Optimization-Based Markov Chain Monte Carlo Methods for Nonlinear Hierarchical Statistical Inverse Problems

Johnathan M. Bardsley, Tiangang Cui

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6 Scopus citations


In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and nonconcave structures in the joint posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parameter-to-observable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomize-then-optimize (RTO) method [J. M. Bardsley et al., SIAM J. Sci. Comput., 42 (2016), pp. A1317-A1347] for exploring the high- or infinite-dimensional parameter space. We first extend the RTO machinery to the Poisson likelihood and discuss the implementation of RTO in the hierarchical setting. Then, by using RTO either as a proposal distribution in a Metropolis-within-Gibbs update or as a biasing distribution in the pseudomarginal MCMC [C. Andrieu and G. O. Roberts, Ann. Statist., 37 (2009), pp. 697-725], we present efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudomarginal MCMC has sampling performance robust to model parameter dimensions. Numerical examples in PDE-constrained inverse problems and positron emission tomography are used to demonstrate the performance of our methods.

Original languageEnglish
Pages (from-to)29-64
Number of pages36
JournalSIAM-ASA Journal on Uncertainty Quantification
Issue number1
StatePublished - 2021


\ast Received by the editors February 12, 2020; accepted for publication (in revised form) October 26, 2020; published electronically January 12, 2021. https://doi.org/10.1137/20M1318365 Funding: The work of the first author was supported by the Gordon Preston Fellowship offered by Monash University. The work of the second author was supported by the Australian Research Council under grant CE140100049. \dagger Department of Mathematical Sciences, University of Montana, Missoula, MT 59812 USA (bardsleyj@ mso.umt.edu). \ddagger Corresponding author. School of Mathematics, Monash University, VIC 3800, Australia (tiangang.cui@ monash.edu).

FundersFunder number
Australian Research CouncilCE140100049
Monash University


    • Hierarchical Bayes
    • Inverse problems
    • Markov chain Monte Carlo
    • Poisson likelihood
    • Positron emission tomography
    • Pseudomarginalization


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