A common problem in statistical inference is to use data to decide between two or more competing<unk> s. Frequentist statistics uses hypothesis tests for this purpose. There are several Bayesian approaches. One approach is through Bayes factors.
The posterior probability of a<unk>  given data, Pr(H|D), is given by Bayes' theorem:
The key data-dependent term Pr(D|H) is a likelihood, and is sometimes called the evidence for<unk>  H; evaluating it correctly is the key to<unk> .
The evidence is usually the normalizing constant or partition function of another inference, namely the inference of the parameters of<unk>  H given the data D.
The plausibility of two different<unk> s H1 and H2, parametrised by<unk>  parameter vectors 
θ
1
{\displaystyle \theta _{1}}
 and 
θ
2
{\displaystyle \theta _{2}}
 is assessed by the Bayes factor given by
Thus the<unk>  does not depend on the parameters used by each<unk> .  Instead, it considers the probability of the<unk>  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<unk>  structure. It thus guards against overfitting.
Another approach is to treat<unk><unk>  as a decision problem, computing the expected value or cost of each<unk>  choice.
Another approach is to use Minimum Message Length (MML).
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