B ayes' Factors

Abstract Bayes' factors play a pivotal role in Bayesian hypothesis testing, representing the factor through which the prior odds between competing hypotheses are transformed to the posterior odds between the hypotheses. Mathematically, Bayes' factors are defined as the ratio of the marginal probability assigned to the data by one hypothesis to the marginal probability assigned to data by the other hypothesis. In many parametric statistical tests, hypotheses are defined by the specification of prior densities on unknown parameters. When the prior density on a parameter concentrates on a single value under both the null and alternative hypotheses (resulting in two ‘simple’ or ‘point’ hypotheses), the Bayes factor equals the likelihood ratio. More generally, Bayes' factors represent the ratio of an averaged likelihood function, averaged with respect to the different prior densities assigned to the unknown parameter under each hypothesis. Inconsistencies in the limiting behaviour of Bayes' factors may arise when hypotheses are defined with respect to prior densities that have overlapping support. Key Concepts Bayes' factors are used in Bayesian hypothesis tests and are the key factors through which experimental data determine the posterior probability assigned to scientific hypotheses. Bayes' factors represent the ratio of the probability assigned by competing hypotheses to a common set of data. When testing whether one of the two simple hypotheses is true, the Bayes factor equals the likelihood ratio between the hypotheses. Bayes' factors equal the posterior odds divided by the prior odds between hypotheses. The natural logarithm of a Bayes' factor is called the weight of evidence.