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The ULA Gaussian proposal is a probability kernel with a positive density

Statement

Assume AC and the hypotheses of Unadjusted Langevin algorithm (ULA). For $γ > 0$ , its proposal family $Q_{γ}$ is a probability kernel on $R^{d}$ and has the everywhere positive finite jointly continuous Lebesgue density $q_{γ} (y ∣ x) = (4 π γ)^{- d / 2} \exp ⁣ (- \frac{∥ y - x + γ \nabla U (x) ∥^{2}}{4 γ}) .$

Facts & Assumptions

Given: AC, a finite $d \geq 1$ , $U \in C^{1} (R^{d})$ , $γ > 0$ , and the proposal in Unadjusted Langevin algorithm (ULA).

[A1]

AC holds as required by the algorithm and multivariate normal law. The Axiom of Choice Unadjusted Langevin algorithm (ULA)

[F1]

For positive-definite covariance, $N_{d} (m, Σ)$ has the displayed Gaussian Lebesgue density. A positive-definite multivariate normal law has its Gaussian density

[F2]

$N_{d} (m, Σ)$ is a probability law when $Σ$ is symmetric positive semidefinite; its coordinates and projections follow the multivariate-normal convention. Multivariate normal law, including singular covariance

[F3]

The definition of a probability kernel requires each section to be a probability measure and each evaluation on a Borel set to be measurable in the starting point. Measure kernel and probability kernel

[F4]

Integration of a nonnegative product-measurable function against a uniformly sigma-finite kernel is measurable in the starting point. Measurability of integration against a kernel

[F5]

Lebesgue measure on $R^{d}$ is sigma-finite in the sense of Finite, sigma-finite, and semifinite measures, with finite mass on bounded boxes. Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ Finite, sigma-finite, and semifinite measures Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure

[F6]

AC implies countable choice, which is the hypothesis of the continuous-map measurability result. AC supplies countable selections and prescribed serial paths The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F7]

A continuous Euclidean map is Borel measurable in the sense of Borel measurable and Lebesgue measurable functions on $R^{n}$ , under countable choice. Continuous functions on Euclidean spaces are Borel measurable Borel measurable and Lebesgue measurable functions on $R^{n}$

[F8]

The Borel sigma-algebra The Borel sigma-algebra of a topological space of $R^{2 d}$ is the product of the two Borel sigma-algebras on $R^{d}$ . The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}

Proof

Given: The hypotheses and conventions above.

1.1A1F1F2given

The covariance $2 γ I_{d}$ is symmetric positive definite. By [F2], the transition law in Unadjusted Langevin algorithm (ULA) is a probability measure for each $x$ . Applying [F1] with $m = x - γ \nabla U (x)$ gives the displayed density; its normalizing factor is finite and positive because $γ > 0$ .

2.1F3F4F5F6F7F8step 1.1∎

The map $(x, y) \mapsto x - y + γ \nabla U (x)$ is continuous, so the displayed $q_{γ} (y ∣ x)$ is Borel on $R^{2 d}$ by [F7], hence product-measurable by [F8]. For a Borel set $A$ , the constant kernel $K (x, B) = λ_{d} (B)$ is uniformly sigma-finite: $[- n, n]^{d}$ exhausts $R^{d}$ and has finite Lebesgue measure by [F5]. Applying [F4] to $f (x, y) = 1_{A} (y) q_{γ} (y ∣ x)$ gives measurability of $x \mapsto \int_{A} q_{γ} (y ∣ x) d y = Q_{γ} (x, A)$ . Together with step 1.1 this is exactly [F3], so $Q_{γ}$ is a probability kernel.

Depends on

Used by

Cited to discharge well-definedness by Unadjusted Langevin algorithm (ULA).

Dependency tree · two levels

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Sources

Alain Durmus and Éric Moulines, Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm, §1, equation (2) (standard reference, not scraped)
Gautam Iyer, Lecture 20: Langevin Monte Carlo, course notes, §3 (standard reference, not scraped)