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

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.,