3.3.1. Metropolis-Adjusted Langevin Algorithm

Previously we have discussed the Metropolis-Hastings algorithm. The idea of the Metropolis-Hastings algorithm is to make a new proposal θ* using random walk. Then this proposal is accepted with probability.

Although this algorithm benefits from desirable theoretical guarantees, the random walk proposal is not efficient, especially when the number of parameters in the model becomes large. Metropolis-adjusted Langevin algorithm (MALA), originally proposed in [34], is designed to solve the same problem—sample from the target distribution. The big advantage of MALA in comparison to Metropolis–Hastings is the construction for the proposal of the candidate parameter θ*. The proposal mechanism for MALA originates from the stochastic differential equation based on Langevin diffusion; the proposal mechanism reads

where we define L(θn)=log(p(θ)) and z∼N(z|0,I) and ϵ—integration step size. Convergence for this proposal is not guaranteed unless we employ a Metropolis acceptance probability after every integration step. For convenience, let us define

then the proposal density can be written as q(θ*|θn)=N(θ*|μ(θn,ϵ),ϵ2I). The standard acceptance probability follows

The type of proposal in Equation (29) is inefficient for strongly correlated parameters θ. To solve this issue, one can use a preconditioning matrix M

Unfortunately there is no principled way to choose matrix M. As we will see later, HMC encounters the same problem. Generally, MALA iterates between two general steps. First, Langevin dynamics is used for the proposals, and it exploits the gradients of the target. Second, the proposals are accepted or rejected similarly to those of the Metropolis–Hastings algorithm.