# American Institute of Mathematical Sciences

February  2013, 7(1): 81-105. doi: 10.3934/ipi.2013.7.81

## Bayesian inverse problems with Monte Carlo forward models

 1 Department of Applied Physics and Applied Mathematics, Columbia University, 200 S. W. Mudd Building, MC 4701, 500 W. 120th Street, New York, NY 10027, United States, United States 2 Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

Received  December 2011 Revised  December 2012 Published  February 2013

The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called $MC^3$ that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.
Citation: Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
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