Bayes factor

Bayesian statistics
Theory
Bayesian probability Probability interpretations Bayes' theorem Bayes' rule · Bayes factor Bayesian inference Bayesian network Prior · Posterior · Likelihood Conjugate prior Hyperparameter · Hyperprior Principle of indifference Principle of maximum entropy Empirical Bayes method Cromwell's rule Bernstein–von Mises theorem Bayesian information criterion Credible interval Maximum a posteriori estimation
Techniques
Bayesian linear regression Bayesian estimator Approximate Bayesian computation

In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing.^[1]^[2] Bayesian model comparison is a method of model selection based on Bayes factors.

1 Definition
2 Interpretation
3 Example
4 See also
5 References
6 External links

Definition

The posterior probability Pr(M|D) of a model M given data D is given by Bayes' theorem:

$\Pr(M|D) = \frac{\Pr(D|M)\Pr(M)}{\Pr(D)}.$

The key data-dependent term Pr(D|M) is a likelihood, and represents the probability that some data is produced under the assumption of this model, M; 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 M given the data D.

Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M₁ and M₂, parametrised by model parameter vectors $\theta_1$ and $\theta_2$ is assessed by the Bayes factor K given by

$K = \frac{\Pr(D|M_1)}{\Pr(D|M_2)} = \frac{\int \Pr(\theta_1|M_1)\Pr(D|\theta_1,M_1)\,d\theta_1} {\int \Pr(\theta_2|M_2)\Pr(D|\theta_2,M_2)\,d\theta_2 }.$

where Pr(D|M_i) is called the marginal likelihood for model i.

If instead of the Bayes factor integral, the likelihood corresponding to the Maximum likelihood estimate of the parameter for each model is used, then the test becomes a classical likelihood-ratio test. Unlike a likelihood-ratio test, this Bayesian model comparison does not depend on any single set of parameters, as it integrates over all parameters in each model (with respect to the respective priors). However, an advantage of the use of Bayes factors is that it automatically, and quite naturally, includes a penalty for including too much model structure.^[3] 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.^[4]

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).

Interpretation

A value of K > 1 means that the data indicate that M₁ is more strongly supported by the data under consideration than M₂. Note that classical hypothesis testing gives one hypothesis (or model) preferred status (the 'null hypothesis'), and only considers evidence against it. Harold Jeffreys gave a scale for interpretation of K:^[5]

K	dB	bits	Strength of evidence
< 1:1	< 0		Negative (supports M₂)
1:1 to 3:1	0 to 5	0 to 1.6	Barely worth mentioning
3:1 to 10:1	5 to 10	1.6 to 3.3	Substantial
10:1 to 30:1	10 to 15	3.3 to 5.0	Strong
30:1 to 100:1	15 to 20	5.0 to 6.6	Very strong
> 100:1	> 20	> 6.6	Decisive

The second column gives the corresponding weights of evidence in decibans (tenths of a power of 10); bits are added in the third column for clarity. According to I. J. Good a change in a weight of evidence of 1 deciban or 1/3 of a bit (i.e. a change in an odds ratio from evens to about 5:4) is about as finely as humans can reasonably perceive their degree of belief in a hypothesis in everyday use.^[6]

The use of Bayes factors or classical hypothesis testing takes place in the context of inference rather than decision-making under uncertainty. That is, we merely wish to find out which hypothesis is true, rather than actually making a decision on the basis of this information. Frequentist statistics draws a strong distinction between these two because classical hypothesis tests are not coherent in the Bayesian sense. Bayesian procedures, including Bayes factors, are coherent, so there is no need to draw such a distinction. Inference is then simply regarded as a special case of decision-making under uncertainty in which the resulting action is to report a value. For decision-making, Bayesian statisticians might use a Bayes factor combined with a prior distribution and a loss function associated with making the wrong choice. In an inference context the loss function would take the form of a scoring rule. Use of a logarithmic score function for example, leads to the expected utility taking the form of the Kullback–Leibler divergence.

Example

Suppose we have a random variable which produces either a success or a failure. We want to compare a model M₁ where the probability of success is q = ½, and another model M₂ where q is completely unknown and we take a prior distribution for q which is uniform on [0,1]. We take a sample of 200, and find 115 successes and 85 failures. The likelihood can be calculated according to the binomial distribution:

${{200 \choose 115}q^{115}(1-q)^{85}}.$

So we have

$P(X=115|M_1)={200 \choose 115}\left({1 \over 2}\right)^{200}=0.005956...,\,$

but

$P(X=115|M_2)=\int_{0}^1{200 \choose 115}q^{115}(1-q)^{85}dq = {1 \over 201} = 0.004975...\,.$

The ratio is then 1.197..., which is "barely worth mentioning" even if it points very slightly towards M₁.

This is not the same as a classical likelihood ratio test, which would have found the maximum likelihood estimate for q, namely ¹¹⁵⁄₂₀₀ = 0.575, and used that to get a ratio of 0.1045... (rather than averaging over all possible q), and so pointing towards M₂. Alternatively, Edwards's "exchange rate" of two units of likelihood per degree of freedom suggests that $M_2$ is preferable (just) to $M_1$ , as $0.1045\ldots = e^{-2.25\ldots}$ and $2.25>2$ : the extra likelihood compensates for the unknown parameter in $M_2$ .

A frequentist hypothesis test of $M_1$ (here considered as a null hypothesis) would have produced a more dramatic result, saying that M₁ could be rejected at the 5% significance level, since the probability of getting 115 or more successes from a sample of 200 if q = ½ is 0.0200..., and as a two-tailed test of getting a figure as extreme as or more extreme than 115 is 0.0400... Note that 115 is more than two standard deviations away from 100.

M₂ is a more complex model than M₁ because it has a free parameter which allows it to model the data more closely. The ability of Bayes factors to take this into account is a reason why Bayesian inference has been put forward as a theoretical justification for and generalisation of Occam's razor, reducing Type I errors.^[7]

References

^ Goodman S (1999). "Toward evidence-based medical statistics. 1: The P value fallacy" (PDF). Ann Intern Med 130 (12): 995–1004. PMID 10383371. http://www.annals.org/cgi/reprint/130/12/995.pdf.
^ Goodman S (1999). "Toward evidence-based medical statistics. 2: The Bayes factor" (PDF). Ann Intern Med 130 (12): 1005–13. PMID 10383350. http://www.annals.org/cgi/reprint/130/12/1005.pdf.
^ Robert E. Kass and Adrian E. Raftery (1995) "Bayes Factors", Journal of the American Statistical Association, Vol. 90, No. 430, p. 791.
^ Toni, T.; Stumpf, M.P.H. (2009). "Simulation-based model selection for dynamical systems in systems and population biology" (PDF). Bioinformatics 26 (1): 104–10. doi:10.1093/bioinformatics/btp619. PMC 2796821. PMID 19880371. http://bioinformatics.oxfordjournals.org/cgi/reprint/26/1/104.pdf.
^ H. Jeffreys, The Theory of Probability (3e), Oxford (1961); p. 432
^ Good, I.J. (1979). "Studies in the History of Probability and Statistics. XXXVII A. M. Turing's statistical work in World War II". Biometrika 66 (2): 393–396. doi:10.1093/biomet/66.2.393. MR 82c:01049.
^ Sharpening Ockham's Razor On a Bayesian Strop

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).
Richard O. Duda, Peter E. Hart, David G. Stork (2000) Pattern classification (2nd edition), Section 9.6.5, p. 487-489, Wiley, ISBN 0-471-05669-3
Chapter 24 in Probability Theory - The logic of science by E. T. Jaynes, 1994.
David J.C. MacKay (2003) Information theory, inference and learning algorithms, CUP, ISBN 0-521-64298-1, (also available online)
Winkler, Robert, Introduction to Bayesian Inference and Decision, 2nd Edition (2003), Probabilistic. ISBN 0-9647938-4-9.

External links

Bayesian critique of classical hypothesis testing
Web-based Bayes-factor calculator for t-tests, regression designs, and binomially distributed data
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, discusses Bayesian model comparison in Chapter 28, p343.