SCM.Factored

For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a natural number $n$ no greater than its number of observed nodes, the observed-prefix value space is the one-point space of the empty assignment when $n=0$, and the product of the preceding prefix space and the value space of the $n$-th observed node when $n$ is positive.

Random values consisting of the latent tuple paired with an observed prefix. This is the state-space of the kernel at step n in the factored construction.

For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a prefix length no greater than the number of its observed nodes, an assignment on that prefix, and a position within the prefix, the observed-prefix coordinate reader has no value when the prefix is empty, and otherwise returns the assigned value at that position, reading recursively from the preceding prefix or directly from its final coordinate.

For a finite collection of distinguishable node labels with measurable value spaces, a structural causal model, a natural number $n$, and evidence that the model has at least $n+1$ observed nodes, the ordered-latent prefix extension map takes a latent assignment, an assignment to the first $n$ observed nodes, and the value of the next observed node, and returns the same latent assignment paired with the resulting length-$n+1$ observed prefix.

For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the kernel giving the distribution of the latent-node values conditional on the fixed-node values assigns the same latent product distribution to every fixed-node assignment.

For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the zero-step prefix kernel maps every fixed-node assignment to the joint distribution of all latent-node values and the unique empty observed-node prefix at prefix length zero.

For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the prefix kernel maps each fixed-node assignment to the joint distribution of all latent-node values and the values of the first specified number of observed nodes. At zero observed nodes it is the zero-step prefix kernel; at each positive prefix length it extends the preceding prefix distribution by the next observed node's deterministic structural equation.

For a structural causal model and an index $n$ for which at least $n+1$ observed vertices exist, the deterministic step function maps a fixed-value assignment together with the latent and already generated observed-prefix values to the value of the $n$-th observed vertex in the model's canonical topological order, by applying that vertex's structural function to the values of its parents.

For a structural causal model and an index $n$ for which at least $n+1$ observed vertices exist, the step kernel is the probability kernel that assigns unit mass to the value of the $n$-th observed vertex produced by its structural function from the fixed values, latent values, and already generated observed-prefix values.

For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that this integer does not exceed the number of observed nodes, the deterministic prefix-state map takes fixed-node values and latent-node values and returns the latent-node values together with the values generated for the first specified number of observed nodes. At zero observed nodes it returns the latent-node values and the unique empty prefix; at each positive prefix length it first forms the preceding prefix and then appends the value given by the next node's structural equation.

For a finite collection of nodes with a measurable outcome space for each node and a structural causal model, the map from a completed latent-and-observed prefix state to an assignment of all random nodes assigns each observed node its completed prefix value and each unobserved node its latent value.

For a finite collection of distinguishable base-variable labels, a SWIG graph, and a graph node, the observed-predecessor set consists exactly of the observed nodes that occur strictly before that node in the graph's topological order.

For a finite collection of distinguishable base-variable labels with a measurable value space attached to each label, a structural causal model, and a nonnegative integer, the prefix node set consists of precisely the first $n$ observed nodes in its canonical topological order; if $n$ is at least the number of observed nodes, it is the full observed-node set.

For an index population, a singleton index in a family of value spaces, and an assignment on that singleton, the singleton-coordinate value is the assignment's value at that index.

For an index population, a singleton index in a family of value spaces, and a value at that index, the singleton assignment is the assignment on the singleton set whose sole coordinate equals that value.

For a structural causal model, an index strictly below its number of observed nodes, a standard Borel and nonempty value space for the observed node at that index, and a countably generated conditioning σ-algebra for the fixed values and preceding observed values, the one-step observational conditional kernel gives the conditional distribution of that node's value given the fixed values and all earlier observed values.

For a structural causal model, the empty-prefix assignment is the unique assignment of values to its empty initial observed-node set.

For a structural causal model, the zero-step observational chain kernel assigns, at every fixed-value setting, unit probability to the unique assignment on the empty observed prefix.

For a structural causal model and an index strictly below its number of observed nodes, the extended prefix assignment maps an assignment on the first $n$ observed nodes together with a value for the next node to the assignment on the first $n+1$ observed nodes that retains the prefix values and appends that value.

For a structural causal model, standard Borel and nonempty value spaces for every observed node, and countably generated conditioning σ-algebras for every observed prefix, the observational chain kernel assigns to every nonnegative integer not exceeding the number of observed nodes the kernel obtained at zero by the point mass on the empty prefix and at each positive index by composing the preceding chain kernel with the next one-node conditional kernel and extending the prefix.

For a structural causal model, standard Borel and nonempty value spaces for every observed node, and countably generated conditioning σ-algebras for every observed prefix, the full observational chain-rule product is the full-length observational chain kernel transported from the complete prefix to the observed-value space.

For a finite collection of nodes with a measurable outcome space for each node, a structural causal model, a nonnegative integer, and proof that the next position exists among the observed nodes, the map producing the values of all parents of the next observed node takes fixed-node values, latent-node values, and a prefix of the preceding observed-node values, and returns the corresponding value for every parent of that next observed node.