An important and challenging problem in Bayesian estimation is dealing with high dimensions. Monte Carlo methods, which have traditionally been the favored tools for extracting samples from Bayesian posteriors, suffer from the curse of dimensionality. The computational burden of a sampler scales with the dimension of the unknown quantity of interest, i.e., the unknown parameters in a Bayesian model. When the number of unknowns becomes too large, Monte Carlo methods become infeasible in practice, and practitioners have to rely on suboptimal methods.
Recent research efforts from the academic community have produced a number of adaptive Monte Carlo methods, which improve a sampler’s performance in high-dimensional scenarios. One of my goals as a researcher is to continue to advance the state of adaptive Monte Carlo to allow for Bayesian inference in problems of the highest complexity. Most of my work involves the design of hybrid methods, which embed optimization-based approaches within the Monte Carlo framework.