Calculation and Applications of Bayesian Evidence in Astrophysics and Particle Physics Phenomenology

Bayesian inference is usually divided into two categories: parameter estimation and model selection. Parameter estimation is mostly performed using Markov Chain Monte Carlo (MCMC) sampling methods, most often based on the standard Metropolis-Hastings algorithm or its variants, such as slice, Gibbs´ and Hamiltonian sampling. These methods can be highly inefficient in exploring multi-modal distributions or ones exhibiting degeneracies between parameters. Moreover, in order to perform Bayesian model selection, estimation of Bayesian evidence is needed which is by definition, a (possibly high dimensional) integration over the entire parameter space. Unfortunately, the computational expense involved in the evaluation of Bayesian evidence is typically an order of magnitude higher than doing the parameter estimation. In this paper, we discuss some techniques, built around the nested sampling framework, that have been developed to efficiently calculate the Bayesian evidence. We also discuss applications of Bayesian evidence in astrophysics and particle physics, in problems involving model selection, source detection, consistency checks between different data-sets, multi-model inference and in determining the constraining power of different observables.