Bayes' Factors

Valen E Johnson, University of Texas M.D. Anderson Cancer Center, Houston, Texas, USA

Published online: March 2011

DOI: 10.1002/9780470015902.a0005851

Full Article on Wiley Online Library

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. In such cases, 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.
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

	Figure 1. Prior densities for binomial success probability under composite hypotheses H₁ (flat line) and H₂ (curved line).
	Figure 2. Bayes' factor as a function of the number of successes y in n=10 trials when testing two composite binomial hypotheses.
	Figure 3. Illustration of prior densities and a likelihood function based on 450 successes in 1000 trials. The likelihood function is the curve with a sharp mode at 0.45.
	Figure 4. Weight of evidence in favour of local and nonlocal hypotheses for normal data. Based on a sample size of 100. The upper curve represents the logarithm of the Bayes factor (i.e. weight of evidence) in favour of H₂^a; the lower curve the weight of evidence in favour of H₂^b.

References
	Berger JO and Pericchi LR (1996) The intrinsic Bayes factor for model selection and prediction. Journal of the American Statistical Association 91: 109–122.
	book Cover JA and Curd M (1998) Philosophy of Science: The Central Issues. New York: WM Norton & Company.
	Johnson VE and Rossell D (2010) On the use of non-local prior densities in Bayesian hypothesis tests. Journal of the Royal Statistical Society, Series B 72(2): 143–170.
Further Reading
	book Jeffreys H (1998) Theory of Probability, 3rd edn. Oxford: Oxford University Press.
	Kass RE and Raftery AE (1995) Bayes factors. Journal of the American Statistical Association 90: 773–795.
	Moreno E, Bertolino F and Racugno W (1998) An intrinsic limiting procedure for model selection and hypotheses testing. Journal of the American Statistical Association 93: 1451–1460.
	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.

Article Title:	Bayes' Factors
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Bayes' Factors

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