This talk introduces the overdamped Langevin stochastic differential equation as a method for sampling from complex probability distributions, with brief historical context from statistical physics. We begin by deriving the infinitesimal generator of the Langevin diffusion and the associated Fokker–Planck equation, which governs the evolution of probability densities. This correspondence allows us to characterize invariant (stationary) distributions and to analyze qualitative dynamical behavior, including probability flow and transition times between modes of the distributions.

Exploiting the special Gibbs form of the stationary distribution, we show how overdamped Langevin dynamics can be used as a practical sampling mechanism for high-dimensional target distributions. We then compare classical Metropolis–Hastings algorithms with Langevin-based methods, highlighting their respective strengths, such as improved scalability with data through gradient information, as well as their limitations, including discretization bias and sensitivity to step size. We conclude with remarks on challenges that arise when applying Langevin-based samplers to latent-variable models, such as latent Dirichlet allocation and tree-structured latent variable models, where other methods such as Variational Inference perform much quicker with great results.