Bayes factor

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. 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. The Bayes factor is a likelihood ratio of the marginal likelihood of two competing hypotheses, usually a null and an alternative. The posterior probability Pr ( M | D ) {displaystyle Pr(M|D)} of a model M given data D is given by Bayes' theorem: The key data-dependent term Pr ( D | M ) {displaystyle Pr(D|M)} represents the probability that some data are produced under the assumption of the 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 θ 1 {displaystyle heta _{1}} and θ 2 {displaystyle heta _{2}} , is assessed by the Bayes factor K given by When the two models are equally probable a priori, so that Pr ( M 1 ) = Pr ( M 2 ) {displaystyle Pr(M_{1})=Pr(M_{2})} , the Bayes factor is equal to the ratio of the posterior probabilities of M1 and M2. If instead of the Bayes factor integral, the likelihood corresponding to the maximum likelihood estimate of the parameter for each statistical model is used, then the test becomes a classical likelihood-ratio test. Unlike a likelihood-ratio test, this Bayesian model comparison does not depend on any single set of parameters, as it integrates over all parameters in each model (with respect to the respective priors). However, an advantage of the use of Bayes factors is that it automatically, and quite naturally, includes a penalty for including too much model structure. It thus guards against overfitting. For models where an explicit version of the likelihood is not available or too costly to evaluate numerically, approximate Bayesian computation can be used for model selection in a Bayesian framework,with the caveat that approximate-Bayesian estimates of Bayes factors are often biased.