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Bayes-reversed transition kernels for a diffusion

Definition

Let $T > 0$ , let $p_{t}$ be a strictly positive probability density on $R^{d}$ for each $t \in [0, T]$ , and let $q (s, x; t, y)$ be a jointly Borel, strictly positive transition density for $0 \leq s < t \leq T$ . Assume the marginal identity $p_{t} (y) = \int_{R^{d}} p_{s} (x) q (s, x; t, y) d x$ and the Chapman–Kolmogorov identity $q (s, x; t, y) = \int_{R^{d}} q (s, x; r, z) q (r, z; t, y) d z$ whenever $s < r < t$ . For $0 \leq u < v \leq T$ define $K_{u, v} (y, A) = \frac{\int_{A} p_{T - v} (x) q (T - v, x; T - u, y) d x}{p_{T - u} (y)} (A \subseteq R^{d} Borel),$ and set $K_{u, u} (y, \cdot) = δ_{y}$ . This is the Bayes-reversed transition probability kernel from reverse time $u$ to reverse time $v$ ; its source state $y$ is the forward-time state at $T - u$ and its output $x$ is the state at $T - v$ .

Definition references

Anderson, Sections 3–4: uses forward transition and marginal densities to describe finite-interval reversal under additional smoothness and uniqueness assumptions.

Tang and Zhao, Section 2.2: states the score-based reverse-drift formula and marks its derivation as heuristic.

Facts & Assumptions

Given: the positive Borel densities and marginal and Chapman–Kolmogorov identities in the definition, the measure-kernel convention, and Tonelli's theorem for nonnegative measurable functions.

[F1]

A probability kernel assigns a probability measure to each source state and has a measurable evaluation function for each Borel target set. Measure kernel and probability kernel

[F2]

The integral of a nonnegative jointly Borel function over one variable is measurable in the remaining variable, and iterated nonnegative integrals may be interchanged. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Verification

1.1F2F1

For fixed $u < v$ and Borel $A$ , the integrand $1_{A} (x) p_{T - v} (x) q (T - v, x; T - u, y)$ is nonnegative and jointly Borel in $(y, x)$ . Tonelli's parameter-integral clause [F2] makes its integral Borel in $y$ ; division by the positive Borel function $p_{T - u} (y)$ preserves measurability. For each fixed $y$ the numerator defines a measure in $A$ by countable additivity of the Lebesgue integral.

2.1F1step 1.1

Taking $A = R^{d}$ in the formula gives $K_{u, v} (y, R^{d}) = p_{T - u} (y) / p_{T - u} (y) = 1$ by the marginal identity, so each $K_{u, v} (y, \cdot)$ is a probability measure. The identity kernel at $u = v$ is also a probability kernel.

3.1F2step 1.1∎

If $u < v < w$ , Tonelli and the two density ratios give $\int K_{v, w} (x, A) K_{u, v} (y, d x) = \frac{\int_{A} p_{T - w} (z) \int q (T - w, z; T - v, x) q (T - v, x; T - u, y) d x d z}{p_{T - u} (y)} .$ The inner integral is $q (T - w, z; T - u, y)$ by forward Chapman–Kolmogorov, so the expression is $K_{u, w} (y, A)$ . If two adjacent times coincide, composition with the identity kernel gives the same equality.

Depends on

Used by

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Sources

B. D. O. Anderson, Reverse-time diffusion equation models, Sections 3–4 (standard reference, not scraped)
Y. Tang and H. Zhao, Score-based diffusion models via stochastic differential equations: a technical tutorial, Section 2.2 (standard reference, not scraped)