Publications

Map Estimators for Bayesian Nonparametrics

Bibliography:

M. Dashti, K. J. H. Law, A.M. Stuart and J. Voss.  Map Estimators for Bayesian Nonparametrics.  Inverse Problems, 29 095017 (2013).

Authors:

M. Dashti, K. J. H. Law, A.M. Stuart and J. Voss

Keywords:

Probability, Bayesian non-parametrics, Inverse problems

Year:

2013

Abstract:

We consider the inverse problem of estimating an unknown function $u$ from noisy measurements $y$ of a known, possibly nonlinear, map $\mathcal{G}$ applied to $u$. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field $\mu_0$. We work under a natural set of conditions on the likelihood which imply the existence of a well-posed posterior measure, $\mu^y$. Under these conditions we show that the {\em maximum a posteriori} (MAP) estimator is well-defined as the minimiser of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency. We also prove a similar result for the case where the observation of $\mathcal{G}(u)$ can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics.

ISSN:

doi:10.1088/0266-5611/29/9/095017