Bayes Factor

The Bayes Factor of model class M₁ to model class M₂ for some set of observations X is the ratio of their associated marginal class likelihoods. The marginal class likelihood of some set of observations is the marginal likelihood of the observations for the model class obtained by marginalizing the joint probability distribution of the observations and the model class parameters by treating the model class parameters as nuisance parameters.

$m_{\theta }(X|M)=\int _{\theta }p(X,\theta |M)\,d\theta =\int _{\theta }p(X|\theta ,M)\,p(\theta |M)\,d\theta$

showing that it may be calculated directly using the joint probability distribution or the conditional probability distribution where the second form of the integral was obtained from the Bayesian Product Rule.