Courses

Abstract: Implementation of standard MCMC algorithms requires one to be able to evaluate an acceptance ratio which depends in particular on the targeted probability distribution. In some situations this cannot be achieved, either because the corresponding posterior density is expensive to evaluate, or because it does not admit an analytical expression. It has been shown that in some circumstances computationally cheap estimators of such acceptance ratios can be used instead and nevertheless lead to correct Markov chain Monte Carlo algorithms. Correctness here means that these noisy algorithms share the same theoretical convergence guarantees enjoyed by standard MCMC algorithms for which the posterior density can be evaluated. The lectures will focus on a presentation of the methodology and a discussion of the theoretical properties of this class of algorithms.

Abstract: This course starts with Metropolis-Hastings chains that take as proposal moves time discretizations of self-adjoint diffusions (e.g. MALA); and then studies MCMC methods that sample the stationary distribution of (and more generally simulate) high-dimensional, non-symmetric diffusions.

Abstract: The need to sample from probability measures in high dimensions arises naturally from the finite dimensional approximation of Bayesian inverse problems and conditioned diffusion processes. For prior measures which are Gaussian and for additive noise diffusions, respectively, these measures are defined via their density with respect to a Gaussian. The lectures will focus on the efficiency of MCMC methods for such problems, and will comprise two main components: (i) the study of standard methods such as RWM, MALA and HMC; (ii) the development of new methods which exploit the underlying Gaussian structure for gains in efficiency.