Title: Manifold MCMC methods for inference in inverse problems with highly informative observations
Authors: Matt Graham - Newcastle University (United Kingdom) [presenting]
Alexandre Thiery - National University of Singapore (Singapore)
Khai Xiang Au - National University of Singapore (Singapore)
Abstract: Inverse problems - inferring the configuration of a model of a physical system given observations - abound in engineering settings. Typically, the inverse problem is ill-posed. In such settings, Bayesian methodology offers a principled approach for combining prior knowledge with observations to infer the posterior distribution of plausible configurations of the model. A particularly challenging setting is where the data are highly informative with a large signal-to-noise ratio. Such observations lead to a posterior which concentrates around a lower-dimensional manifold embedded in the model configuration space. This high-fidelity observation regime is common in engineering settings where often the measurement process is carefully designed to minimise the effects of noise. Existing Markov chain Monte Carlo (MCMC) methods struggle in this regime, requiring an increasing computational effort as the signal-to-noise ratio grows. We will present a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher-dimensional space. In contrast to the original posterior, this lifted distribution remains diffuse in the limit of vanishing observation noise. By leveraging the geometry of this lifted posterior, we propose an MCMC method which remains efficient as the signal-to-noise ratio increases, allowing complex simulator models to be efficiently calibrated against high-fidelity observations.