TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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MALA preserves its target probability exactly

Statement

Assume AC and the hypotheses of Metropolis-adjusted Langevin algorithm (MALA), with $d \geq 1$ , $U \in C^{1} (R^{d})$ , $0 < Z_{U} = \int e^{- U (x)} d x < \infty$ , and $γ > 0$ . The MALA transition kernel $P_{γ}$ is reversible with respect to $π_{U} (d x) = Z_{U}^{- 1} e^{- U (x)} d x,$ and therefore $π_{U}$ is invariant. This conclusion asserts neither irreducibility nor uniqueness, convergence, a rate, nor a central limit theorem.

Facts & Assumptions

Given: AC; the target and proposal specified by Metropolis-adjusted Langevin algorithm (MALA).

[A1]

AC supplies the accepted-flow Radon–Nikodym version in the general Metropolis–Hastings construction. The Axiom of Choice Metropolis–Hastings kernel

[F1]

MALA uses the ULA Gaussian proposal and explicit common-density acceptance ratio, with rejection mass placed at $x$ . Metropolis-adjusted Langevin algorithm (MALA)

[F2]

The proposal $Q_{γ}$ is a probability kernel with finite, positive, jointly measurable density $q_{γ} (y ∣ x)$ . The ULA Gaussian proposal is a probability kernel with a positive density

[F3]

The target density $p_{U} (x) = Z_{U}^{- 1} e^{- U (x)}$ is finite, positive, Borel measurable, and integrates to one, hence defines a probability measure in the sense of Probability measures and probability spaces. Unadjusted Langevin algorithm (ULA) Probability measures and probability spaces

[F4]

For a target density $p$ and jointly measurable proposal density $q$ against a common sigma-finite measure, the pointwise minimum ratio is a version of the accepted-flow Radon–Nikodym derivative. The common-density Metropolis–Hastings ratio

[F5]

A Metropolis–Hastings kernel built from a target probability and proposal probability kernel is reversible and leaves the target invariant. Metropolis–Hastings flow is reversible and invariant

[F6]

Lebesgue measure on $R^{d}$ is sigma-finite in the sense of Finite, sigma-finite, and semifinite measures. 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

[F7]

The Euclidean Borel space, with Borel sigma-algebra The Borel sigma-algebra of a topological space, is standard Borel in the sense of Standard Borel spaces, as required by the general Metropolis–Hastings suppliers. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel Standard Borel spaces The Borel sigma-algebra of a topological space

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3F4F6F7

By [F3], $π_{U}$ is a probability with positive finite density $p_{U}$ . By [F2], $Q_{γ}$ is a probability kernel with jointly measurable finite density $q_{γ}$ relative to the sigma-finite Lebesgue measure [F6]. The state-space hypothesis [F7] is also met. Thus the common-density ratio result [F4] applies at every pair $(x, y)$ with $p_{U} (x) q_{γ} (y ∣ x) > 0$ , which holds because both factors are strictly positive. It identifies the MALA acceptance function in [F1] with a version of the accepted-flow derivative; the chosen pointwise version is measurable and belongs to $[0, 1]$ .

2.1F1F5step 1.1∎

Step 1.1 and [F1] show that $P_{γ}$ is precisely the Metropolis–Hastings transition built from target $π_{U}$ , proposal $Q_{γ}$ , and an accepted-flow derivative version, including its rejected self-transition. Applying [F5] gives reversibility and invariance of $π_{U}$ . No ergodicity or convergence premise is among the hypotheses, so no such conclusion follows here.

Depends on

Used by

MALA corrects the Gaussian ULA proposal with a Hastings ratio Example

Cited to discharge well-definedness by Metropolis-adjusted Langevin algorithm (MALA).

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Sources

Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms, §2.1 and Proposition 2, printed pp. 5–6 (standard reference, not scraped)
Gautam Iyer, Lecture 20: Langevin Monte Carlo, course notes, §4 (standard reference, not scraped)