The University of Montana

Department of Mathematical Sciences

Technical report #28/2010

An Efficient MCMC Method for Uncertainty
Quantification in Inverse Problems

Johnatha M. Bardsley, Univ. of Montana

Department of Mathematical Sciences

University of Montana

Missoula, Montana 59812

USA

E-mail: bardsleyj@mso.umt.edu

**Abstract**

The connection between Bayesian statistics and the technique of
regularization for inverse problems has been given significant attention in recent years.
For example, Bayes’ law is frequently used as motivation for variational regularization
methods of Tikhonov type. In this setting, the regularization function corresponds to
the negative-log of the prior probability density; the fit-to-data function corresponds
to the negative-log of the likelihood; and the regularized solution corresponds to
the maximizer of the posterior density, known as the maximum a posteriori (MAP)
estimator. While a great deal of attention has been focused on the development of
techniques for efficient computation of MAP estimators (regularized solutions), less
explored is the problem of uncertainty quantification, which corresponds to the problem
of determining the shape, at least to some degree, of the posterior density in high
probability regions. One way to do this is to sample from the posterior density using
a Markov chain Monte Carlo (MCMC) method. In this paper, we present an MCMC
method for use on linear inverse problems with independent and identically distributed
Gaussian noise and Gaussian priors (quadratic regularization functions). From the
MCMC samples, an estimator (regularized solution), and measures of variability in
the estimator, are computed. Additionally, samples of the noise and prior precision
parameters are computed, making regularization parameter selection unnecessary.
**Keywords:** inverse problems, regularization, image reconstruction, Bayesian inference,
Markov chain Monte Carlo, uncertainty quantification.

**MSC numbers:** 15A29, 62F15, 65F22, 94A08

**Download Technical Report:** Pdf (179 KB)