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Trans-dimensional inverse problems, model comparison and the evidence

Date

2006

Authors

Sambridge, Malcolm
Gallagher, K
Jackson, Andrew
Rickwood, Peter

Journal Title

Journal ISSN

Volume Title

Publisher

Blackwell Publishing Ltd

Abstract

In most geophysical inverse problems the properties of interest are parametrized using a fixed number of unknowns. In some cases arguments can be used to bound the maximum number of parameters that need to be considered. In others the number of unknowns is set at some arbitrary value and regularization is used to encourage simple, non-extravagant models. In recent times variable or self-adaptive parametrizations have gained in popularity. Rarely, however, is the number of unknowns itself directly treated as an unknown. This situation leads to a trans-dimensional inverse problem, that is, one where the dimension of the parameter space is a variable to be solved for. This paper discusses trans-dimensional inverse problems from the Bayesian viewpoint. A particular type of Markov chain Monte Carlo (MCMC) sampling algorithm is highlighted which allows probabilistic sampling in variable dimension spaces. A quantity termed the evidence or marginal likelihood plays a key role in this type of problem. It is shown that once evidence calculations are performed, the results of complex variable dimension sampling algorithms can be replicated with simple and more familiar fixed dimensional MCMC sampling techniques. Numerical examples are used to illustrate the main points. The evidence can be difficult to calculate, especially in high-dimensional non-linear inverse problems. Nevertheless some general strategies are discussed and analytical expressions given for certain linear problems.

Description

Keywords

Keywords: algorithm; geophysical method; inverse problem; Markov chain; Monte Carlo analysis; numerical model; parameterization; probability Evidence; Inverse problems; Model comparison; Parametrization

Citation

Source

Geophysical Journal International

Type

Journal article

Book Title

Entity type

Access Statement

License Rights

DOI

10.1111/j.1365-246X.2006.03155.x

Restricted until

2037-12-31