Computational Statistics
Table of Contents
1. Metropolis-Adjusted Langevin Algorithm
- MALA, Langevin Monte Carlo (LMC)
Given a probability distribution \( \pi \) on \( \mathbb{R}^d \), under the overdamped Langevin Itô diffusion \[ \dot{X} = \nabla \log \pi(X) + \sqrt{2} \dot{W}, \] \( f_{X_t}(x) \to \pi \) as \( t\to \infty \).
The approximate sample paths of the Langevin diffusion can be generated by many discrete-time methods. One of the simplest is the Euler-Maruyama method with a fixed time step \( \tau >0 \): \[ x_{k+1} := x_k + \tau \nabla \log \pi(x_k) + \sqrt{2\tau} \xi_k \] where \( \xi_k \) are independently drawn from a multivariate normal distribution with mean 0 and identity covariance matrix.