Langevin Algorithm

Definition

The Langevian algorithm (Langevin MCMC) is an algorithm for generating samples from a probability distribution for which we have access to the gradient of its Score Function.

The Langevian updating process is defined as $x_{t+1}=x_t-\gamma \nabla \ln p(x_t)+\sqrt{2\gamma }{ϵ}_t$ where $x_t$ is the sample at iteration $t$, $\gamma >0$ is the step-size, $\nabla \ln p(x_t)$ is the score of the target-distribution, and ${ϵ}_t\sim N(0,I)$ is a Gaussian noise.

Starting from an initial point $x_0$, we iteratively apply this updating to generate a sequence of samples