Mathematical Basis for Physical Inference

While the axiomatic introduction of a probability distribution over a space is common, its use for making predictions, using physical theories and prior knowledge, suffers from a lack of formalization. We propose to introduce, in the space of all probability distributions, two operations, the OR and the AND operation, that bring to the space the necessary structure for making inferences on possible values of physical parameters. While physical theories are often asumed to be analytical, we argue that consistent inference needs to replace analytical theories by probability distributions over the parameter space, and we propose a systematic way of obtaining such "theoretical correlations", using the OR operation on the results of physical experiments. Predicting the outcome of an experiment or solving "inverse problems" are then examples of the use of the AND operation. This leads to a simple and complete mathematical basis for general physical inference.Comment: 24 pages, 4 figure

Informed Proposal Monte Carlo

Any search or sampling algorithm for solution of inverse problems needs guidance to be efficient. Many algorithms collect and apply information about the problem on the fly, and much improvement has been made in this way. However, as a consequence of the the No-Free-Lunch Theorem, the only way we can ensure a significantly better performance of search and sampling algorithms is to build in as much information about the problem as possible. In the special case of Markov Chain Monte Carlo sampling (MCMC) we review how this is done through the choice of proposal distribution, and we show how this way of adding more information about the problem can be made particularly efficient when based on an approximate physics model of the problem. A highly nonlinear inverse scattering problem with a high-dimensional model space serves as an illustration of the gain of efficiency through this approach

Inconsistency and Acausality of Model Selection in Bayesian Inverse Problems

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.Comment: The paper replaces arXiv:2308.05858v1 which contained incorrectly normalized distributions in two important counterexamples on hierarchical Bayes and trans-dimensional inversion. In the new, corrected version of the paper (where a key counter-example on transdimensional inversion is further expanded) the conclusions remain the same as in the original pape