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(Introduction) Edit of equations and discussion of meaning of second equation |
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The posterior probability of a model given data, Pr(''H''|''D''), is given by [[Bayes' theorem]]:
:<math>Pr(
The key data-dependent term Pr(''D''|''H'') is a [[likelihood function|likelihood]], and is sometimes called the evidence for model ''H''; evaluating it correctly is the key to Bayesian model comparison.
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}.
</math>
Thus the Bayesian model comparison does not depend on the parameters used by each model. Instead, it considers the probability of the model considering all possible parameter values. Alternatively, the [[Maximum likelihood estimate]] could be used for each of the parameters.
An advantage of the use of [[Bayes factors]] is that it automatically, and quite naturally, includes a penalty for including too much model structure. It thus guards against [[overfitting]].
Another approach is to treat model comparison as a [[Decision theory#Choice under uncertainty|decision problem]], computing the expected value or cost of each model choice.
Another approach is to use [[Minimum Message Length]] ([[Minimum_Message_Length|MML]]).
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