## Bayes' Factors

Valen E Johnson, Texas A&M University, College Station, Texas, USA

Published online: November 2016

DOI: 10.1002/9780470015902.a0005851.pub2

### Abstract

Bayes' factors play a pivotal role in Bayesian hypothesis testing, representing the factor through which the prior odds between
competing hypotheses are transformed to the posterior odds between the hypotheses. Mathematically, Bayes' factors are defined
as the ratio of the marginal probability assigned to the data by one hypothesis to the marginal probability assigned to data
by the other hypothesis. In many parametric statistical tests, hypotheses are defined by the specification of prior densities
on unknown parameters. When the prior density on a parameter concentrates on a single value under both the null and alternative
hypotheses (resulting in two ‘simple’ or ‘point’ hypotheses), the Bayes factor equals the likelihood ratio. More generally,
Bayes' factors represent the ratio of an averaged likelihood function, averaged with respect to the different prior densities
assigned to the unknown parameter under each hypothesis. Inconsistencies in the limiting behaviour of Bayes' factors may arise
when hypotheses are defined with respect to prior densities that have overlapping support.

#### Key Concepts

- Bayes' factors are used in Bayesian hypothesis tests and are the key factors through which experimental data determine the
posterior probability assigned to scientific hypotheses.
- Bayes' factors represent the ratio of the probability assigned by competing hypotheses to a common set of data.
- When testing whether one of the two simple hypotheses is true, the Bayes factor equals the likelihood ratio between the hypotheses.
- Bayes' factors equal the posterior odds divided by the prior odds between hypotheses.
- The natural logarithm of a Bayes' factor is called the weight of evidence.

Keywords: Bayesian hypothesis test; integrated likelihood; intrinsic prior; likelihood ratio; local prior density; marginal likelihood; nonlocal prior density; weight of evidence

References

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Further Reading

Berger JO and Delampady M (1987) Testing precise hypotheses. Statistical Science 2: 317–335.

Edwards W, Lindman H and Savage LJ (1963) Bayesian statistical inference for psychological research. Psychological Review 70: 193–242.

Goodman SN (1999) Toward evidence‐based medical statistics. 2: The Bayes factor. Annals of Internal Medicine 130: 1005–1013.

Kass RE and Raftery AE (1995) Bayes factors. Journal of the American Statistical Association 90: 773–795.

O'Hagan A (1995) Fractional Bayes factors for model comparison. Journal of the Royal Statistical Society, Series B 57: 99–118.

O'Hagan A (1997) Properties of intrinsic and fractional Bayes factors. Test 6: 101–118.