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An exact triangular transport disintegrates to the posterior

Statement

Assume the Axiom of Choice. Let $π$ be a probability measure on $R^{d} \times R^{m}$ , where $d, m \geq 1$ , and let $π_{Y} (B) : = π (R^{d} \times B)$ be its second marginal. Let $η$ be a probability measure on $R^{d}$ , and let $y \mapsto π_{y}$ be a probability kernel satisfying $π (A \times B) = \int_{B} π_{y} (A) π_{Y} (d y)$ for all Borel $A \subseteq R^{d}$ and $B \subseteq R^{m}$ . Suppose $T : R^{d} \times R^{m} \to R^{d} \times R^{m}$ is Borel and triangular, $T (x, y) = (T_{y} (x), y)$ , with each $T_{y}$ a Borel bijection, and define the image measure by $T_{#} μ (E) : = μ (T^{- 1} (E))$ . If $T_{#} (η \otimes π_{Y}) = π,$ then $T_{y}_{#} η = π_{y}$ as probability measures for $π_{Y}$ -almost every $y$ . By disintegration on standard Borel spaces, such a kernel $π_{y}$ exists and is unique outside one $π_{Y}$ -null set. In a proper Bayesian model, when $π$ is the joint parameter/data law, this conditional kernel is the posterior kernel; the conclusion says an exact triangular transport maps the reference law to the data-conditioned posterior for almost every datum.

For the triangular terminal flow in the preceding ODE lemma, joint continuity supplies the Borel map and each fiber is a bijection. The theorem requires only the exact joint pushforward identity; it does not require a density or a Jacobian formula.

Statement references

The ODE flow result supplies a jointly continuous triangular terminal map with bijective fibers.

The Bayesian model definition identifies the conditional kernel of the proper joint law as a posterior version.

Facts & Assumptions

Given: The Axiom of Choice, the joint law $π$ , its second marginal $π_{Y}$ , the reference probability $η$ , the supplied probability kernel $π_{y}$ , and the Borel triangular map and exact joint pushforward identity in the Statement.

[A1]

Assume the Axiom of Choice for standard-Borel disintegration existence and simultaneous almost-everywhere uniqueness; the choice-implication theorem gives Dependent Choice and Countable Choice from AC for the ODE supplier. The Axiom of Choice, AC implies DC implies countable choice, Disintegration of a joint law on standard borel spaces

[F1]

Finite-dimensional Euclidean spaces with their Borel sigma-algebras are standard Borel. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel

[F2]

For a probability measure on a product of standard-Borel spaces, a probability-kernel disintegration satisfies the rectangle identity and is unique outside one common marginal-null set. The theorem also gives the corresponding iterated-integral identity for nonnegative product-measurable functions. Disintegration of a joint law on standard borel spaces

[F3]

A probability kernel has a probability-measure section at every parameter, and each fixed-set evaluation is measurable in the parameter. Measure kernel and probability kernel

[F4]

Borel coordinate maps and finite coordinate permutations are measurable for the relevant finite-product Borel sigma-algebras, which agree with the product sigma-algebras. The product sigma-algebra and its finite iterates, A measurable function between measurable spaces, Coordinate maps and permutations are measurable on finite products

[F5]

Integrating a nonnegative product-measurable function against a probability kernel gives a measurable function of the parameter. Measurability of integration against a kernel

[F6]

The product of probability measures has the rectangle formula, and Tonelli identifies its nonnegative integral with either iterated integral. For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F7]

Under the Countable Choice and Dependent Choice supplied by [A1], the stated ODE hypotheses give a jointly continuous triangular terminal flow with bijective fibers; the optional C $^{1}$ density clause is separate. Global Lipschitz ODE flows and the C1 density formula

[F8]

In a proper Bayesian model on standard-Borel parameter and data spaces, the data-indexed conditional posterior kernel is a version of the posterior and is unique as a probability measure outside one data-marginal-null set. Misfit evidence and the potential posterior law

Proof

Given: [A1], [F1]–[F8], and the hypotheses and notation in the Statement.

1.1A1F1F2F3F4F5givenconstruct

By [A1], [F1], and [F2], the supplied $π_{y}$ is a disintegration of $π$ given its second coordinate. For Borel $A \subseteq R^{d}$ , define $κ_{y} (A) : = η ({x : T_{y} (x) \in A}) = \int_{R^{d}} 1_{A} (T_{y} (x)) η (d x)$ . Since $T_{y}$ is Borel, each $κ_{y}$ is the image probability measure $T_{y}_{#} η$ . For fixed $A$ , [F4] makes $(y, x) \mapsto 1_{A} (T_{y} (x))$ product-measurable; applying [F5] to the constant probability kernel $y \mapsto η$ makes $y \mapsto κ_{y} (A)$ measurable. Thus [F3] shows that $κ$ is a probability kernel.

2.1F2F4F6step 1.1givenalgebra

Fix Borel $A \subseteq R^{d}$ and $B \subseteq R^{m}$ . By [F6] and the definition of $κ_{y}$ , $\int_{B} κ_{y} (A) π_{Y} (d y) = (η \otimes π_{Y}) ({(x, y) : T_{y} (x) \in A, y \in B}) = (η \otimes π_{Y}) (T^{- 1} (A \times B))$ . The exact joint pushforward identity makes this $π (A \times B)$ , and the disintegration identity in the Statement makes it $\int_{B} π_{y} (A) π_{Y} (d y)$ . Hence $κ$ also disintegrates $π$ .

3.1F2step 2.1given

Apply the simultaneous almost-everywhere uniqueness in [F2] to the disintegration kernels $κ$ and $π_{y}$ . There is one Borel $π_{Y}$ -null set $N$ such that for every $y \notin N$ , $κ_{y} = π_{y}$ as measures on every Borel subset of $R^{d}$ . Since $κ_{y} = T_{y}_{#} η$ , this proves the asserted fiber identity.

4.1F7F8step 1.1step 3.1given∎

For a terminal map satisfying the ODE flow hypotheses, [F7] gives joint continuity and bijectivity of every fiber, so the exact pushforward case is covered by steps 1.1–3.1. By [F8], when $π$ is the proper Bayesian parameter/data joint law, the conditional kernel is a posterior version. No step uses the separate C $^{1}$ clause or its density formula.

Depends on

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Sources

Rick Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)
S. R. S. Varadhan, Probability Theory, Chapter 4 (standard reference, not scraped)