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Dimension-Robust MCMC in Bayesian Inverse Problems

Chen, Victor and Dunlop, Matthew M. and Papaspiliopoulos, Omiros and Stuart, Andrew M. (2018) Dimension-Robust MCMC in Bayesian Inverse Problems. . (Unpublished)

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The methodology developed in this article is motivated by a wide range of prediction and uncertainty quantification problems that arise in Statistics, Machine Learning and Applied Mathematics, such as non-parametric regression, multi-class classification and inversion of partial differential equations. One popular formulation of such problems is as Bayesian inverse problems, where a prior distribution is used to regularize inference on a high-dimensional latent state, typically a function or a field. It is common that such priors are non-Gaussian, for example piecewise-constant or heavy-tailed, and/or hierarchical, in the sense of involving a further set of low-dimensional parameters, which, for example, control the scale or smoothness of the latent state. In this formulation prediction and uncertainty quantification relies on efficient exploration of the posterior distribution of latent states and parameters. This article introduces a framework for efficient MCMC sampling in Bayesian inverse problems that capitalizes upon two fundamental ideas in MCMC, non-centred parameterisations of hierarchical models and dimension-robust samplers for latent Gaussian processes. Using a range of diverse applications we showcase that the proposed framework is dimension-robust, that is, the efficiency of the MCMC sampling does not deteriorate as the dimension of the latent state gets higher. We showcase the full potential of the machinery we develop in the article in semi-supervised multi-class classification, where our sampling algorithm is used within an active learning framework to guide the selection of input data to manually label in order to achieve high predictive accuracy with a minimal number of labelled data.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Dunlop, Matthew M.0000-0001-7718-3755
Additional Information:MMD and AMS are supported by AFOSR Grant FA9550-17-1-0185 and ONR Grant N00014-17-1-2079.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-17-1-0185
Office of Naval Research (ONR)N00014-17-1-2079
Record Number:CaltechAUTHORS:20190404-111029769
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:94459
Deposited By: George Porter
Deposited On:04 Apr 2019 19:41
Last Modified:03 Oct 2019 21:04

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