# American Institute of Mathematical Sciences

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
##### References:
 [1] Simon Arridge, et al., Approximation errors and model reduction with an application in optical diffusion tomography,, Inverse Problems, 22 (2006). [2] Guillaume Bal, Anthony Davis and Ian Langmore, A hybrid (Monte Carlo/deterministic) approach for multi-dimensional radiation transport,, J. Computational Physics, 230 (2011), 7723. [3] George Casella and Robert Berger, "Statistical Inference,", Duxbury, (2002). [4] Jin Chen and Xavier Intes, Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,, Optics Express, 17 (2009). [5] J. Andrés Christen and Colin Fox, Markov chain Monte Carlo using an approximation,, Journal of Computational and Graphical Statistics, 14 (2005), 795. doi: 10.1198/106186005X76983. [6] Rick Durrett, "Probability: Theory and Examples,", third edition, (2005). [7] Yalchin Efendiev, Thomas Hou and W. Luo, Preconditioning Markov chain Monte Carlo simulations using coarse-scale models,, SIAM J. Sci. Comput., 28 (2006), 776. doi: 10.1137/050628568. [8] Charles J. Geyer, Practical Markov chain Monte Carlo,, Statistical Science, 7 (1992), 473. [9] Carole K. Hayakawa and Jerome Spanier, Perturbation Monte Carlo methods for the solution of inverse problems,, in, (2004), 227. [10] Carole K. Hayakawa, Jerome Spanier, and Frédéric Bevilacqua, et al., Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,, Optics Letters, 26 (2001), 1333. [11] Jari P. Kaipio and Erkki Somersalo, "Statistical and Computational Inverse Problems,", Applied Mathematical Sciences, 160 (2005). [12] Jari P. Kaipio and Erkki Somersalo, Statistical inverse problems: Discretization, model reduction, and inverse crimes,, Journal of Computational and Applied Mathematics, 198 (2007), 493. doi: 10.1016/j.cam.2005.09.027. [13] Ian Langmore, Anthony Davis and Guillaume Bal, Multi-pixel retrieval of structural and optical parameters in a 2D scene with a path-recycling Monte Carlo forward model and a new Bayesian inference engine,, IEEE TGRS, (2012). [14] Jun S. Liu, "Monte Carlo Strategies in Scientific Computing,", Springer Series in Statistics, (2008). [15] David Moulton, Colin Fox and Daniil Svyatskiy, Multilevel approximations in sample-based inversion from the Dirichlet-to-Neumann map,, Journal of Physics: Conference Series, (2008). [16] Hanna K. Pikkarainen, State estimation approach to nonstationary inverse problems: Discretization error and filtering problem,, Inverse Problems, 22 (2006), 365. doi: 10.1088/0266-5611/22/1/020. [17] Christian Robert and George Casella, "Monte Carlo Statistical Methods,", Second edition, (2004). [18] Luke Tierney, Markov chains for exploring posterior distributions,, The Annals of Statistics, 22 (1994), 1701. doi: 10.1214/aos/1176325750.

show all references

##### References:
 [1] Simon Arridge, et al., Approximation errors and model reduction with an application in optical diffusion tomography,, Inverse Problems, 22 (2006). [2] Guillaume Bal, Anthony Davis and Ian Langmore, A hybrid (Monte Carlo/deterministic) approach for multi-dimensional radiation transport,, J. Computational Physics, 230 (2011), 7723. [3] George Casella and Robert Berger, "Statistical Inference,", Duxbury, (2002). [4] Jin Chen and Xavier Intes, Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,, Optics Express, 17 (2009). [5] J. Andrés Christen and Colin Fox, Markov chain Monte Carlo using an approximation,, Journal of Computational and Graphical Statistics, 14 (2005), 795. doi: 10.1198/106186005X76983. [6] Rick Durrett, "Probability: Theory and Examples,", third edition, (2005). [7] Yalchin Efendiev, Thomas Hou and W. Luo, Preconditioning Markov chain Monte Carlo simulations using coarse-scale models,, SIAM J. Sci. Comput., 28 (2006), 776. doi: 10.1137/050628568. [8] Charles J. Geyer, Practical Markov chain Monte Carlo,, Statistical Science, 7 (1992), 473. [9] Carole K. Hayakawa and Jerome Spanier, Perturbation Monte Carlo methods for the solution of inverse problems,, in, (2004), 227. [10] Carole K. Hayakawa, Jerome Spanier, and Frédéric Bevilacqua, et al., Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,, Optics Letters, 26 (2001), 1333. [11] Jari P. Kaipio and Erkki Somersalo, "Statistical and Computational Inverse Problems,", Applied Mathematical Sciences, 160 (2005). [12] Jari P. Kaipio and Erkki Somersalo, Statistical inverse problems: Discretization, model reduction, and inverse crimes,, Journal of Computational and Applied Mathematics, 198 (2007), 493. doi: 10.1016/j.cam.2005.09.027. [13] Ian Langmore, Anthony Davis and Guillaume Bal, Multi-pixel retrieval of structural and optical parameters in a 2D scene with a path-recycling Monte Carlo forward model and a new Bayesian inference engine,, IEEE TGRS, (2012). [14] Jun S. Liu, "Monte Carlo Strategies in Scientific Computing,", Springer Series in Statistics, (2008). [15] David Moulton, Colin Fox and Daniil Svyatskiy, Multilevel approximations in sample-based inversion from the Dirichlet-to-Neumann map,, Journal of Physics: Conference Series, (2008). [16] Hanna K. Pikkarainen, State estimation approach to nonstationary inverse problems: Discretization error and filtering problem,, Inverse Problems, 22 (2006), 365. doi: 10.1088/0266-5611/22/1/020. [17] Christian Robert and George Casella, "Monte Carlo Statistical Methods,", Second edition, (2004). [18] Luke Tierney, Markov chains for exploring posterior distributions,, The Annals of Statistics, 22 (1994), 1701. doi: 10.1214/aos/1176325750.
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