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MALA corrects the Gaussian ULA proposal with a Hastings ratio

Example

Let $U (x) = x^{2} / 2$ and $π = N (0, 1)$ on $R$ . For every $γ > 0$ , the MALA proposal is $q_{γ} (y ∣ x) = N ((1 - γ) x, 2 γ)$ , and its logarithmic Hastings ratio is $\log \frac{p (y) q_{γ} (x ∣ y)}{p (x) q_{γ} (y ∣ x)} = \frac{γ}{4} (x^{2} - y^{2}) .$ Thus the acceptance probability is $\min {1, e^{γ (x^{2} - y^{2}) / 4}}$ , and the resulting rejected-self-transition kernel preserves $N (0, 1)$ .

Facts & Assumptions

Given: AC, the standard Gaussian target, and the MALA construction.

[A1]

AC is assumed for the Gaussian law and the general Metropolis–Hastings construction. The Axiom of Choice Metropolis-adjusted Langevin algorithm (MALA)

[F1]

The ULA Gaussian proposal has density $q_{γ} (y ∣ x) = (4 π γ)^{- 1 / 2} \exp (- (y - (1 - γ) x)^{2} / (4 γ))$ . The ULA Gaussian proposal is a probability kernel with a positive density

[F2]

The standard Gaussian target density is proportional to $e^{- x^{2} / 2}$ . Standard normal and normal laws

[F3]

The MALA accept/reject kernel uses $\min (1, p (y) q (x ∣ y) / (p (x) q (y ∣ x)))$ and places every rejected proposal at its current state; this common-density ratio is a version of the accepted-flow derivative. Metropolis-adjusted Langevin algorithm (MALA) The common-density Metropolis–Hastings ratio

[F4]

The MALA kernel is reversible with respect to its target and therefore invariant. MALA preserves its target probability exactly

Verification

Given: $U (x) = x^{2} / 2$ , $γ > 0$ , and [F1]–[F4].

1.1F1F2algebra

Write $a = 1 - γ$ . The Gaussian normalizing constants cancel, and direct expansion gives $\log \frac{q_{γ} (x ∣ y)}{q_{γ} (y ∣ x)} = \frac{(y - a x)^{2} - (x - a y)^{2}}{4 γ} = \frac{2 - γ}{4} (y^{2} - x^{2}) .$ The target log-density ratio is $\log (p (y) / p (x)) = - (y^{2} - x^{2}) / 2$ , so adding the two terms yields $γ (x^{2} - y^{2}) / 4$ .

2.1F1F3step 1.1

By [F3], the acceptance probability is $\min {1, e^{γ (x^{2} - y^{2}) / 4}}$ and rejected proposals remain at $x$ . At $γ = 1$ , the proposal is independent $N (0, 2)$ , but its reverse-to-forward density ratio is still $q_{1} (x) / q_{1} (y)$ and is included in the computed exponent.

3.1A1F1F3F4step 2.1∎

The proposal is a probability kernel with positive density by [F1], and [F3] identifies the displayed acceptance rule with the MALA specialization of Metropolis–Hastings. Applying [F4] proves exact invariance of $N (0, 1)$ for every $γ > 0$ .

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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)