Editing Marginal likelihood (section)

=== Bayesian model comparison ===
In [[Bayesian model comparison]], the marginalized variables <math>\theta</math> are parameters for a particular type of model, and the remaining variable <math>M</math> is the identity of the model itself. In this case, the marginalized likelihood is the probability of the data given the model type, not assuming any particular model parameters. Writing <math>\theta</math> for the model parameters, the marginal likelihood for the model ''M'' is

:<math> p(\mathbf{X}\mid M) = \int p(\mathbf{X}\mid\theta, M) \, p(\theta\mid M) \, \operatorname{d}\!\theta </math>

It is in this context that the term ''model evidence'' is normally used.  This quantity is important because the posterior odds ratio for a model ''M''<sub>1</sub> against another model ''M''<sub>2</sub> involves a ratio of marginal likelihoods, called the [[Bayes factor]]:

:<math> \frac{p(M_1\mid \mathbf{X})}{p(M_2\mid \mathbf{X})} = \frac{p(M_1)}{p(M_2)} \, \frac{p(\mathbf{X}\mid M_1)}{p(\mathbf{X}\mid M_2)} </math>

which can be stated schematically as

:posterior [[odds]] = prior odds × [[Bayes factor]]