Quotient-Law Inference for Latent-Class Mean Effects at Collisions

The latent class and treatment arms satisfy $$\min_{1\le u\le k,\ t\in\{0,1\}} P(U=u,T=t)\ge \pi_0 .$$

The reference- and target-proxy feature matrices satisfy $$\min\{\sigma_k(A_0),\sigma_k(A_1),\sigma_k(B)\}\ge \sigma_0 .$$

Fix natural numbers $k,d_x,d_z$ and real constants $L,\pi_0,\sigma_0$. Assume:

• (Latent dimension.) $2\le k$, $k\le d_x$, and $k\le d_z$.
• (Bound and margins.) $1\le L$, $0<\pi_0\le (2k)^{-1}$, and $0<\sigma_0\le 1$.

Let $L_\tau=4L\sqrt{d_z}/\sigma_0$, and let $\mathcal K=\overline{\mathscr S}$ be the summary closure from Definition P-8. For two summaries $$s=(M_0,M_1,N_0,N_1,m_X), \qquad q=(M'_0,M'_1,N'_0,N'_1,m'_X),$$ write $$d_S(s,q) = \|M_0-M'_0\|+\|M_1-M'_1\| +\|N_0-N'_0\|+\|N_1-N'_1\| +\Bigl(\sum_i (m_X(i)-m'_X(i))^2\Bigr)^{1/2}.$$ There exists a constant $C_{\mathrm{mod}}>0$ such that, for every pair of full-data laws $P,Q\in\mathcal M$, $$W_1(\nu_P,\nu_Q)\le C_{\mathrm{mod}}\,d_S(S(P),S(Q)).$$ Moreover, there is a map $$\overline F:\mathcal K\to \mathcal P_k([-L_\tau,L_\tau])$$ such that $\overline F$ is continuous and Borel measurable, $$W_1(\overline F(q),\overline F(q')) \le C_{\mathrm{mod}}\,d_S(q,q') \qquad\text{for all }q,q'\in\mathcal K,$$ and $$\overline F(S(Q))=\nu_Q \qquad\text{for every }Q\in\mathcal M.$$ This map is unique among continuous maps $G:\mathcal K\to\mathcal P_k([-L_\tau,L_\tau])$ that obey $$W_1(G(q),G(q')) \le C_{\mathrm{mod}}\,d_S(q,q') \qquad\text{for all }q,q'\in\mathcal K$$ and satisfy $G(S(Q))=\nu_Q$ for every $Q\in\mathcal M$: every such $G$ satisfies $G(q)=\overline F(q)$ for all $q\in\mathcal K$.

Fix integers $k,d_x,d_z$ and constants $L,\pi_0,\sigma_0$. Suppose that

• (Dimensions.) $2\le k$, $k\le d_x$, and $k\le d_z$.
• (Uniform bounds.) $1\le L$, $0<\pi_0\le 1/(2k)$, and $0<\sigma_0\le 1$.

Then there is a constant $C=C(k,d_x,d_z,L,\pi_0,\sigma_0)>0$ such that, for every $n\ge 1$, there exist a structured-lattice estimator $\widehat\lambda_n$ taking values in $\mathcal P_k([-L_\tau,L_\tau])$, with $$L_\tau=4L\sqrt{d_z}/\sigma_0,$$ and a nearest-summary repair estimator $\widehat\nu_n=\overline F(\Pi(\widehat S_n))$ from Definition P-9, such that $\widehat\lambda_n$ is the first lexicographic minimizer over a finite exhaustive structured lattice of well-formed candidates $\vartheta$ for the criterion $$J_n(\vartheta;\widehat S_n),$$ with tie-breaking by the lattice order and output $\lambda_\vartheta$. Both estimators are Borel sample maps. For every $\eta$ satisfying $0<\eta<1/2$, every $P\in\mathcal M$, and $Q_P^{(n)}$-sample, $$Q_P^{(n)}\!\left\{ \max\!\left\{ W_1(\widehat\lambda_n,\nu_P), W_1(\widehat\nu_n,\nu_P) \right\} > C\sqrt{\frac{\log(C/\eta)}{n}} \right\}\le \eta .$$ Moreover, for each treatment arm $t\in\{0,1\}$, $$Q_P^{(n)}\{N_{t,n}=0\}\le (1-k\pi_0)^n .$$

Fix integers $k,d_x,d_z$ and constants $\pi_0,\sigma_0$ such that $$2\le k,\qquad k\le d_x,\qquad k\le d_z,\qquad 0<\pi_0\le {1\over 2k},\qquad 0<\sigma_0\le 1 .$$ There exists a simultaneous-concentration constant $C_0\ge 1$ such that, for every boundedness radius $L\ge 1$, there is a constant $C_{\mathrm{mod}}>0$ for which the prescribed structured-lattice constant $C_{\mathrm{lat}}$ is positive and the following holds. For every sample size $n\ge 1$, there exist:

• (Structured lattice estimator.) a measurable estimator $\widehat\lambda_n$ taking values in $\mathcal P_k([-L_\tau,L_\tau])$, generated by a finite exhaustive list of well-formed structured-lattice candidates, ordered lexicographically, with $\widehat\lambda_n$ equal to the effect law of the first lexicographic minimizer of the structured-lattice criterion;
• (Summary repair data.) a continuous extension $\overline F$ on the summary closure $\mathcal K$, a measurable nearest-summary selector $\Pi$ with its empty-closure fallback, and the associated measurable repaired summary-to-law map.

For every miscoverage level $\alpha$ satisfying $$0<\alpha< {1\over 2},$$ let $\mathcal C_{n,\alpha}$, $R_{n,\alpha}$, and $\mathcal C^{\mathrm{alg}}_{n,\alpha}$ be the confidence-set objects of Definition P-10 formed from this repair data and this structured-lattice estimator, with $m_\star=\pi_0$ and center $\widehat\lambda_n$. Then, on every sample, both $\mathcal C_{n,\alpha}$ and $\mathcal C^{\mathrm{alg}}_{n,\alpha}$ are nonempty, and $\mathcal C^{\mathrm{alg}}_{n,\alpha}$ has the finite constrained transport representation $$\nu\in\mathcal C^{\mathrm{alg}}_{n,\alpha} \quad\Longleftrightarrow\quad \operatorname{AtomFloor}(\pi_0,\nu)\ \text{and there exists a transport plan }\gamma\text{ from }\nu\text{ to }\widehat\lambda_n \text{ with cost at most }R_{n,\alpha}.$$ Moreover, for every full-data probability law $P\in\mathcal M$ in Definition P-1, $$1-\alpha \le Q_P^{(n)} \left\{ \nu_P\in\mathcal C_{n,\alpha}\cap\mathcal C^{\mathrm{alg}}_{n,\alpha} \right\}.$$ Finally, on every sample, $$\operatorname{diam}_{W_1}(\mathcal C_{n,\alpha}) \le 4C_{\mathrm{mod}}\, r_{n,\alpha}, \qquad \operatorname{diam}_{W_1}(\mathcal C^{\mathrm{alg}}_{n,\alpha}) \le 2R_{n,\alpha}.$$

Let $k,d_x,d_z\in\mathbb N$, $\pi_0\in\mathbb R$, and $\sigma_0\in\mathbb R$. Suppose that

• (Dimensions.) $2\le k$, $k\le d_x$, and $k\le d_z$.
• (Latent mass.) $0<\pi_0\le 1/(2k)$.
• (Rank margin.) $0<\sigma_0\le 1$.

Then there is a simultaneous-concentration constant $C_0\ge 1$ such that, for every $L\ge 1$, if $$L_\tau=\frac{4L\sqrt{d_z}}{\sigma_0},$$ and $$\begin{gathered} s_0=\pi_0\sigma_0^2,\qquad c_V=4\sqrt{d_xk},\qquad K_D=\frac{4\sqrt{k}LL_\tau}{\sigma_0},\\ A_D=\frac{8}{3s_0}+\frac{32L}{s_0^2},\qquad K_f=\frac{16\sqrt{d_x}kL^2}{\sigma_0^2},\\ c_D=2K_Dc_V+\frac{4k\sqrt{k}LL_\tau}{\sigma_0^2} +\frac{2\sqrt{k}L}{\sigma_0}+\frac{kL_\tau}{\sigma_0},\\ c_m=2\sqrt{k}Lc_V+k+kL,\qquad c_b=k+2\sqrt{k}Lc_V,\\ c_{\mathrm{grid}}=c_D+c_m+c_b,\qquad B_{\mathrm{lat}}=2(A_D+1)+c_{\mathrm{grid}}, \end{gathered}$$ then the prescribed structured-lattice stability constant $$C_{\mathrm{lat}} = \max\left\{ \frac{8L_\tau}{s_0}, \, (L_\tau+K_D+LK_f)B_{\mathrm{lat}} \right\}$$ is positive. For every $n\ge 1$, there exist a measurable structured-lattice estimator $\widehat\lambda_n$ and repair data consisting of a continuous extension on $\mathcal K$ and a Borel nearest-summary selector, such that $\widehat\lambda_n$ is obtained from an exhaustive lexicographically ordered finite list of well-formed structured-lattice candidates by choosing the first minimizer of $J_n(\vartheta;\widehat S_n)$. For every $\alpha$ with $0<\alpha<1/2$, the estimator has atom floor $\pi_0$. Moreover, for every $P\in\mathcal M$ in Definition P-1 and every sample, $$W_1\!\left(\widehat\lambda_n,\nu_P\right) \le C_{\mathrm{lat}} \left\{ d_S\!\left(\widehat S_n,S(P)\right)+\frac{1}{\sqrt n} \right\}.$$ Define $$r_{n,\alpha} = C_0L\sqrt{\frac{\log(C_0/\alpha)}{n}}, \qquad R_{n,\alpha} = C_{\mathrm{lat}}\left(r_{n,\alpha}+\frac{1}{\sqrt n}\right), \qquad \rho_{n,\alpha}=\frac{R_{n,\alpha}}{\pi_0},$$ with $\widehat S_n$ as in Definition P-3, and set $$E_P=\left\{\omega: d_S\!\left(\widehat S_n(\omega),S(P)\right)\le r_{n,\alpha} \right\}.$$ Then $$Q_P^{(n)}(E_P)\ge 1-\alpha.$$ For every sample $\omega\in E_P$, the cluster report $\mathfrak R_{n,\alpha}$ centered at $\widehat\lambda_n(\omega)$ satisfies the following properties for the representative atomic law $\nu_P$:

• (Partition.) Each support point $x\in\operatorname{supp}(\nu_P)$ belongs to a unique empirical component $C\in\widehat{\mathscr K}_{n,\alpha}$ with $x\in K_C(P)$.
• (Association.) For every $C\in\widehat{\mathscr K}_{n,\alpha}$, the set $K_C(P)$ is nonempty and is contained in $\operatorname{supp}(\nu_P)$.
• (Support interval.) If $C\in\widehat{\mathscr K}_{n,\alpha}$ and $x\in K_C(P)$, then $x$ lies in the reported support interval for $C$.
• (Mass interval.) If $C\in\widehat{\mathscr K}_{n,\alpha}$, then the reported mass interval $I_C$ contains $\nu_P(K_C(P))$.
• (Singleton width.) If $C\in\widehat{\mathscr K}_{n,\alpha}$ and $\operatorname{card}(K_C(P))=1$, then the reported support interval for $C$ has length at most $4\rho_{n,\alpha}$, and $$\operatorname{length}(I_C) \le \frac{8}{\pi_0} \min\left\{1,\frac{R_{n,\alpha}}{\delta(P)}\right\}.$$
• (External-gap width.) If $C\in\widehat{\mathscr K}_{n,\alpha}$ and $\Delta_C(P)=+\infty$, then $\operatorname{length}(I_C)=0$. For every $C\in\widehat{\mathscr K}_{n,\alpha}$, $$\operatorname{length}(I_C) \le \frac{8}{\pi_0} \min\left\{1,\frac{R_{n,\alpha}}{\Delta_C(P)}\right\},$$ with the displayed right side read as zero when $\Delta_C(P)=+\infty$.
• (Constrained representation.) The computable class $\mathcal C^{\mathrm{alg}}_{n,\alpha}$ has the finite constrained-program representation $$\xi\in\mathcal C^{\mathrm{alg}}_{n,\alpha} \quad\Longleftrightarrow\quad \operatorname{AtomFloor}(\pi_0,\xi) \ \text{and}\ \exists\gamma\ \text{with transport cost at most } R_{n,\alpha} \text{ from }\xi\text{ to }\widehat\lambda_n(\omega).$$

Let $k,d_x,d_z\in\mathbb N$ and $L,\pi_0,\sigma_0\in\mathbb R$. Assume:

• (Latent size.) $2\le k$.
• (Proxy dimensions.) $k\le d_x$ and $k\le d_z$.
• (Envelope.) $1\le L$.
• (Latent-arm margin.) $0<\pi_0\le (2k)^{-1}$.
• (Rank margin.) $0<\sigma_0\le 1$.

Then there is a constant $C=C(k,d_x,d_z,L,\pi_0,\sigma_0)>0$ such that, for every $n\ge1$, one can choose the simplex-valued measurable ordered-weight estimator $\widehat p_n^{\uparrow}$ from Definition P-12 so that, for every gap scale $g$ satisfying $0<g\le1/4$ and every full-data probability law $P\in\mathcal M(g)$, $$E_{Q_P^{(n)}}\!\left[\left\|\widehat p_n^{\uparrow}-p^{\uparrow}(P)\right\|_1\right] \le C\min\left\{1,\left(\sqrt n\,g\right)^{-1}\right\}.$$

Fix $c_{\mathrm{loc}}\in(0,1)$. There exist constants $a,c,C\in\mathbb R$ such that $$0<a\le \frac18,\qquad c>0,\qquad C>0,$$ and, for every integer $n\ge 1$, the following statements hold for the explicit submodel $k=d_x=d_z=2$, $L=2$, $\pi_0=1/10$, and $\sigma_0=1/10$.

• (Quotient-law risk.) For every measurable quotient-law estimator $\widetilde\nu_n$ with support radius $$L_\tau=\frac{4L\sqrt{d_z}}{\sigma_0},$$ there is a law $P$ in the local quotient-law experiment $\mathcal L_n^\nu$ of Definition P-14 such that $$P=P_0^{\mathrm{wit}} \quad\text{or}\quad P=P_{a/\sqrt n}^{\mathrm{wit}},$$ with $P_\varepsilon^{\mathrm{wit}}$ as in Definition P-13, and $$\frac{c}{\sqrt n} \le E_{Q_P^{(n)}}\!\left[W_1(\widetilde\nu_n,\nu_P)\right].$$
• (Ordered-weight risk.) For every $g\in(0,1/4]$ and every measurable ordered-weight estimator $\widetilde p_n$ taking values in the simplex, there is a law $P$ in the local labeled-weight experiment $\mathcal L_{n,g}^{p}$ of Definition P-15 such that $$P=P_{g,0} \quad\text{or}\quad P=P_{g,h(n,g)}, \qquad h(n,g)=a\min\left\{1,\frac{1}{\sqrt n\,g}\right\},$$ and $$c\min\left\{1,\frac{1}{\sqrt n\,g}\right\} \le E_{Q_P^{(n)}}\!\left[\|\widetilde p_n-p^{\uparrow}(P)\|_1\right].$$
• (Quotient-law certificate.) The laws $P_0^{\mathrm{wit}}$ and $P_{a/\sqrt n}^{\mathrm{wit}}$ belong to the corresponding local quotient-law experiments, and their observed-data marginals and quotient laws satisfy $$D_{\mathrm{KL}}\!\left((P_{a/\sqrt n}^{\mathrm{wit}})_O\Vert (P_0^{\mathrm{wit}})_O\right) \le \frac{C}{n}, \qquad \frac{c}{\sqrt n} \le W_1\!\left(\nu_{P_0^{\mathrm{wit}}},\nu_{P_{a/\sqrt n}^{\mathrm{wit}}}\right).$$
• (Labeled-path certificate.) For every $g\in(0,1/4]$, with $$h=h(n,g)=a\min\left\{1,\frac{1}{\sqrt n\,g}\right\},$$ the displacement $h$ lies in the tangent-amplitude domain, the laws $P_{g,h}$ and $P_{g,0}$ belong to the corresponding local labeled-weight experiments, and $$D_{\mathrm{KL}}\!\left((P_{g,h})_O\Vert (P_{g,0})_O\right) \le C g^2h^2, \qquad \sum_i\left|p_i^{\uparrow}(P_{g,h})-p_i^{\uparrow}(P_{g,0})\right| = 2|h|.$$