  In this paper we review the concepts of Bayesian evidence and Bayes factors,
also known as log odds ratios, and their application to model selection. The
theory is presented along with a discussion of analytic, approximate and
numerical techniques. Specific attention is paid to the Laplace approximation,
variational Bayes, importance sampling, thermodynamic integration, and nested
sampling and its recent variants. Analogies to statistical physics, from which
many of these techniques originate, are discussed in order to provide readers
with deeper insights that may lead to new techniques. The utility of Bayesian
model testing in the domain sciences is demonstrated by presenting four
specific practical examples considered within the context of signal processing
in the areas of signal detection, sensor characterization, scientific model
selection and molecular force characterization.

Methodology;Instrumentation and Methods for Astrophysics;Applications;Computation;Machine Learning