# Bayesian model comparison

A common problem in statistical inference is to use data to determine which of two competing models is the truth. Frequentist statistics uses hypothesis tests for this purpose. There are several Bayesian approaches. One approach is through Bayes factors.

The posterior probability of a model given data, P(H|D), is given by Bayes' theorem:

P(H|D) = P(D|H)P(H)/P(D)

The key data-dependent term P(D|H) is a likelihood, and is sometimes called the evidence for model H; evaluating it correctly is the key to Bayesian model comparison.

The evidence is usually the normalizing constant or partition function of another inference, namely the inference of the parameters of model H given the data D.

The plausibility of two different models H1 and H2, parametrised by model parameter vectors [itex] \theta_1 [itex] and [itex] \theta_2 [itex] is assessed by the Bayes factor given by

[itex] \frac{P(D|H2)}{P(D|H1)}

= \frac{\int P(\theta_2|H2)P(D|\theta_2,H2)\,d\theta_2} {\int P(\theta_1|H1)P(D|\theta_1,H1)\,d\theta_1 }. [itex]

Another approach is to treat model comparison as a decision problem.

## References

• Gelman, A., Carlin, J.,Stern, H. and Rubin, D. Bayesian Data Analysis. Chapman and Hall/CRC.(1995)
• Bernardo, J., and Smith, A.F.M., Bayesian Theory. John Wiley. (1994)
• Lee, P.M. Bayesian Statistics. Arnold.(1989).
• Denison, D.G.T., Holmes, C.C., Mallick, B.K., Smith, A.F.M., Bayesian Methods for Nonlinear Classification and Regression. John Wiley. (2002).

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