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

Inconsistency and Acausality of Model Selection in Bayesian Inverse Problems