Bayes factor

Bayesian methods

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This is an alternative to classical hypothesis testing. You can select models based on Bayes factor.

The aim of the Bayes factor is to quantify the support for a model over another, regardless of whether these models are correct.

When the two models are equally probable a priori, so that $\Pr(M_{1})=\Pr(M_{2})$ , the Bayes factor is equal to the ratio of the posterior probabilities of M₁ and M₂.

When the priors are different, it is the following ratio. Pr(M₁|D) is simply the aforementioned posterior probability of M₁.

{\frac {\Pr(M_{1}|D)}{\Pr(M_{2}|D)}}{\frac {\Pr(M_{2})}{\Pr(M_{1})}}}

an advantage of the use of Bayes factors is that it automatically, and quite naturally, includes a penalty for including too much model structure.[6] It thus guards against overfitting. For models where an explicit version of the likelihood is not available or too costly to evaluate numerically, approximate Bayesian computation can be used for model selection in a Bayesian framework,[7] with the caveat that approximate-Bayesian estimates of Bayes factors are often biased.[8]

Other approaches are:

• to treat model comparison as a decision problem, computing the expected value or cost of each model choice;
• to use minimum message length (MML).