__Computational Methods for Bayesian Inverse Problems__

A. Stuart

**Abstract**

Many problems in the physical sciences require the determination of an unknown field from a finite set of indirect measurements.
Examples include oceanography, oil recovery, water resource management and weather forecasting. The Bayesian approach to these problems is natural for many reasons, including the under-determined and ill-posed nature of the inversion, the noise in the data and the uncertainty in the differential equation models used to describe complex mutiscale physics. The object of interest in the Bayesian approach is the posterior probability distribution on the unknown field.

However the Bayesian approach presents a computationally formidable task as it results in the need to probe a probability measure on function space. The talk will start by briefly overviewing the state-of-the-art in MCMC methods for the exploration of such probability measures [1]. It will then highlight two classes of methods which aim to simplfy the computational task based on approximating the probability measure by a Dirac measure or by a Gaussian. The motivation for doing this is partly related to the idea of Bayesian posterior consistency: the fact that, in the large data or small noise limits, the posterior distribution concentrates near the true value of the unknown field underlying the data [2].

The idea of approximating the posterior measure by a Dirac is the maximum a posteriori (MAP) estimator which, in words, computes the mostly likely point under the posterior probability distribution. It will be shown how to make sense of this idea in infinite dimensions, resulting in a problem from the calculus of variations; posterior consistency of the MAP estimator will also be studied [3]. We will then study approximation of the posterior measure by a Gaussian, looking for the closest Gaussian with respect to the Kullback-Leibler divergence. Again we show how to make sense of this in infinite dimensions, and we describe computational methods for the problem, based on the Robbins-Monro algorithm [4].

**Bibliography**

[1] S.L. Cotter, G.O. Roberts, A.M. Stuart and D. White,
MCMC methods for functions: modifying old algorithms
to make them faster, Statistical Science, to appear.
http://arxiv.org/abs/1202.0709

[2] S. Agapiou, S. Larsson and A.M. Stuart, Posterior
consistency of the Bayesian approach to linear ill-posed
inverse problems, Stochastic Processes and Applications, to appear.
http://arxiv.org/abs/1203.5753

[3] M. Dashti, K.J.H. Law, A.M. Stuart and J. Voss, MAP
estimators and posterior consistency in Bayesian nonparametric
inverse problems, Inverse Problems, submitted.
http://arxiv.org/abs/1303.4795

[4] F.J. Pinski, G. Simpson, A.M. Stuart and H. Weber, Kullback-Leibler approximations for measures on infinite dimensional spaces, in preparation.