Likelihood is the denominator of Eq. (1) and represents the probability of the **data** given the model. The marginal Likelihood has the desirable qualities of rewarding models that match the **data** well and penalizing models that are overly complex (i.e. have too many degrees of freedom or parameters). The marginal Likelihood is given by,
P (D|M ) = Ω Θ P (D|Θ, M) P (Θ|M ) dΘ,(2)
where Ω Θ represents the complete parameter space. Note that by integrating over the parameters in the model we are able to compare Bayesian Evidences for two distinct models in a meaningful way, even if they contain different numbers of parameters. This ratio of the Bayesian Evidence for two models is called the Bayes Factor and is given by
R = P (M A |D) P (M B |D) = P (D|M A ) P (M A ) P (D|M B ) P (M B ) ,(3)
where M A and M B are the two models under consideration [15]. Kass and Raftery [21] provide a commonly used guide to interpret the Bayes factor, in which a value in the range 3 − 20 indicates a positive strength of evidence for the preference of one model versus another and a factor in the range 20 − 150 represents strong evidence to prefer one model over another. Unfortunately because of the integral over parameter space Eq. (2) is notoriously difficult to evaluate. Only with recent advances in sampling techniques has the model Evidence become an appealing option for model selection. We briefly discuss some selected sampling approaches in the following section.