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**Bayes Factors and Model Uncertainty**

**Robert E. Kass and Adrian E. Raftery **

### Abstract:

*In a 1935 paper, and in his book **Theory of Probability*, Jeffreys
developed a methodology for quantifying the evidence in favor of a
scientific theory. The centerpiece was a number, now called the *Bayes factor*, which is the posterior odds of the null hypothesis when
the prior probability on the null is one-half. Although there has been
much discussion of Bayesian hypothesis testing in the context of
criticism of -values, less attention has been given to the Bayes
factor as a practical tool of applied statistics. In this paper we
review and discuss the uses of Bayes factors in the context of five
scientific applications.
The points we emphasize are:

- from Jeffreys's Bayesian point of view,
the purpose of hypothesis testing is to evaluate the evidence in favor
of a scientific theory;

- Bayes factors offer a way of evaluating
evidence *in favor of* a null hypothesis;

- Bayes factors provide a way of incorporating
external information into the evaluation of
evidence about a hypothesis;

- Bayes factors are very general, and do
not require alternative models to be nested;

- several techniques are
available for computing Bayes factors, including asymptotic
approximations which are easy to compute using the output from
standard packages that maximize likelihoods;

- in ``non-standard''
statistical models that do not satisfy common regularity conditions,
it can be technically simpler to calculate Bayes factors than to
derive non-Bayesian significance tests;

- the Schwarz criterion (or BIC)
gives a crude approximation to the logarithm of the Bayes factor,
which is easy to use and does not require evaluation of prior
distributions;

- when one is interested in estimation or prediction,
Bayes factors may be converted to weights to be attached to various
models so that a composite estimate or prediction may be obtained that
takes account of structural or model uncertainty;

- algorithms have been proposed that allow model uncertainty to be taken
into account when the class of models initially considered is very
large;

- Bayes factors are useful for guiding an evolutionary
model-building process;

- and, finally, it is important, and feasible,
to assess the sensitivity of conclusions to the prior distributions
used.

*KEY WORDS:*
Bayesian hypothesis tests;
BIC;
Importance sampling;
Laplace method;
Markov chain Monte Carlo;
Model selection;
Monte Carlo integration;
Posterior model probabilities;
Posterior odds;
Sensitivity analysis;
Quadrature;
Schwarz criterion;
Strength of evidence.

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