Projects per year
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.
- Hierarchical Bayes
- Inverse problems
- Markov chain Monte Carlo
- Poisson likelihood
- Positron emission tomography
- 1 Finished
ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights
Hall, P., Bartlett, P., Bean, N., Burrage, K., DeGier, J., Delaigle, A., Forrester, P., Geweke, J., Kohn, R., Kroese, D., Mengersen, K. L., Pettit, A., Pollett, P., Roughan, M., Ryan, L. M., Taylor, P., Turner, I., Wand, M., Garoni, T., Smith-Miles, K. A., Caley, M., Churches, T., Elazar, D., Gupta, A., Harch, B., Tam, S., Weegberg, K., Willinger, W. & Hyndman, R.
Australian Research Council (ARC), Monash University – Internal Department Contribution, University of Melbourne, Queensland University of Technology (QUT), University of Adelaide, University of New South Wales (UNSW), University of Queensland , University of Technology (UTS) Sydney, Monash University – Internal University Contribution, Monash University – Internal Faculty Contribution, Monash University – Internal School Contribution, Roads Corporation (trading as VicRoads) (Victoria)
1/01/17 → 31/12/21