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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]].  

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 <math>\theta</math> as [[nuisance parameters]], i.e.,

::<math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math>

::<math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math>

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]].

 

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

[[Category:Probability]]

[[Category:Probability]]