Efficient sampling of non log-concave posterior distributions with mixture of noises

TL;DR

This paper introduces a novel MCMC sampling method combining PMALA and MTM kernels to efficiently explore complex, non-log-concave, multimodal posterior distributions in challenging inverse problems with noisy and censored data.

Contribution

It develops an advanced Bayesian inference algorithm tailored for intractable, multimodal posteriors in complex inverse problems, overcoming challenges of non-log-concavity and large Lipschitz constants.

Findings

Successfully applied to classical multimodal distributions.

Demonstrated effectiveness on a realistic astronomical inverse problem.

Provided credible uncertainty quantification in complex inverse settings.

Abstract

This paper focuses on a challenging class of inverse problems that is often encountered in applications. The forward model is a complex non-linear black-box, potentially non-injective, whose outputs cover multiple decades in amplitude. Observations are supposed to be simultaneously damaged by additive and multiplicative noises and censorship. As needed in many applications, the aim of this work is to provide uncertainty quantification on top of parameter estimates. The resulting log-likelihood is intractable and potentially non-log-concave. An adapted Bayesian approach is proposed to provide credibility intervals along with point estimates. An MCMC algorithm is proposed to deal with the multimodal posterior distribution, even in a situation where there is no global Lipschitz constant (or it is very large). It combines two kernels, namely an improved version of (Preconditioned Metropolis Adjusted Langevin) PMALA and a Multiple Try Metropolis (MTM) kernel. Whenever smooth, its gradient admits a Lipschitz constant too large to be exploited in the inference process. This sampler addresses all the challenges induced by the complex form of the likelihood. The proposed method is illustrated on classical test multimodal distributions as well as on a challenging and realistic inverse problem in astronomy.