rationals. For given $n,m$ , partition the labeled indices, in their original order, into blocks of sizes $M_0=\lfloor n/3\rfloor,\qquad n_p=\left\lfloor\frac{n-M_0}{2}\right\rfloor, \qquad n_f=n-M_0-n_p,$ and partition the auxiliary indices into blocks of sizes $m_p=\lfloor m/2\rfloor$ and $m_f=m-m_p$ . The outcome block is the first labeled block; the pilot and factorial pools have sizes M_p=n_p+m_p,\qquad M_f=n_f+m_f,\qquad u=M_0/8,\quad t_p=M_p/8,\quad t=M_f/8. \tag{C2} If $n<24$ , declare the calibration predicate $\mathsf C_\epsilon(n,m)$ false. If $n\ge24$ , put $N=n+m$ , $\ell=\log(en)$ , $h_n=1+\lceil\log_2 n\rceil$ , L=\max\{2,\lfloor c^\circ h_n\rfloor\},\qquad B=\frac{H_\epsilon L}{\min(t_p,t)},\qquad k_0=\lfloor t_pB/4\rfloor, \tag{C3} and let $\mathsf C_\epsilon(n,m)$ be the conjunction of the following finite tests: \frac1{32}\le\frac un\le\frac1{24},\qquad \frac1{32}\le\frac{t_p}{N},\frac tN\le\frac3{32},\qquad \frac13\le\frac t{t_p}\le3,\qquad L\ge\frac{c^\circ h_n}{2}, \tag{C4} t_pB\ge8,\qquad tB\ge L,\qquad L^4A_\star^{2L}\le n,\qquad L^6A_\star^{2L}\le n, \tag{C5} \frac{t_pB}{4}\ge15L+12h_n,\qquad \left(1+\bar\epsilon\frac t{t_p}\right)^{k_0}\ge n^{12}, \qquad \bar\epsilon tB\ge12h_n. \tag{C6} These comparisons are a finite decidable computation from $(n,m,\epsilon)$ : $h_n$ is the binary length of $n$ , $\bar\epsilon$ is dyadic, and every remaining quantity and comparison is rational. If $\mathsf C_\epsilon(n,m)$ is false, set $\widehat\tau^{\mathrm{mix}}_{n,m,d}=0$ . If it is true, independently average over prefix lengths $D_0\sim\operatorname{Poi}(u)$ , $D_p\sim\operatorname{Poi}(t_p)$ , and $D_f\sim\operatorname{Poi}(t)$ , returning zero whenever a prefix exceeds its pool. For a valid prefix, let $J_x$ be the pilot count and call $x$ light when $J_x\le k_0$ . Define, with polynomial continuation at zero, $H_L(z)=\frac{1-T_L(1-2z)}{2L^2},\qquad E_L(z)=\frac{H_L(z)}z,\qquad G_L(z)=\frac{1-E_L(z)}z,$ and expand $W_{a,L}(s_0,s_1)=\frac{s_0+s_1}{B}G_L(s_a/B) =\sum_{r_0,r_1}c_{a,r_0,r_1}s_0^{r_0}s_1^{r_1}.$ For factorial-pool counts $K_{ax}$ , outcome-block success counts $S_{ax}$ , and falling factorials $(k)_r$ , put $\widehat W_{a,L,x}=\sum_{r_0,r_1}c_{a,r_0,r_1} \frac{(K_{0x})_{r_0}(K_{1x})_{r_1}}{t^{r_0+r_1}}, \qquad \widehat W^H_{a,x}=\frac{K_{0x}+K_{1x}+1}{K_{ax}+1},$ $Z_x=\mathbf1\{J_x\le k_0\} \left(\frac{S_{1x}}u\widehat W_{1,L,x}-\frac{S_{0x}}u\widehat W_{0,L,x}\right) +\mathbf1\{J_x>k_0\} \left(\frac{S_{1x}}u\widehat W^H_{1,x}-\frac{S_{0x}}u\widehat W^H_{0,x}\right).$ The valid-prefix output is $\operatorname{clip}_{[-1,1]}(\sum_xZ_x)$ . Finally $\widehat\tau^{\mathrm{mix}}_{n,m,d}$ is the conditional expectation of this output over the three prefix lengths given $(\mathcal L_n,\mathcal U_m)$ . It is a finite weighted sum of rationally generated polynomial coefficients, observed counts, and computable Poisson weights, and hence is a total computable measurable map.
Definition P-7 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalPriorOf Let $N=n+m$ and $\ell_n=\log(en)$ . In the regime $\min\{1,d^2/(N^2\ell_n^2)\}\le C_\epsilon/n$ , let $\Pi_0,\Pi_1$ be point masses at the two laws concentrated on $X=1$ , with $e_1=1/2$ , $\mu_{01}=0$ , and $\mu_{11}=1/2\mp h_n$ , where $h_n=c_\epsilon/\sqrt n\le1/4$ . In the complementary regime use the following approximation-dual construction. Put $L=\lceil C_\epsilon^\circ\ell_n\rceil$ , $B=b_\epsilon L/N$ , $a=\gamma_\epsilon B/L^2$ , and $\kappa=(1-2\epsilon)/\epsilon$ , with the constants calibrated in \mathrm{lem:common\mbox{-}marginal\mbox{-}poisson\mbox{-}prior}. On $I=[a/\kappa,B]$ , let $\sigma$ be the finite signed measure of total variation one furnished by lem:finite-alternation-duality, which annihilates $1,p,\ldots,p^L$ and satisfies $\int p/(p+a)\,d\sigma(p)\ge c_\epsilon>0$ . Write $h=d\sigma/d|\sigma|$ , and define $\nu(dp)=\frac{a}{p+a}|\sigma|(dp)\quad(p\in I), \qquad \nu(\{0\})=1-\int_I\frac{a}{p+a}|\sigma|(dp).$ Choose $k=\min\{d-1,\lfloor c_\epsilon^\circ NL\rfloor\}$ rare cells. Draw their raw masses independently from $\nu$ , put $s_i=\epsilon(p_i+a)$ when $p_i>0$ , and set $e_i=s_i/p_i$ . The reservoir raw mass is $p_*=1-kE_\nu[p]$ , with propensity $1/2$ and zero outcomes. Divide every raw cell mass by $S=p_*+\sum_{i=1}^kp_i$ . Under $\Pi_1$ , put $\mu_{1i}=(1+h(p_i))/2$ ; under $\Pi_0$ , put $\mu_{1i}=(1-h(p_i))/2$ ; under both put $\mu_{0i}=0$ , and assign identical legal values at raw mass zero. Remaining alphabet cells have mass zero. In either branch $\Pi_0$ and $\Pi_1$ are probability measures on $\mathcal M_{d,\epsilon}$ and induce exactly the same random $P_{XA}$ .
Definition P-8 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.finiteSmallInstance Let $\epsilon=1/4$ , $d=2$ , $p_1=p_2=1/2$ , $e_1=e_2=1/2$ , and $\mu_{01}=\mu_{02}=0$ . For $n=m=1$ , independence of the samples gives $E\!\left[2\mathbf 1\{\mathcal L_1=((1,1,1))\}\left(\mathbf 1\{\mathcal U_1=((1,0))\}+\mathbf 1\{\mathcal U_1=((1,1))\}\right)\right]=2q_{11}p_1.$ At $\mu_{11}=1$ , this equals $1/4$ , while the treated cell target is $p_1\mu_{11}=1/2$ . Define $\Pi_0^{\mathrm{small}}$ to put unit mass on the law with $\mu_{11}=\mu_{12}=0$ , and define $\Pi_1^{\mathrm{small}}$ to put unit mass on the law with $\mu_{11}=1$ and $\mu_{12}=0$ . Under both priors, $P_{XA}(x,a)=1/4$ for all $(x,a)$ , whereas $\tau(P)=0$ under $\Pi_0^{\mathrm{small}}$ and $\tau(P)=1/2$ under $\Pi_1^{\mathrm{small}}$ ; consequently the auxiliary laws agree for every $m$ , and the total variation distance between the two labeled laws is $1-(3/4)^n$ . These identities constitute $\mathfrak C_{\mathrm{small}}$ .
Definition P-9 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalExperiment $\mathcal E^{\mathrm{known}}_{n,d,\epsilon}:=\{(P^{\otimes n},P_{XA}):P\in\mathcal M_{d,\epsilon},\ n\ge1,\ d\ge2\}$ , where the first coordinate generates $\mathcal L_n$ and the second coordinate is the exact $2d$ -cell table observed as deterministic side information.
Definition P-10 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRisk $R_\epsilon^{\mathrm{known}}(n,d):=\inf_{T\in\mathcal T^{\mathrm{known}}_{n,d}}\sup_{P\in\mathcal M_{d,\epsilon}}E_{P^{\otimes n}}[(T(\mathcal L_n,P_{XA})-\tau(P))^2]$ .

Assumptions (6)

Assumption A-1 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.Overlap $\epsilon\le e_x\le1-\epsilon$ for every $x$ with $p_x>0$ .
Assumption A-2 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.LabeledIid $\mathcal L_n\sim P^{\otimes n}$ .
Assumption A-3 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.AuxiliaryIid $\mathcal U_m\sim P_{XA}^{\otimes m}$ .
Assumption A-4 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.SampleIndependence $\mathcal L_n\perp\!\!\!\perp\mathcal U_m$ .
Assumption A-5 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.Consistency $Y=Y(A)$ almost surely.
Assumption A-6 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.ConditionalExchangeability $(Y(0),Y(1))\perp\!\!\!\perp A\mid X$ .

Lemmas (29)

Lemma L-1 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_count_risk Let $u,t>0$ . In the independent Poisson experiment with outcome-success counts $S_{ax}\sim\operatorname{Poi}(u q_{ax})$ and marginal counts $K_{ax}\sim\operatorname{Poi}(t s_{ax})$ , define $Z^H=\sum_{x=1}^d\left\{\frac{S_{1x}}u\frac{K_{0x}+K_{1x}+1}{K_{1x}+1}- \frac{S_{0x}}u\frac{K_{0x}+K_{1x}+1}{K_{0x}+1}\right\}.$ Uniformly over $P\in\mathcal M_{d,\epsilon}$ , $|E_PZ^H-\tau(P)|\le C_\epsilon\min\{1,d/t\}, \qquad \operatorname{Var}_P(Z^H)\le C_\epsilon(u^{-1}+t^{-1}).$
Lemma L-2 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.chebyshev_factorial_certificate For every integer $L\ge2$ , the displayed polynomials $H_L,E_L,G_L$ are well defined, $G_L$ has degree at most $L-2$ , and on $0\le z\le1$ , $0\le E_L(z)\le\min\{1,(L^2z)^{-1}\},\qquad zG_L(z)+E_L(z)=1.$ There is a universal $A>1$ such that the factorial lift $\widehat W_{a,L}$ of $W_{a,L}$ satisfies, whenever $tB\ge L$ , $E\widehat W_{a,L}^{2}\le A^L\{1+p/B\}^{2L},$ and the right side can be replaced by $A^L$ when $p\le B$ .
Lemma L-3 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_hybrid_risk Use the split sizes, intensities, constants $A_\star=7056$ , $c^\circ=1/256$ , dyadic certificate $\bar\epsilon$ , integer $H_\epsilon$ , and finite predicate $\mathsf C_\epsilon(n,m)$ in \mathrm{def:mixed\mbox{-}estimator\mbox{-}handle}. In the independent Poisson experiment with outcome intensity $u$ , pilot intensity $t_p$ , and factorial intensity $t$ , apply the displayed hybrid statistic when $\mathsf C_\epsilon(n,m)$ holds and output zero otherwise. There is a finite $C_\epsilon$ , depending only on $\epsilon$ , such that, uniformly over $P\in\mathcal M_{d,\epsilon}$ and simultaneously for every $n\ge1$ , $m\ge0$ , and $d\ge2$ , its clipped output $Z^{\mathrm{mix}}$ obeys E_P[(Z^{\mathrm{mix}}-\tau(P))^2] \le C_\epsilon\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2\log^2(en)}\right\}. \tag{C14} The calibration and the statistic are finite and computable; only the risk constant $C_\epsilon$ is existential.
Lemma L-4 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.fixed_sample_hybrid_transfer The independent-Poisson hybrid statistic of \mathrm{lem:poisson\mbox{-}hybrid\mbox{-}risk} has a total deterministic implementation from exactly $n$ labeled and $m$ auxiliary observations. Its risk is at most twice the Poisson risk plus $C/n$ , uniformly over the observation experiment.
Lemma L-5 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.finite_alternation_duality Fix $0<\epsilon<1/2$ and put $\kappa=(1-2\epsilon)/\epsilon$ . There are positive constants $\gamma_\epsilon,c_\epsilon$ such that, for every integer $L\ge2$ , every $B>0$ , and $a=\gamma_\epsilon B/L^2$ , the interval $I=[a/\kappa,B]$ is nonempty and there is a finite signed measure $\sigma$ on $I$ satisfying $|\sigma|(I)=1,\qquad \int_Ip^j\,d\sigma(p)=0\quad(0\le j\le L), \qquad \int_I\frac p{p+a}\,d\sigma(p)\ge c_\epsilon.$
Lemma L-7 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.fixed_sample_common_marginal_transfer There are fixed numerical constants $C_{\mathrm{fixed}},c_{\mathrm{risk}}>0$ with the following property. For every $0<\epsilon<1/2$ and every positive variance-bound constant $C_\epsilon$ in the uniform-intensity exact-witness interface, there is a concentration threshold $c_{\epsilon,C_\epsilon}>0$ . For every $n\ge1$ , $m\ge0$ , and every recipe carrying that exact witness at intensities $u=C_{\mathrm{fixed}}n$ and $v=C_{\mathrm{fixed}}(n+m)$ , let $\Delta$ be its actual raw-center separation. If that recipe's raw mixture laws have total variation at most $1/8$ and $kBa\le c_{\epsilon,C_\epsilon}\Delta^2$ , then $R_\epsilon(n,m,d)\ge c_{\mathrm{risk}}\Delta^2.$ This implication holds for every qualifying exact-witness recipe, including both the exact $b_\epsilon L/(u+v)$ specialization and the C55 bandwidth supplied through the uniform-intensity inequality; the transfer does not select a recipe. It preserves the identical random $P_{XA}$ law and holds for the fixed, unequal sample sizes.
Lemma L-8 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.explicit_chebyshev_calibration Let $T_j$ be the Chebyshev polynomials, put $Q_j(z)=T_j(1-2z)$ , and let $\|\cdot\|_1$ denote coefficient $\ell^1$ -norm. For every integer $L\ge2$ , the polynomials $H_L(z)=\frac{1-Q_L(z)}{2L^2},\qquad E_L(z)=H_L(z)/z, \qquad G_L(z)=\{1-E_L(z)\}/z$ are well defined by polynomial continuation and satisfy \|Q_L\|_1\le7^L,\qquad \|(z_0+z_1)G_L(z_a)\|_1\le14^L. \tag{C7} Moreover, if $K_b\sim\operatorname{Poi}(ts_b)$ independently, $p=s_0+s_1$ , and $\widehat W_{a,L}$ is the factorial lift of $(z_0+z_1)G_L(z_a)$ with $z_b=s_b/B$ , then, whenever $tB\ge L$ , the explicit integer $A_\star=7056$ satisfies E\widehat W_{a,L}^{2}\le A_\star^L(1+p/B)^{2L}, \qquad p\le B\Longrightarrow E\widehat W_{a,L}^{2}\le A_\star^L. \tag{C8} On $0\le z\le1$ , one also has 0\le E_L(z)\le\min\{1,(L^2z)^{-1}\},\qquad zG_L(z)+E_L(z)=1. \tag{C9}
Lemma L-9 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametric_label_floor There is a positive numerical $c$ , independent of $m,d$ and depending on $\epsilon$ only through the requirement $\epsilon<1/2$ , such that for every $n\ge1$ , $m\ge0$ , and $d\ge2$ , R_\epsilon(n,m,d)\ge c/n. \tag{C39} The two hypotheses have the same treatment--covariate marginal, so the bound is entirely label-gated.
Lemma L-10 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.common_marginal_uniform_intensity There are positive constants $b_\epsilon,\gamma_\epsilon,c_\epsilon,C_\epsilon$ and $0<\rho_\epsilon<1$ with the following property. Let $L\ge2$ , let $u,v\ge0$ , put $M=u+v>0$ , and choose $B>0$ , $a=\gamma_\epsilon B/L^2$ , and an integer $k\le d-1$ such that MB\le b_\epsilon L,\qquad ka\le c_\epsilon. \tag{C42} Apply the finite-atom recipe in \mathrm{def:common\mbox{-}marginal\mbox{-}prior\mbox{-}handle} to $k$ rare cells at scale $(a,B,L)$ . The normalized laws lie in $\mathcal M_{d,\epsilon}$ , the two priors induce exactly the same distribution of the full random table $P_{XA}$ , their raw target centers are separated by at least $c_\epsilon ka$ , and their raw total-mass and target variances are at most $C_\epsilon kBa$ . In the raw Poisson experiment of labeled intensity $u$ and auxiliary intensity $v$ , the two mixture laws have total variation at most C_\epsilon uka\rho_\epsilon^L. \tag{C43}
Lemma lem:poisson-inverse-marginal-energy-scalar verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_marginal_energy_with_outcome_rates Let $0<\epsilon\le1$ , $u>0$ , $p\ge0$ , and $w\ge0$ , and let $q_0,q_1\in[0,p]$ . For $a\in\{0,1\}$ , let $\lambda_a$ satisfy $\epsilon w\le\lambda_a\le w$ , and let $r_{2,a},r_{4,a}\ge0$ satisfy $(1+\lambda_a)^2r_{2,a}\le8$ and $(1+\lambda_a)^4r_{4,a}\le96$ . Put $c_{\mathrm{mar}}=8\epsilon^{-2}+96(\epsilon^{-2}+\epsilon^{-3})$ . Then $\lambda_1\bigl\{(q_1u^{-1}+q_1^2)(\lambda_0+\lambda_0^2)r_{4,1}+(q_0u^{-1}+q_0^2)r_{2,0}\bigr\} +\lambda_0\bigl\{(q_0u^{-1}+q_0^2)(\lambda_1+\lambda_1^2)r_{4,0}+(q_1u^{-1}+q_1^2)r_{2,1}\bigr\} \le (pu^{-1}+p^2)\,\frac{2c_{\mathrm{mar}}}{1+w}.$
Lemma lem:poisson-prefix-count-law verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.finitePrefixCounts_map_eq_poissonCountLaw Let $P$ be a probability law on $[d]\times\{0,1\}^2$ , let $u,t\ge0$ , and write $q_{ax}=P(x,a,1)$ and $s_{ax}=P(x,a,0)+P(x,a,1)$ . Under the prefix law $\mathbb P^{\mathrm{pref}}_{P,u,t}$ , draw independently an outcome-marked prefix consisting of $D_0\sim\operatorname{Poi}(u)$ independent records from $P$ and a treatment--covariate prefix consisting of $D_f\sim\operatorname{Poi}(t)$ independent records from $P_{XA}$ ; let $S_{ax}$ be the number of outcome-marked prefix records equal to $(x,a,1)$ and $K_{ax}$ the number of treatment--covariate prefix records equal to $(x,a)$ . Then the count array $(S_{ax},K_{ax})_{x\in[d],\,a\in\{0,1\}}$ has independent coordinates with $S_{ax}\sim\operatorname{Poi}(u\,q_{ax}),\qquad K_{ax}\sim\operatorname{Poi}(t\,s_{ax});$ that is, the pushforward of $\mathbb P^{\mathrm{pref}}_{P,u,t}$ under the count map is the independent Poisson count law used for $Z^H$ .
Lemma lem:inverse-count-poisson-prefix-transfer verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverseCountEstimator_risk_le_poissonPrefixes Let $n,m,d\ge0$ be integers and $P\in\mathcal M_{d,\epsilon}$ . Put $M_0=\lfloor n/3\rfloor$ , $n_p=\lfloor(n-M_0)/2\rfloor$ , $n_f=n-M_0-n_p$ , $m_f=m-\lfloor m/2\rfloor$ , $u=M_0/8$ , and $t=(n_f+m_f)/8$ , with the convention $S/u=0$ when $u=0$ . Let $\widehat\tau^{\mathrm{IC}}_{n,m,d}$ be the inverse-count estimator: the weighted sum, over prefix lengths $0\le r_0\le M_0$ and $0\le r_f\le n_f+m_f$ with weights $\Pr\{\operatorname{Poi}(u)=r_0\}\Pr\{\operatorname{Poi}(t)=r_f\}$ , of $\operatorname{clip}_{[-1,1]}(Z^H)$ computed from the first $r_0$ records of the outcome block (the first $M_0$ labeled records) and the first $r_f$ records of the factorial pool (the treatment--covariate parts of the last $n_f$ labeled records together with the last $m_f$ auxiliary records), where $Z^H=\sum_{x=1}^d\left\{\frac{S_{1x}}u\frac{K_{0x}+K_{1x}+1}{K_{1x}+1}-\frac{S_{0x}}u\frac{K_{0x}+K_{1x}+1}{K_{0x}+1}\right\},$ $S_{ax}$ counts outcome-block prefix records equal to $(x,a,1)$ , and $K_{ax}$ counts factorial-pool prefix records with $(X,A)=(x,a)$ . Under the prefix law $\mathbb P^{\mathrm{pref}}_{P,u,t}$ , draw independently an outcome-marked prefix consisting of $D_0\sim\operatorname{Poi}(u)$ independent records from $P$ and a treatment--covariate prefix consisting of $D_f\sim\operatorname{Poi}(t)$ independent records from $P_{XA}$ ; let $S_{ax}$ be the number of outcome-marked prefix records equal to $(x,a,1)$ and $K_{ax}$ the number of treatment--covariate prefix records equal to $(x,a)$ . Let $\widehat\tau^{\mathrm{IC,pref}}_{u,t}=\operatorname{clip}_{[-1,1]}(Z^H)$ be computed from these prefix counts. Then $E_{P^{\otimes n}\otimes P_{XA}^{\otimes m}}\!\left[\bigl(\widehat\tau^{\mathrm{IC}}_{n,m,d}-\tau(P)\bigr)^2\right] \le E_{\mathbb P^{\mathrm{pref}}_{P,u,t}}\!\left[\bigl(\widehat\tau^{\mathrm{IC,pref}}_{u,t}-\tau(P)\bigr)^2\right] +4\Bigl[\Pr\{\operatorname{Poi}(u)>M_0\}+\Pr\{\operatorname{Poi}(t)>n_f+m_f\}\Bigr].$
Lemma lem:shifted-chebyshev-quotient-coefficient-bound verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.coefficientL1_chebG_le Let $L\ge1$ be an integer, let $T_L$ be the Chebyshev polynomial of degree $L$ , and let $H_L(z)=\frac{1-T_L(1-2z)}{2L^2},\qquad E_L(z)=\frac{H_L(z)}z,\qquad G_L(z)=\frac{1-E_L(z)}z$ be the polynomials obtained by polynomial continuation at zero, as in \mathrm{def:mixed\mbox{-}estimator\mbox{-}handle}. If $\|\cdot\|_1$ denotes the coefficient $\ell^1$ -norm of a real polynomial, then $\|G_L\|_1\le 6^L .$
Lemma lem:finite-side-information-convergence verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalGeneric_tendsto Let $0<\epsilon<1/2$ , $n\ge0$ , and $d\ge2$ . Let $\Theta_{\epsilon,d}=\bigl\{(\pi,\varrho,\omega):\pi\in\mathsf{Prob}([d]),\ \varrho_x\in[\epsilon,1-\epsilon],\ \omega_{\alpha x}\in[0,1]\ \text{for }x\in[d],\ \alpha\in\{0,1\}\bigr\},$ and for $\theta=(\pi,\varrho,\omega)\in\Theta_{\epsilon,d}$ put $P_\theta(x,\alpha,y)=\pi_x\varrho_x^{\alpha}(1-\varrho_x)^{1-\alpha}\omega_{\alpha x}^{y}(1-\omega_{\alpha x})^{1-y}$ , let $\lambda_\theta=P_\theta^{\otimes n}$ be the law of the labeled sample $Z\in([d]\times\{0,1\}^2)^n$ , let $\xi_\theta(x,\alpha)=\sum_{y\in\{0,1\}}P_\theta(x,\alpha,y)$ , and let $\upsilon_\theta=\sum_{x\in[d]}\pi_x(\omega_{1x}-\omega_{0x})$ . For $m\ge0$ define $\mathfrak R_m=\inf_{\varphi}\sup_{\theta\in\Theta_{\epsilon,d}}\int\bigl(\varphi(Z,U)-\upsilon_\theta\bigr)^2\,d\bigl(\lambda_\theta\otimes\xi_\theta^{\otimes m}\bigr)(Z,U),$ where $\varphi$ ranges over all maps $([d]\times\{0,1\}^2)^n\times([d]\times\{0,1\})^m\to[-1,1]$ , and define $\mathfrak R_\star=\inf_{\psi}\sup_{\theta\in\Theta_{\epsilon,d}}\int\bigl(\psi(Z,\xi_\theta)-\upsilon_\theta\bigr)^2\,d\lambda_\theta(Z),$ where $\psi$ ranges over all maps from $([d]\times\{0,1\}^2)^n\times\mathsf{Prob}([d]\times\{0,1\})$ to $[-1,1]$ . Then $\lim_{m\to\infty}\mathfrak R_m=\mathfrak R_\star .$
Lemma lem:common-marginal-recipe-transfer verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.fixed_sample_common_marginal_recipe_transfer_rule Let $0<\epsilon<1/2$ , and fix a common-marginal calibration at overlap level $\epsilon$ , a variance-bound constant $C_\epsilon>0$ , and a real number $\rho_\epsilon$ . Let $n\ge1$ , $m\ge0$ , $d\ge2$ , integers $L,k\ge0$ , and reals $B,a$ be given, and let $\mathcal R$ be a common-marginal rare-cell recipe with $k$ rare cells at scale $(a,B,L)$ carrying the exact Poisson-prior witness with constants $(C_\epsilon,\rho_\epsilon)$ at labeled intensity $u=65536\,n$ and auxiliary intensity $v=65536\,(n+m)$ . Let $\theta_1$ and $\theta_0$ be the alternative and null raw prior centers of $\mathcal R$ , and put $\Delta=|\theta_1-\theta_0|$ . If the two raw mixture laws of $\mathcal R$ have total variation distance at most $1/8$ and $kBa\le\frac{\Delta^2}{65536\,C_\epsilon},$ then $R_\epsilon(n,m,d)\ge\frac{\Delta^2}{65536}.$
Lemma lem:annotation-frontier-strict-improvement verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotation_frontier_strict_improvement_iff Let $(m_n)_{n\ge1}$ and $(d_n)_{n\ge1}$ be sequences of nonnegative integers with $d_n\ge2$ for every $n\ge1$ , and put $N_n=n+m_n$ and $\ell_n=\log(en)$ . Then, as $n\to\infty$ , $r_\epsilon(n,m_n,d_n)=o\!\left(r_\epsilon(n,0,d_n)\right)$ if and only if both $\frac{d_n}{\sqrt n\,\ell_n}\to\infty \quad\text{and}\quad \frac{d_n/(N_n\ell_n)}{\min\{1,d_n/(n\ell_n)\}}\to0.$
Lemma lem:common-marginal-recipe-target-concentration verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_prior_target_concentration Fix $\epsilon\in\mathbb R$ , a common-marginal calibration at overlap level $\epsilon$ with dual-gap constant $c_\epsilon$ , reals $C_\epsilon,\rho_\epsilon,u,v,B,a$ , and integers $n,m,d,L,k\ge0$ . Let $\mathcal R$ be a common-marginal rare-cell recipe with $k$ rare cells at scale $(a,B,L)$ , with atom-generator probability measure $\Gamma$ on $w\in\mathbb R^{k}$ , branch- $b$ laws $P_b(w)$ , raw targets $r_b(w)$ , raw total mass $S(w)$ , and branch priors $\Pi_b=\Gamma\circ P_b^{-1}$ for $b\in\{0,1\}$ ; put $\theta_b=\int r_b\,d\Gamma$ . Assume the recipe bounds hold at intensities $(u,v)$ with constants $(C_\epsilon,\rho_\epsilon)$ : $\mathcal R$ uses the finite-atom construction, both priors are supported in $\mathcal M_{d,\epsilon}$ and induce the same distribution of the random table $P_{XA}$ , $c_\epsilon ka\le|\theta_1-\theta_0|$ , $\int S\,d\Gamma=1$ , $S,r_0,r_1\in L^2(\Gamma)$ with $\operatorname{Var}_\Gamma(S)\le C_\epsilon kBa$ and $\max_b\operatorname{Var}_\Gamma(r_b)\le C_\epsilon kBa$ , and the two raw mixture laws at labeled intensity $u$ and auxiliary intensity $v$ have total variation distance at most $C_\epsilon uka\rho_\epsilon^L$ . Then for each branch $b\in\{0,1\}$ and every $0<\Delta\le2$ , $\Pi_b\!\left\{P:\ |\tau(P)-\theta_b|>\frac{\Delta}{4}\right\}\le\frac{512\,C_\epsilon kBa}{\Delta^2}.$
Lemma lem:common-marginal-recipe-scale-tail verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_prior_scale_tail Fix $\epsilon\in\mathbb R$ , a common-marginal calibration at overlap level $\epsilon$ with dual-gap constant $c_\epsilon$ , reals $C_\epsilon,\rho_\epsilon,u,v,B,a$ , and integers $n,m,d,L,k\ge0$ . Let $\mathcal R$ be a common-marginal rare-cell recipe with $k$ rare cells at scale $(a,B,L)$ , with atom-generator probability measure $\Gamma$ on $w\in\mathbb R^{k}$ , branch- $b$ laws $P_b(w)$ , raw targets $r_b(w)$ , raw total mass $S(w)$ , and branch priors $\Pi_b=\Gamma\circ P_b^{-1}$ for $b\in\{0,1\}$ ; put $\theta_b=\int r_b\,d\Gamma$ . Assume the recipe bounds hold at intensities $(u,v)$ with constants $(C_\epsilon,\rho_\epsilon)$ : $\mathcal R$ uses the finite-atom construction, both priors are supported in $\mathcal M_{d,\epsilon}$ and induce the same distribution of the random table $P_{XA}$ , $c_\epsilon ka\le|\theta_1-\theta_0|$ , $\int S\,d\Gamma=1$ , $S,r_0,r_1\in L^2(\Gamma)$ with $\operatorname{Var}_\Gamma(S)\le C_\epsilon kBa$ and $\max_b\operatorname{Var}_\Gamma(r_b)\le C_\epsilon kBa$ , and the two raw mixture laws at labeled intensity $u$ and auxiliary intensity $v$ have total variation distance at most $C_\epsilon uka\rho_\epsilon^L$ . Let $s_{\mathcal R}$ be the recipe raw scale, a measurable real function of the observed-data law with $s_{\mathcal R}(P_b(w))=S(w)$ for $\Gamma$ -almost every $w$ , and put $s_{\mathcal R,+}=\max\{s_{\mathcal R},0\}$ . Then for each branch $b\in\{0,1\}$ , $\Pi_b\!\left\{P:\ s_{\mathcal R,+}(P)<\frac12\right\}\le 4\,C_\epsilon kBa.$
Lemma lem:common-marginal-canonical-certificate-family verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.common_marginal_canonical_certificate_family There are positive numerical constants $C_{\mathrm{fixed}}$ and $c_{\mathrm{risk}}$ with the following property. For every $0<\epsilon<1/2$ there are a calibration in the sense of def:common-marginal-prior-handle, with bandwidth constants $b_0,b_\epsilon$ , dual-gap constant $c_\epsilon$ , and dimension constant $C_d$ , together with real numbers $C_\epsilon>0$ , $\rho_\epsilon\in(0,1)$ , $c_{\mathrm{small}}>0$ , and $c_{\mathrm{floor}}$ with $0<c_{\mathrm{floor}}\le c_{\mathrm{risk}}$ , such that the following two assertions hold. For a common-marginal prior handle $H$ at $(n,m,d)$ , write $L,k,B,a$ for its integers and reals, $\theta_0(H),\theta_1(H)$ for the null and alternative raw prior centers of its recipe, $\Delta_H=c_\epsilon ka$ , and $D(n,m,d)=\min\{1,d^2/((n+m)^2\log(en)^2)\}$ ; say that the recipe of $H$ carries the exact witness if it carries the exact common-marginal Poisson-prior witness with bandwidth $B=b_0L/(n+m)$ , constants $(C_\epsilon,\rho_\epsilon)$ , labeled intensity $u=C_{\mathrm{fixed}}n$ , and auxiliary intensity $v=C_{\mathrm{fixed}}(n+m)$ . (i) Transfer at these constants: for all $n\ge1$ , $m\ge0$ , $d\ge2$ and every handle $H$ at $(n,m,d)$ whose recipe carries the exact witness, whose two raw mixture laws have total variation distance at most $1/8$ , and which satisfies $kBa\le c_{\mathrm{small}}|\theta_1(H)-\theta_0(H)|^2$ , one has $c_{\mathrm{risk}}|\theta_1(H)-\theta_0(H)|^2\le R_\epsilon(n,m,d)$ . (ii) Certificate family: for all $n\ge1$ , $m\ge0$ , $d\ge2$ there is a handle $H$ at $(n,m,d)$ such that $L\ge2$ , $2C_{\mathrm{fixed}}b_0\le b_\epsilon$ , $(C_{\mathrm{fixed}}n+C_{\mathrm{fixed}}(n+m))B\le b_\epsilon L$ , the priors $\Pi_0(H)$ and $\Pi_1(H)$ induce the same distribution of the random table $P_{XA}$ , and $c_{\mathrm{floor}}/n\le R_\epsilon(n,m,d)$ ; if $H$ uses the finite-atom construction, then moreover $ka\le c_\epsilon$ , the recipe carries the exact witness, its recipe bounds hold at intensities $(u,v)$ with constants $(C_\epsilon,\rho_\epsilon)$ (both priors supported in $\mathcal M_{d,\epsilon}$ , identical random $P_{XA}$ law, raw-center separation at least $c_\epsilon ka$ , raw total mass of generator mean one, square-integrable raw total mass and raw targets with generator variances at most $C_\epsilon kBa$ , and raw mixture total variation at most $C_\epsilon uka\rho_\epsilon^L$ ), its two raw mixture laws have total variation distance at most $1/8$ , $\Delta_H\le|\theta_1(H)-\theta_0(H)|$ , $kBa\le c_{\mathrm{small}}\Delta_H^2$ , $c_{\mathrm{floor}}\Delta_H^2\le R_\epsilon(n,m,d)$ , and $C_dD(n,m,d)\le\Delta_H^2$ ; if $H$ does not use the finite-atom construction, then $C_dD(n,m,d)\le c_{\mathrm{floor}}/n$ .
Lemma lem:calibrated-envelope-absorption verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.calibrated_envelope_absorption_exists Let $\epsilon>0$ and $C_0>0$ . There is a constant $C>0$ , depending only on $\epsilon$ and $C_0$ , with the following property. Let $n\ge1$ , $m\ge0$ , and $d\ge2$ be integers for which the calibration predicate $\mathsf C_\epsilon(n,m)$ of the hybrid estimator holds, and let $L=L(n)$ , $B=B(n,m,\epsilon)$ , $u=M_0/8$ , $t=M_f/8$ , and $A_\star=7056$ be the calibrated polynomial degree, light-cell scale, outcome-block and factorial-pool Poisson intensities, and Chebyshev-factorial constant of that estimator. With $N=n+m$ and $\ell_n=\log(en)$ as in the frontier rate, $4A_\star^L\Bigl\{\frac{\min\{1,dB\}}{u}+dB^2\Bigr\}+C_0\bigl(u^{-1}+t^{-1}\bigr)+\frac{8dB^2}{\epsilon^2L^4}+\frac4{\epsilon t}+414\,n^{-12}+\Bigl(\frac{2dB}{\epsilon L^2}+6n^{-12}\Bigr)^2\le C\Bigl(\frac1n+\frac{d^2}{N^2\ell_n^2}\Bigr).$
Lemma lem:finite-side-measurable-comparison verified Causalean.Stat.Minimax.FiniteSideInformation.measurableExactTableMinimaxValue_le_empiricalSideMinimaxValue Let $\Theta$ be a nonempty parameter set, let $\mathcal Z$ and $\mathcal C$ be nonempty finite sets, and let $\beta_-\le\beta_+$ be real numbers. For each $\theta\in\Theta$ , let $\lambda_\theta$ be a probability vector on $\mathcal Z$ , let $\xi_\theta$ be a probability vector on $\mathcal C$ , and let $\upsilon_\theta\in[\beta_-,\beta_+]$ . For each integer $m\ge0$ define the empirical-side value $\mathfrak R_m=\inf_{\varphi:\mathcal Z\times\mathcal C^m\to[\beta_-,\beta_+]}\ \sup_{\theta\in\Theta}\ \sum_{z\in\mathcal Z}\lambda_\theta(z)\sum_{v\in\mathcal C^m}\Bigl(\prod_{j=1}^m\xi_\theta(v_j)\Bigr)\bigl(\varphi(z,v)-\upsilon_\theta\bigr)^2,$ and define the measurable ambient exact-table value $\widetilde{\mathfrak R}_\star=\inf_{\psi}\ \sup_{\theta\in\Theta}\ \sum_{z\in\mathcal Z}\lambda_\theta(z)\bigl(\psi(z,\xi_\theta)-\upsilon_\theta\bigr)^2,$ where $\psi$ ranges over all maps $\mathcal Z\times\mathbb R^{\mathcal C}\to[\beta_-,\beta_+]$ whose section $w\mapsto\psi(z,w)$ is Borel measurable for each $z\in\mathcal Z$ . Then $\widetilde{\mathfrak R}_\star\le\mathfrak R_m$ for every $m\ge0$ .
Lemma lem:hybrid-light-branch-mean-bound verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.sum_light_branchMean_sq_le Let $\epsilon>0$ , let $P\in\mathcal M_{d,\epsilon}$ , let $u,t,B>0$ be reals, and let $L\ge2$ be an integer. For each covariate cell $x\in[d]$ , let $S_{0x},S_{1x},K_{0x},K_{1x}$ be independent Poisson counts with $S_{ax}\sim\operatorname{Poi}(u\,q_{ax})$ and $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ , and let $P_x=\frac{S_{1x}}u\widehat W_{1,L,x}-\frac{S_{0x}}u\widehat W_{0,L,x}$ and $H_x=\frac{S_{1x}}u\widehat W^H_{1,x}-\frac{S_{0x}}u\widehat W^H_{0,x}$ be the light polynomial and heavy inverse-count cell contributions of the hybrid estimator, with the factorial lift $\widehat W_{a,L,x}$ taken at degree $L$ , scale $B$ , and intensity $t$ . Then the light-cell branch means satisfy $\sum_{x:\,p_x\le B}\bigl(\mathbb E_P P_x-\mathbb E_P H_x\bigr)^2\le\frac{8dB^2}{\epsilon^2L^4}+\frac4{\epsilon t}.$
Lemma lem:poisson-inverse-arm-mean-formula verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_mean_formula Let $P$ be an observed-data law on $[d]\times\{0,1\}^2$ , let $x\in[d]$ and $a\in\{0,1\}$ , and let $u>0$ and $t>0$ . Write $p_x$ , $s_{ax}$ , and $q_{ax}$ for the cell mass, treatment-cell mass, and outcome-success mass of $P$ at $x$ , and $\mu_{ax}=q_{ax}/s_{ax}$ for the outcome regression (set to $0$ when $s_{ax}=0$ ). Let $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ and $K_{1-a,x}\sim\operatorname{Poi}(t\,s_{1-a,x})$ be independent factorial-pool counts. Then the inverse-count arm mean, with outcome-block Poisson mean $u\,q_{ax}$ rescaled by $u$ , satisfies $\frac{u\,q_{ax}}{u}\,\mathbb E\!\left[1+\frac{K_{1-a,x}}{K_{ax}+1}\right] =p_x\mu_{ax}-(p_x-s_{ax})\,\mu_{ax}\,e^{-t s_{ax}}.$
Lemma lem:heavy-light-second-moment-sum verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.sum_heavy_light_secondMoment_le Let $\epsilon>0$ , let $P$ be an observed-data law on $[d]\times\{0,1\}^2$ , and let $n\ge1$ and $m\ge0$ be integers for which the calibration predicate $\mathsf C_\epsilon(n,m)$ of the hybrid estimator holds, with outcome, pilot, and factorial intensities $u=M_0/8$ , $t_p=M_p/8$ , $t=M_f/8$ , degree $L=L(n)$ , light-cell scale $B=B(n,m,\epsilon)$ , and pilot threshold $k_0=k_0(n,m,\epsilon)$ . For each cell $x\in[d]$ , let $\mu_x$ be the product law under which $S_{ax}\sim\operatorname{Poi}(u\,q_{ax})$ and $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ , $a\in\{0,1\}$ , are independent; let $P_x=\frac{S_{1x}}u\widehat W_{1,L,x}-\frac{S_{0x}}u\widehat W_{0,L,x}$ be the light polynomial cell contribution of the hybrid estimator, with the factorial lift taken at degree $L$ , scale $B$ , and intensity $t$ ; and let $\pi_x=\mathbb P\{J_x\le k_0\}$ , where the pilot count $J_x=J_{0x}+J_{1x}$ has independent components $J_{ax}\sim\operatorname{Poi}(t_p\,s_{ax})$ . Then the light-branch second moments over heavy cells satisfy $\sum_{x:\,p_x>B}\pi_x\,\mathbb E_{\mu_x}\!\left[P_x^2\right]\le132\,n^{-12}.$
Lemma lem:hybrid-heavy-branch-mean-bound verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.sum_heavy_branchMean_sq_le Let $\epsilon>0$ , let $P\in\mathcal M_{d,\epsilon}$ , and let $n\ge1$ and $m\ge0$ be integers for which the calibration predicate $\mathsf C_\epsilon(n,m)$ of the hybrid estimator holds, with outcome, pilot, and factorial intensities $u=M_0/8$ , $t_p=M_p/8$ , $t=M_f/8$ , degree $L=L(n)$ , light-cell scale $B=B(n,m,\epsilon)$ , and pilot threshold $k_0=k_0(n,m,\epsilon)$ . For each cell $x\in[d]$ , let $P_x$ and $H_x$ be the light polynomial and heavy inverse-count cell contributions of the hybrid estimator (factorial lift at degree $L$ , scale $B$ , intensity $t$ ) under independent counts $S_{ax}\sim\operatorname{Poi}(u\,q_{ax})$ and $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ , and let $\pi_x=\mathbb P\{J_x\le k_0\}$ and $\eta_x=\mathbb P\{J_x>k_0\}$ , where the pilot count $J_x=J_{0x}+J_{1x}$ has independent components $J_{ax}\sim\operatorname{Poi}(t_p\,s_{ax})$ . Then the heavy-cell branch-mean term satisfies $\sum_{x:\,p_x>B}\pi_x\eta_x\bigl(\mathbb E_P P_x-\mathbb E_P H_x\bigr)^2\le282\,n^{-12}.$
Lemma lem:calibrated-hybrid-variance-bound verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.calibrated_hybrid_variance_le Let $0<\epsilon<1/2$ . There is a constant $C_{\mathrm{var}}>0$ , depending only on $\epsilon$ , with the following property. Let $d$ be an alphabet size, let $P\in\mathcal M_{d,\epsilon}$ , and let $n\ge1$ and $m\ge0$ be integers for which the calibration predicate $\mathsf C_\epsilon(n,m)$ of the hybrid estimator holds, with outcome, pilot, and factorial intensities $u=M_0/8$ , $t_p=M_p/8$ , $t=M_f/8$ , degree $L=L(n)$ , light-cell scale $B=B(n,m,\epsilon)$ , and pilot threshold $k_0=k_0(n,m,\epsilon)$ . Under the independent Poisson count law in which, independently across cells $x\in[d]$ and arms $a\in\{0,1\}$ , $S_{ax}\sim\operatorname{Poi}(u\,q_{ax})$ , $J_{ax}\sim\operatorname{Poi}(t_p\,s_{ax})$ , and $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ , let $P_x$ and $H_x$ be the light polynomial and heavy inverse-count cell contributions of the hybrid estimator (factorial lift at degree $L$ , scale $B$ , intensity $t$ ), let $J_x=J_{0x}+J_{1x}$ be the pilot count, and let $\widetilde Z=\sum_{x\in[d]}\bigl(\mathbf 1\{J_x\le k_0\}P_x+\mathbf 1\{J_x>k_0\}H_x\bigr)$ be the unclipped hybrid statistic. Then $\operatorname{Var}_P(\widetilde Z)\le 4A_\star^L\left\{\frac{\min\{1,dB\}}u+dB^2\right\}+C_{\mathrm{var}}\left(u^{-1}+t^{-1}\right)+\frac{8dB^2}{\epsilon^2L^4}+\frac4{\epsilon t}+414\,n^{-12}.$
Lemma lem:calibrated-hybrid-mean-bias verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.calibrated_hybrid_mean_bias_le Let $0<\epsilon<1/2$ , let $d$ be an alphabet size, let $P\in\mathcal M_{d,\epsilon}$ , and let $n\ge1$ and $m\ge0$ be integers for which the calibration predicate $\mathsf C_\epsilon(n,m)$ of the hybrid estimator holds, with outcome, pilot, and factorial intensities $u=M_0/8$ , $t_p=M_p/8$ , $t=M_f/8$ , degree $L=L(n)$ , light-cell scale $B=B(n,m,\epsilon)$ , and pilot threshold $k_0=k_0(n,m,\epsilon)$ . Under the independent Poisson count law in which, independently across cells $x\in[d]$ and arms $a\in\{0,1\}$ , $S_{ax}\sim\operatorname{Poi}(u\,q_{ax})$ , $J_{ax}\sim\operatorname{Poi}(t_p\,s_{ax})$ , and $K_{ax}\sim\operatorname{Poi}(t\,s_{ax})$ , let $P_x$ and $H_x$ be the light polynomial and heavy inverse-count cell contributions of the hybrid estimator (factorial lift at degree $L$ , scale $B$ , intensity $t$ ), let $J_x=J_{0x}+J_{1x}$ be the pilot count, and let $\widetilde Z=\sum_{x\in[d]}\bigl(\mathbf 1\{J_x\le k_0\}P_x+\mathbf 1\{J_x>k_0\}H_x\bigr)$ be the unclipped hybrid statistic. Then $\left|\mathbb E_P\widetilde Z-\tau(P)\right|\le\frac{2dB}{\epsilon L^2}+6\,n^{-12}.$
Lemma lem:finite-product-padding-identity verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.pi_prod_map_finitePadding Let $I$ and $J$ be finite index sets, let $\mathsf Y$ be a measurable space with measurable singletons, let $f:I\to J$ be injective, and let $r\in J$ satisfy $r\ne f(i)$ for every $i\in I$ . Let $\mu_i$ ( $i\in I$ ) and $\nu$ be probability measures on $\mathsf Y$ , and fix $y_0\in\mathsf Y$ . Define the padding map $\mathsf A:\mathsf Y^I\times\mathsf Y\to\mathsf Y^J$ by $\mathsf A(z,y)(r)=y$ , $\mathsf A(z,y)(f(i))=z_i$ for $i\in I$ (the index $i$ being unique by injectivity), and $\mathsf A(z,y)(j)=y_0$ for $j\notin\{r\}\cup f(I)$ , and define coordinate laws $\mathsf H_r=\nu$ , $\mathsf H_{f(i)}=\mu_i$ for $i\in I$ , and $\mathsf H_j=\delta_{y_0}$ for $j\notin\{r\}\cup f(I)$ . Then $\mathsf A_\#\Bigl(\Bigl(\bigotimes_{i\in I}\mu_i\Bigr)\otimes\nu\Bigr)=\bigotimes_{j\in J}\mathsf H_j .$
Lemma lem:finite-side-cover-approximate-decision verified Causalean.Stat.Minimax.FiniteSideInformation.finiteCover_approximateDecision Let $\Theta$ be a nonempty compact topological space, let $\mathsf Z$ be a finite set, let $\mathsf C$ be a finite nonempty set, and let $l\le u$ . For each $\theta\in\Theta$ let $\Lambda_\theta$ be a probability vector on $\mathsf Z$ , let $\Xi_\theta$ be a probability vector on $\mathsf C$ , and let $\upsilon_\theta\in[l,u]$ , where $\theta\mapsto\Lambda_\theta(z)$ , $\theta\mapsto\Xi_\theta(c)$ , and $\theta\mapsto\upsilon_\theta$ are continuous for every $z\in\mathsf Z$ and $c\in\mathsf C$ . For probability vectors $\nu,\nu'$ on $\mathsf C$ put $D_\infty(\nu,\nu')=\max_{c\in\mathsf C}|\nu(c)-\nu'(c)|$ , and define the exact-side minimax value $\mathfrak R_\star=\inf_{\psi}\sup_{\theta\in\Theta}\sum_{z\in\mathsf Z}\Lambda_\theta(z)\bigl(\psi(z,\Xi_\theta)-\upsilon_\theta\bigr)^2,$ where $\psi$ ranges over all maps $\mathsf Z\times\mathsf{Prob}(\mathsf C)\to[l,u]$ . Then for every $\delta>0$ there are an integer $k\ge1$ , centers $\theta^\circ_1,\ldots,\theta^\circ_k\in\Theta$ , decisions $\beta_1,\ldots,\beta_k:\mathsf Z\to[l,u]$ , radii $\operatorname{rad}_1,\ldots,\operatorname{rad}_k$ , and a margin $\eta>0$ with $\eta<\operatorname{rad}_j$ for every $j$ , such that every $\theta\in\Theta$ admits some $j$ with $D_\infty(\Xi_\theta,\Xi_{\theta^\circ_j})<\operatorname{rad}_j-\eta$ , and, for every $j$ and every $\theta\in\Theta$ , $D_\infty(\Xi_\theta,\Xi_{\theta^\circ_j})<\operatorname{rad}_j$ implies $\sum_{z\in\mathsf Z}\Lambda_\theta(z)\bigl(\beta_j(z)-\upsilon_\theta\bigr)^2\le\mathfrak R_\star+\delta .$

Theorems (7)

Theorem T-1 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.known_marginal_boundary There exist $0<c_\epsilon\le C_\epsilon<\infty$ such that, uniformly for $n\ge1$ and $d\ge2$ , $\frac{c_\epsilon}{n}\le R_\epsilon^{\mathrm{known}}(n,d)\le\frac{C_\epsilon}{n}.$
Theorem T-2 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.uniform_mixed_upper With $r_\epsilon(n,m,d)=\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2\log^2(en)}\right\},$ the total computable estimator $\widehat\tau^{\mathrm{mix}}_{n,m,d}$ in \mathrm{def:mixed\mbox{-}estimator\mbox{-}handle} satisfies, for a constant depending only on $\epsilon$ , $\sup_{P\in\mathcal M_{d,\epsilon}} E_{P^{\otimes n}\otimes P_{XA}^{\otimes m}} [\{\widehat\tau^{\mathrm{mix}}_{n,m,d}-\tau(P)\}^2] \le C_\epsilon r_\epsilon(n,m,d)$ simultaneously for every $n\ge1$ , $m\ge0$ , and $d\ge2$ .
Theorem T-3 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.common_marginal_converse For the explicit rate $r_\epsilon(n,m,d)=\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2\log^2(en)}\right\},$ the regime-dependent priors $\Pi_0,\Pi_1$ in \mathrm{def:common\mbox{-}marginal\mbox{-}prior\mbox{-}handle} induce exactly the same distribution of the entire random table $P_{XA}$ and imply, for a constant depending only on $\epsilon$ , $R_\epsilon(n,m,d)\ge c_\epsilon r_\epsilon(n,m,d)$ for every $n\ge1$ , $m\ge0$ , and $d\ge2$ , including the constant-risk regime.
Theorem T-4 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.sharp_annotation_frontier Uniformly over every $n\ge1$ , $m\ge0$ , and $d\ge2$ , $c_\epsilon\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2\log^2(en)}\right\} \le R_\epsilon(n,m,d)\le C_\epsilon\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2\log^2(en)}\right\}.$ At $m=0$ this is $r_\epsilon(n,0,d)=\min\!\left\{1,\frac1n+\frac{d^2}{n^2\log^2(en)}\right\},$ the endpoint recorded in $\texttt{stat\_discrete\_ate\_minimax\_loggap}$ . For every fixed $n,d$ , $\lim_{m\to\infty}R_\epsilon(n,m,d)=R_\epsilon^{\mathrm{known}}(n,d).$ Along arbitrary sequences $n\to\infty$ , with $N_n=n+m_n$ and $\ell_n=\log(en)$ , the following are necessary and sufficient: $R_\epsilon(n,m_n,d_n)\to0 \quad\Longleftrightarrow\quad \frac{d_n}{N_n\ell_n}\to0;$ $R_\epsilon(n,m_n,d_n)=O(n^{-1}) \quad\Longleftrightarrow\quad d_n=O\!\left(\frac{N_n\ell_n}{\sqrt n}\right);$ and $R_\epsilon(n,m_n,d_n)=o\{R_\epsilon(n,0,d_n)\}$ if and only if both $\frac{d_n}{\sqrt n\,\ell_n}\to\infty \quad\text{and}\quad \frac{d_n/(N_n\ell_n)}{\min\{1,d_n/(n\ell_n)\}}\to0.$
Theorem T-5 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverse_count_baseline There is a finite $C_\epsilon$ , depending only on $\epsilon$ , such that for all $n\ge1$ , $m\ge0$ , and $d\ge2$ , $R_\epsilon(n,m,d)\le C_\epsilon\min\!\left\{1,\frac1n+\frac{d^2}{(n+m)^2}\right\}.$ The witnessing estimator is the clipped and derandomized Poisson-prefix version of the all-heavy inverse-count statistic in \mathrm{lem:poisson\mbox{-}inverse\mbox{-}count\mbox{-}risk}.
Theorem T-6 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.binary_alphabet_parametric_rate For $d=2$ there are constants $0<c_\epsilon\le C_\epsilon<\infty$ such that, uniformly over $n\ge1$ and $m\ge0$ , $\frac{c_\epsilon}{n}\le R_\epsilon(n,m,2)\le\frac{C_\epsilon}{n}.$
Theorem T-7 verified CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.known_marginal_limit For every fixed $n\ge1$ and $d\ge2$ , $\lim_{m\to\infty}R_\epsilon(n,m,d)=R_\epsilon^{\mathrm{known}}(n,d).$

Auxiliary Lean lemmas (machine-checked) (187)

Definition AnnotationFrontierConsequences —
Definition BlockSizes —
Definition calibrationPredicate —
Definition ClassLaw —
Definition DiscreteLaw —
Definition randomAuxTableLaw —
Definition twoSampleMSE —
Definition annotationLaw —
Definition inverseCountEstimator —
Definition CommonMarginalRecipe —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorNullLaw —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorPertLaw —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorNullLaw_jointMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorPertLaw_jointMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorNullLaw_model —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorPertLaw_model —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorNullLaw_ate —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloorPertLaw_ate —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.parametricFloor_auxMarginal_eq —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotationMinimaxRisk_ge_two_model_tv —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalScore —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalEstimator —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.measurable_auxTable_sample_eval —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalScore_sample_measurable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalEstimator_measurable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalEstimator_mem —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.cellMass_eq_armMass_add —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.jointMass_nonneg —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.armMass_nonneg —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.cellMass_nonneg —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.jointMass_le_armMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.cellMass_sum_eq_one —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginal_arm_bounds —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalScore_mean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalScore_sq_integral_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalScore_variance_le —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRawEstimator —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRawEstimator_measurable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRawEstimator_mean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRawEstimator_variance_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.integral_sq_error_eq_variance_of_mean_eq —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRisk_upper —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.knownMarginalRisk_ge_two_model_tv —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalPriorOf_isProbabilityMeasure —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.map_auxMarginal_commonMarginalPriorOf_eq —
Definition CommonMarginalCalibration —
Definition commonMarginalParametricRegime —
Definition CommonMarginalPriorHandle —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poissonPrefixMass_le_one —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.jointMass_mem_unitInterval —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.cellMass_mem_unitInterval —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.outcomeMean_mem_unitInterval —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.observedMarginal_jointMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.ateFunctional_observedMarginal_eq_po_contrast —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_bias_envelope —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_bias_sum_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_bias_sum_le_min —
Definition CommonMarginalBounds —
Definition CommonMarginalCanonicalCertificate —
Definition CommonMarginalPoissonPriorWitness —
Definition CommonMarginalExactPoissonPriorWitness —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.markedMass_eq_armMass_mul_outcomeMean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.cellMass_sum_eq_one_annotation —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.overlap_armMass_bounds —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.poisson_natCast_memLp_two —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.shifted_reciprocal_memLp_two —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.shifted_reciprocal_pair_memLp_two —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.poisson_inverse_weight_second_moment_scalar —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.poisson_inverse_weight_poincare_energy_scalar —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_bias_min_rescale —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverse_count_bias_aggregate —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.PoissonReciprocal.poisson_natCast_first_moment —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_weight_mean_mul_rate —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_expectation_scalar —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_bias_envelope —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_cell_variance_sum_bound —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverse_count_outcome_addOne —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverse_count_marginal_addOne —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverse_count_variance_aggregate —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.inverseCountStatistic_memLp_and_addOne —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poissonCountLaw_eq_nestedPairedPoissonMeasure —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.varianceUnder_eq_variance —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poissonCountLaw_integral_cell —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poissonCountLaw_integral_arm —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_cross_moment_true —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_cross_moment_false —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_cross_moment —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_outcome_mean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_mean_formula —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_mean_envelope —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_arm_means_bias_aggregate —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poissonInverseArmMean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_count_bias_of_expectation_eq —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.nestedPaired_count_fst_le_total_annotation —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.nestedPaired_count_snd_le_total_annotation —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_weight_mean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_count_expectation_eq_arm_means —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_natCast_second_moment_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_weight_second_moment_bound —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_weight_second_moment_cell —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_outcome_coordinate_energy_bound —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_marginal_cross_square_moment —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_marginal_reciprocal_square_moment —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_bidirectional_poincare_energy_scalar —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_marginal_energy_with_outcome_rates —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_marginal_cell_energy_bound —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullAtomMass —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeAtomMass —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullLaw —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeLaw —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.integral_pi_fin_one —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.tvDist_le_one_sub_common —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternative_auxMarginal —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonAtom —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonSample —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonNullMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonAlternativeMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonProductMassNull —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonProductMassAlternative —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAtomEqOfCommon —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallCommonRestrictEq —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidNullAtom —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidNullSample —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidNullMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidAlternativeMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidProductMassNull —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAvoidProductMassAlternative —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallLabeledTV —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullCellMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeCellMass —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullPropensity —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativePropensity —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullControlMean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeControlMean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullTreatedMean —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeTreatedMeanFirst —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeTreatedMeanSecond —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullAuxTable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeAuxTable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallNullTarget —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeTarget —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeAuxIntegral —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeLabelIntegral —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.smallAlternativeExpectationIdentity —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.sqRisk_eq_varianceUnder_add_bias_sq —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.poisson_inverse_count_sqRisk_bound —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.logEN_pos_of_one_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.frontierRate_sequence_formula —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotation_frontier_consistency_iff —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotation_frontier_parametric_iff —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.positive_two_sided_bounds_isTheta —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.minimaxRisk_le_measurable_estimator —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.min_one_sq_nonnegative —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.strict_min_rates_iff_ratios —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.supervised_floor_ratio_iff —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.annotation_frontier_strict_improvement_iff —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_explicit_parametric_floor —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.fixed_sample_common_marginal_transfer_rule —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_mass_concentration —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_target_concentration —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_mass_concentration_at —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_prior_scale_tail —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_rawPriorCenter_abs_le_one —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_normalizedTarget_event —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_prior_target_concentration —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_raw_bayesSquared_eq_bayesDecision —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_raw_fuzzy_lower —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_poisson_lower_tail —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalLabeledMarkKernel —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalAuxMarkKernel —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalOrderedRawKernel —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_orderedRaw_isRandomScale —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalRawCountEquiv —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_predictive_reconstruction_contraction —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalTransferClip —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalTransferLoss —
Definition CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalFixedKernel —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalTransferLoss_measurable —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalTransferLoss_le_four —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_clippedLoss_lintegral —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginalTransferClip_sq_sub_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_clipping_mse_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_classGated_worstCase_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_mse_bddAbove —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_classGated_minimax_le —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_ofReal_minimax_le_classGated_worstCase —
Lemma CausalSmith.Stat.SemisupervisedDiscreteAteAnnotationFrontier.commonMarginal_ofReal_minimax_le_classGated —

Symbol realizations (Lean @realizes) (40)

X verified realized by Obs, ConditionalExchangeability
A verified realized by Obs, Consistency, ConditionalExchangeability
Y verified realized by Obs, Consistency
P verified realized by DiscreteLaw
p_x verified realized by cellMass, Overlap
s_{ax} verified realized by armMass
e_x verified realized by propensity, Overlap
mu_{ax} verified realized by outcomeMean
q_{ax} verified realized by markedMass
P_{XA} verified realized by auxMarginal, AuxTable
$\mathcal L_n$ verified realized by labeledProductLaw
$\mathcal U_m$ verified realized by auxProductLaw
$\mathcal T_{n,m,d}$ verified realized by Sample
n verified realized by variable@148
m verified realized by variable@148
d verified realized by variable@148, ModelClass
$\epsilon$ verified realized by variable@152, ModelClass
x verified realized by variable@154
a verified realized by variable@155
Y(a) verified realized by PotentialLaw, Consistency, ConditionalExchangeability
$\mathcal M_{d,\epsilon}$ verified realized by ModelClass
$\tau(P)$ verified realized by ateFunctional
$\mathcal E_{n,m,d,\epsilon}$ verified realized by annotationExperiment
$R_\epsilon(n,m,d)$ verified realized by minimaxRisk
N verified realized by frontierRate
$r_\epsilon(n,m,d)$ verified realized by frontierRate
$\mathcal E^{\mathrm{known}}_{n,d,\epsilon}$ verified realized by knownMarginalExperiment
$\mathcal T^{\mathrm{known}}_{n,d}$ verified realized by knownMarginalRisk
$R_\epsilon^{\mathrm{known}}(n,d)$ verified realized by knownMarginalRisk
$\Pi_0$ verified realized by commonMarginalPriorOf
$\Pi_1$ verified realized by commonMarginalPriorOf
$\widehat\tau^{\mathrm{mix}}_{n,m,d}$ verified realized by mixedEstimator
$\Pi_0^{\mathrm{small}}$ verified realized by finiteSmallInstance
$\Pi_1^{\mathrm{small}}$ verified realized by finiteSmallInstance
$\mathfrak C_{\mathrm{small}}$ verified realized by finiteSmallInstance
$c_\epsilon$ verified realized by common_marginal_converse, known_marginal_boundary, sharp_annotation_frontier
$C_\epsilon$ verified realized by known_marginal_boundary, sharp_annotation_frontier
$(m_n)_{n\ge1}$ verified realized by AnnotationFrontierConsequences
$(d_n)_{n\ge1}$ verified realized by AnnotationFrontierConsequences
$\Pi_a$ verified realized by commonMarginalPriorOf

Proof map 36 statements · 42 edges

Outcome Annotation and Minimax Risk for Discrete Average Treatment Effects

Abstract

Introduction

Related work

Model and experiment

Main minimax results

The hybrid estimator

Proof roadmap.

Interpretation and extensions

Limitations and future work.

Appendices

Upper-bound ingredients

Uniform finite-calibration input.

Common-marginal lower bounds

One-coordinate marked-Poisson input.

Known-marginal experiment and verification note

Finite-cover selection input.

Verification note.

Proofs of the main results

References