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Bayes-reversed Gaussian smoothing kernels

Definition

Assume AC. Fix $T > 0$ , a compactly supported Borel probability measure $μ$ on $R^{d}$ , a continuously differentiable function $g : [0, T] \to (0, \infty)$ bounded above and away from zero, and $v (t) = \int_{0}^{t} g (r)^{2} d r$ . For $t > 0$ put $p_{t} (y) = \int_{R^{d}} ϕ_{v (t)} (y - z) μ (d z), q (s, x; t, y) = ϕ_{v (t) - v (s)} (y - x) (0 \leq s < t \leq T),$ where $ϕ_{a}$ is the centered Gaussian density with covariance $a I_{d}$ . Put $p_{0} = μ$ as a probability measure, not as a Lebesgue density. For $0 \leq u < v < T$ define $K_{u, v} (y, d x) = \frac{p_{T - v} (x) q (T - v, x; T - u, y)}{p_{T - u} (y)} d x,$ and for $0 \leq u < T$ at the singular data endpoint define $K_{u, T} (y, A) = \frac{\int_{A} q (0, x; T - u, y) μ (d x)}{p_{T - u} (y)} (A Borel) .$ Set $K_{u, u} (y, \cdot) = δ_{y}$ . These are the Bayes-reversed kernels of the additive Gaussian smoothing process, including the posterior kernel at the original data time; no density for $μ$ is required.

Definition references

Gaussian-smoothed law and transition density: supplies the explicit positive-time marginal density, Gaussian transition density, and forward marginal and Chapman–Kolmogorov identities.

Anderson, Sections 3–4: context for Bayes reversal from forward transition densities.

Facts & Assumptions

Given: AC, $μ$ , $g$ , $v$ , $p_{t}$ , and $q$ in the definition.

[A1]

The Gaussian smoothing supplier assumes AC, compact support, $p_{t} > 0$ for $t > 0$ , the displayed transition density, the forward marginal identity, and Chapman–Kolmogorov; it defines $p_{0} = μ$ without asserting a density there. The Axiom of Choice Gaussian-smoothed data law and transition density

[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]

Nonnegative joint Borel densities can be integrated in either order; parameter integrals are measurable. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Verification

1.1F2A1F1

For $u < v < T$ , the integrand $1_{A} (x) p_{T - v} (x) q (T - v, x; T - u, y)$ is jointly Borel and nonnegative; Tonelli [F2] makes its integral measurable in $y$ , and countable additivity in $A$ follows from countable additivity of the integral. For $v = T$ , the integrand $1_{A} (x) q (0, x; T - u, y)$ is jointly Borel and nonnegative relative to the finite measure $μ$ , so the same parameter-integral argument applies. All denominators are positive because $T - u > 0$ .

2.1A1F1step 1.1

If $v < T$ , the forward marginal identity [A1] gives $\int p_{T - v} (x) q (T - v, x; T - u, y) d x = p_{T - u} (y)$ ; if $v = T$ , the same identity with initial measure $μ$ gives $\int q (0, x; T - u, y) μ (d x) = p_{T - u} (y)$ . Thus both formulas have total mass one. The equal-time kernel is a probability measure by definition.

3.1A1F2step 1.1∎

For $u < v < w < T$ , composing the density ratios cancels $p_{T - v} (x)$ , and Tonelli followed by forward Chapman–Kolmogorov [A1] gives $K_{u, v} K_{v, w} = K_{u, w}$ . If $w = T$ , the same computation leaves $μ (d z)$ times $\int q (0, z; T - v, x) q (T - v, x; T - u, y) d x = q (0, z; T - u, y),$ so the endpoint formula gives $K_{u, v} K_{v, T} = K_{u, T}$ . Cases with equal adjacent times use the identity kernel.

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)