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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]].
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]]. Thus the ''Bayes factor'' for model class M<sub>1</sub> with parameters <math>\theta_1</math> to model class M<sub>2</sub> with parameters <math>\theta_2</math> is
::<math>BF(X,M_1,M_2) = \frac{m_{\theta_1}(X|M_1)}{m_{\theta_2}(X|M_2)} </math>
In determining which model class has the largest ratio when compared against all others, the model class or [[mathematical model]] which best explains the data is determined. It is then left to determine the best inference as to the values of that models parameters via [[parameter estimation]].
[[Category:Probability]]
[[Category:Probability]]