A PathIntegral Approach to Bayesian Inference for Inverse Problems Using the Semiclassical Approximation
Abstract
We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations (ODE, PDE), a problem which is typically illposed. Regularization of these problems using $L^2$ function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem  namely whether the subjective choice of regularization is compatible with prior knowledge. Using pathintegral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative pathintegral approaches, while offering alternatives to computational approaches like MarkovChainMonteCarlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a pathintegral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.
 Publication:

Journal of Statistical Physics
 Pub Date:
 November 2014
 DOI:
 10.1007/s109550141059y
 arXiv:
 arXiv:1312.2974
 Bibcode:
 2014JSP...157..582C
 Keywords:

 Inverse problems;
 Bayesian inference;
 Field theory;
 Path integral;
 Potential theory;
 Semiclassical approximation;
 Physics  Data Analysis;
 Statistics and Probability;
 High Energy Physics  Theory;
 Mathematics  Optimization and Control;
 Mathematics  Statistics Theory;
 Quantitative Biology  Quantitative Methods
 EPrint:
 Fixed some spelling errors and the author affiliations. This is the version accepted for publication by J Stat Phys