Bayesian model comparison

In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing.Bayesian model comparison is a method of model selection based on Bayes factors. The models under consideration are statistical models. The aim of the Bayes factor is to quantify the support for a model over another, regardless of whether these models are correct. The technical definition of "support" in the context of Bayesian inference is described below.

Simply put, the Bayes factor is a ratio of the likelihood probability of two competing hypotheses, usually a null and an alternative.

The posterior probability Pr(M|D) of a model M given data D is given by Bayes' theorem:

The key data-dependent term Pr(D|M) is a likelihood, and represents the probability that some data are produced under the assumption of this model, M; evaluating it correctly is the key to Bayesian model comparison.

Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M₁ and M₂, parametrised by model parameter vectors ${\displaystyle \theta _{1}}$ and ${\displaystyle \theta _{2}}$ is assessed by the Bayes factor K given by

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