Metropolis-adjusted Langevin algorithm (MALA)

Assume AC (The Axiom of Choice) and the target-potential hypotheses of Unadjusted Langevin algorithm (ULA). For a step size $\gamma >0$, use the ULA Gaussian proposal $Q_{\gamma }(x,dy)=q_{\gamma }(y\mid x)dy,q_{\gamma }(y\mid x)=(4\pi \gamma {)}^{-d\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{\mathrm{\parallel }y-x+\gamma \mathrm{\nabla }U(x){\mathrm{\parallel }}^2}{4\gamma }),$ whose kernel properties are in The ULA Gaussian proposal is a probability kernel with a positive density. The target density is $p_U(x)=Z_U^{-1}{e}^{-U(x)}$. At state $x$, propose $y$ from $Q_{\gamma }(x,\cdot )$, set ${\alpha }_{\gamma }(x,y):=\min \{1,\frac{p_U(y)q_{\gamma }(x\mid y)}{p_U(x)q_{\gamma }(y\mid x)}\},$ and use the Metropolis–Hastings transition kernel $P_{\gamma }(x,A)={\int }_A{\alpha }_{\gamma }(x,y)Q_{\gamma }(x,dy)+(1-\int {\alpha }_{\gamma }(x,z)Q_{\gamma }(x,dz)){\mathbf{1}}_A(x)\mathrm{.}$ Thus every rejected proposal remains at the current state. The target and proposal densities are strictly positive, so the displayed ratio has no zero-denominator case. The ratio is the common-density accepted-flow version from The common-density Metropolis–Hastings ratio; this procedure is the general Metropolis–Hastings algorithm Metropolis–Hastings kernel specialized to the ULA proposal. The Euclidean Borel space is standard Borel in the sense of Standard Borel spaces, by Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel, as required by that general Metropolis–Hastings procedure.