Inconsistency and Acausality of Model Selection in Bayesian Inverse Problems

TL;DR

This paper critically examines the inconsistencies and causality violations in Bayesian inverse problems, highlighting issues with conditional probabilities, model selection, and prior assumptions, and calls for a re-evaluation of Bayesian practices.

Contribution

It reveals fundamental inconsistencies in Bayesian inverse methods, especially regarding conditional densities and model selection, proposing the need for revised interpretation and methodology.

Findings

Conditional densities vary with parameterization, leading to inconsistent results.

Model selection metrics like evidence may not accurately reflect data fit.

Hyperparameters in hierarchical models often represent posterior, not prior, information.

Abstract

Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, when applied to inverse problems in physical sciences, Bayesian formulations suffer from a number of inconsistencies that are often overlooked. A well known, but mostly neglected, difficulty is connected to the notion of conditional probability densities. Borel, and later Kolmogorov's (1933/1956), found that the traditional definition of conditional densities is incomplete: In different parameterizations it leads to different results. We will show an example where two apparently correct procedures applied to the same problem lead to two widely different results. Another type of inconsistency involves violation of causality. This problem is found in model selection strategies in Bayesian inversion, such as Hierarchical Bayes and Trans-Dimensional Inversion where so-called hyperparameters are included as variables to control either the number (or type) of unknowns, or the prior uncertainties on data or model parameters. For Hierarchical Bayes we demonstrate that the calculated 'prior' distributions of data or model parameters are not prior-, but posterior information. In fact, the calculated 'standard deviations' of the data are a measure of the inability of the forward function to model the data, rather than uncertainties of the data. For trans-dimensional inverse problems we show that the so-called evidence is, in fact, not a measure of the success of fitting the data for the given choice (or number) of parameters, as often claimed. We also find that the notion of Natural Parsimony is ill-defined, because of its dependence on the parameter prior. Based on this study, we find that careful rethinking of Bayesian inversion practices is required, with special emphasis on ways of avoiding the Borel-Kolmogorov inconsistency, and on the way we interpret model selection results.