In the first part of the talk we consider the phenomenon of noise-level robustness of Markov chain Monte Carlo methods. Motivated by Bayesian inference with highly informative data we analyze the performance of random walk-like Metropolis-Hastings algorithms for approximate sampling of increasingly concentrating target distributions. We focus on Gaussian proposals which use a Hessian-based approximation of the target covariance. By means of pushforward transition kernels we show that for Gaussian target measures the spectral gap of the corresponding Metropolis-Hastings algorithm is independent of the concentration of the posterior, i.e., the noise level in the observational data that is used for Bayesian inference. Moreover, by exploiting the convergence of the concentrating posteriors to their Laplace approximation we extend the analysis to non-Gaussian target measures which either concentrate around a single point or along a linear manifold. In particular, in that setting we show that the average acceptance rate as well as the expected squared jump distance of suitable Metropolis-Hastings Markov chains do not deteriorate as the posterior concentrates.
In the second part we further exploit the concept of pushforward Markov kernels and derive dimension-independent Markov chain Monte Carlo methods for approximate sampling of posterior measures defined on high-dimensional spheres. Such measures occur, for instance, naturally in Bayesian level set inversion. Assuming an angular central Gaussian prior which models antipodally-symmetric directional data we exploit existing dimension-independent samplers in the ambient Hilbert space such as the preconditioned Crank Nicolson Metropolis and the elliptical slice sampler and derive MCMC algorithms on the sphere, which inherit reversibility and spectral gap properties from samplers in linear spaces.
This talk presents result obtained with Claudia Schillings (FU Berlin), Daniel Rudolf (U Passau), Philipp Wacker (U Erlangen-Nürnberg), Han Cheng Lie (U Potsdam) and Tim Sullivan (U Warwick).