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Standard Markov chain Monte Carlo methods struggle to explore distributionsthat are concentrated in the neighbourhood of low-dimensional structures. Thesepathologies naturally occur in a number of situations. For example, they arecommon to Bayesian inverse problem modelling and Bayesian neural networks, whenobservational data are highly informative, or when a subset of the statisticalparameters of interest are non-identifiable. In this paper, we propose astrategy that transforms the original sampling problem into the task ofexploring a distribution supported on a manifold embedded in a higherdimensional space; in contrast to the original posterior this lifteddistribution remains diffuse in the vanishing noise limit. We employ aconstrained Hamiltonian Monte Carlo method which exploits the manifold geometryof this lifted distribution, to perform efficient approximate inference. Wedemonstrate in several numerical experiments that, contrarily to competingapproaches, the sampling efficiency of our proposed methodology does notdegenerate as the target distribution to be explored concentrates near lowdimensional structures.

Manifold lifting: scaling MCMC to the vanishing noise regime