Bayes factor

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. The models under consideration are statistical models.^[3] The aim of the Bayes factor is to quantify the support for a model over another, regardless of whether these models are correct.^[4] The technical definition of "support" in the context of Bayesian inference is described below.

Definition

The Bayes factor is a ratio of the likelihood probability of two competing hypotheses, usually a null and an alternative.^[5]

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 the likelihood of the model M in view of the data D, and represents the probability that some data are produced under the assumption of the model M; evaluating it correctly is the key to Bayesian model comparison.

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}}}={\frac {\Pr(M_{1}|D)}{\Pr(M_{2}|D)}}{\frac {\Pr(M_{2})}{\Pr(M_{1})}}.

When the two models are equally probable a priori, so that $\Pr(M_{1})=\Pr(M_{2})$ , the Bayes factor is equal to the ratio of posterior probabilities of M₁ and M₂. If instead of the Bayes factor integral, the likelihood corresponding to the maximum likelihood estimate of the parameter for each statistical 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.^[6] 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,^[7] with the caveat that approximate-Bayesian estimates of Bayes factors are often biased.^[8]

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 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:^[9]

K	dHart	bits	Strength of evidence
< 10⁰	< 0		negative (supports M₂)
10⁰ to 10^1/2	0 to 5	0 to 1.6	barely worth mentioning
10^1/2 to 10¹	5 to 10	1.6 to 3.3	substantial
10¹ to 10^3/2	10 to 15	3.3 to 5.0	strong
10^3/2 to 10²	15 to 20	5.0 to 6.6	very strong
> 10²	> 20	> 6.6	decisive

The second column gives the corresponding weights of evidence in decihartleys (also known as decibans); 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.^[10]

An alternative table, widely cited, is provided by Kass and Raftery (1995):^[6]

2 ln K	K	Strength of evidence
0 to 2	1 to 3	not worth more than a bare mention
2 to 6	3 to 20	positive
6 to 10	20 to 150	strong
>10	>150	very strong

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 that 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 unknown and we take a prior distribution for q that 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}}}.

Thus we have

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

but

P(X=115\mid 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.

A frequentist hypothesis test of M₁ (here considered as a null hypothesis) would have produced a very different result. Such a test says that M₁ should 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. Thus, whereas a frequentist hypothesis test would yield significant results at the 5% significance level, the Bayes factor hardly considers this to be an extreme result. Note, however, that a non-uniform prior (for example one that reflects the fact that you expect the number of success and failures to be of the same order of magnitude) could result in a Bayes factor that is more in agreement with the frequentist hypothesis test.

A classical likelihood-ratio test would have found the maximum likelihood estimate for q, namely ¹¹⁵⁄₂₀₀ = 0.575, whence

\textstyle P(X=115\mid M_{2})={{200 \choose 115}q^{{115}}(1-q)^{{85}}}=0.056991

(rather than averaging over all possible q). That gives a likelihood ratio of 0.1045 and points towards M₂.

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.^[11]

On the other hand, the modern method of relative likelihood takes into account the number of free parameters in the models, unlike the classical likelihood ratio. The relative likelihood method could be applied as follows. Model M₁ has 0 parameters, and so its AIC value is 2·0 − 2·ln(0.005956) = 10.2467. Model M₂ has 1 parameter, and so its AIC value is 2·1 − 2·ln(0.056991) = 7.7297. Hence M₁ is about exp((7.7297 − 10.2467)/2) = 0.284 times as probable as M₂ to minimize the information loss. Thus M₂ is slightly preferred, but M₁ cannot be excluded.

Notes

↑ Goodman S (1999). "Toward evidence-based medical statistics. 1: The P value fallacy" (PDF). Ann Intern Med. 130 (12): 995–1004. doi:10.7326/0003-4819-130-12-199906150-00008. PMID 10383371.
↑ Goodman S (1999). "Toward evidence-based medical statistics. 2: The Bayes factor" (PDF). Ann Intern Med. 130 (12): 1005–13. doi:10.7326/0003-4819-130-12-199906150-00019. PMID 10383350.
↑ Morey, Richard D.; Romeijn, Jan-Willem; Rouder, Jeffrey N. (2016). "The philosophy of Bayes factors and the quantification of statistical evidence". Journal of Mathematical Psychology. 72: 6–18. doi:10.1016/j.jmp.2015.11.001.
↑ Ly, Alexander; Verhagen, Josine; Wagenmakers, Eric-Jan (2016). "Harold Jeffreys's default Bayes factor hypothesis tests: Explanation, extension, and application in psychology". Journal of Mathematical Psychology. 72: 19–32. doi:10.1016/j.jmp.2015.06.004.
↑ Good, Phillip; Hardin, James (July 23, 2012). Common errors in statistics (and how to avoid them) (4th ed.). Hoboken, New Jersey: John Wiley & Sons, Inc. pp. 129–131. ISBN 978-1118294390.
1 2 Robert E. Kass & Adrian E. Raftery (1995). "Bayes Factors" (PDF). Journal of the American Statistical Association. 90 (430): 791. doi:10.2307/2291091.
↑ 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. arXiv:0911.1705. doi:10.1093/bioinformatics/btp619. PMC 2796821. PMID 19880371.
↑ Robert, C.P.; J. Cornuet; J. Marin & N.S. Pillai (2011). "Lack of confidence in approximate Bayesian computation model choice". Proceedings of the National Academy of Sciences. 108 (37): 15112–15117. Bibcode:2011PNAS..10815112R. doi:10.1073/pnas.1102900108. PMC 3174657. PMID 21876135.
↑ H. Jeffreys (1961). The Theory of Probability (3rd ed.). Oxford. 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 0548210.
↑ Sharpening Ockham's Razor On a Bayesian Strop

References

Bernardo, J.; Smith, A. F. M. (1994). Bayesian Theory. John Wiley. ISBN 0-471-92416-4.
Denison, D. G. T.; Holmes, C. C.; Mallick, B. K.; Smith, A. F. M. (2002). Bayesian Methods for Nonlinear Classification and Regression. John Wiley. ISBN 0-471-49036-9.
Duda, Richard O.; Hart, Peter E.; Stork, David G. (2000). "Section 9.6.5". Pattern classification (2nd ed.). Wiley. pp. 487–489. ISBN 0-471-05669-3.
Gelman, A.; Carlin, J.; Stern, H.; Rubin, D. (1995). Bayesian Data Analysis. London: Chapman & Hall. ISBN 0-412-03991-5.
Jaynes, E. T. (1994), Probability Theory: the logic of science, chapter 24.
Lee, P. M. (2012). Bayesian Statistics: an introduction. Wiley. ISBN 9781118332573.
Winkler, Robert (2003). Introduction to Bayesian Inference and Decision (2nd ed.). Probabilistic. ISBN 0-9647938-4-9.

External links

BayesFactor —an R package for computing Bayes factors in common research designs
Bayes Factor Calculators —web-based version of much of the BayesFactor package

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[Goodman1999a-1] Goodman S (1999). "Toward evidence-based medical statistics. 1: The P value fallacy" (PDF). Ann Intern Med. 130 (12): 995–1004. doi:10.7326/0003-4819-130-12-199906150-00008. PMID 10383371.

[Goodman1999b-2] Goodman S (1999). "Toward evidence-based medical statistics. 2: The Bayes factor" (PDF). Ann Intern Med. 130 (12): 1005–13. doi:10.7326/0003-4819-130-12-199906150-00019. PMID 10383350.

[3] Morey, Richard D.; Romeijn, Jan-Willem; Rouder, Jeffrey N. (2016). "The philosophy of Bayes factors and the quantification of statistical evidence". Journal of Mathematical Psychology. 72: 6–18. doi:10.1016/j.jmp.2015.11.001.

[4] Ly, Alexander; Verhagen, Josine; Wagenmakers, Eric-Jan (2016). "Harold Jeffreys's default Bayes factor hypothesis tests: Explanation, extension, and application in psychology". Journal of Mathematical Psychology. 72: 19–32. doi:10.1016/j.jmp.2015.06.004.

[5] Good, Phillip; Hardin, James (July 23, 2012). Common errors in statistics (and how to avoid them) (4th ed.). Hoboken, New Jersey: John Wiley & Sons, Inc. pp. 129–131. ISBN 978-1118294390.

[kassraftery1995-6] 1 2 Robert E. Kass & Adrian E. Raftery (1995). "Bayes Factors" (PDF). Journal of the American Statistical Association. 90 (430): 791. doi:10.2307/2291091.

[Toni2009b-7] 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. arXiv:0911.1705. doi:10.1093/bioinformatics/btp619. PMC 2796821. PMID 19880371.

[Robert2011-8] Robert, C.P.; J. Cornuet; J. Marin & N.S. Pillai (2011). "Lack of confidence in approximate Bayesian computation model choice". Proceedings of the National Academy of Sciences. 108 (37): 15112–15117. Bibcode:2011PNAS..10815112R. doi:10.1073/pnas.1102900108. PMC 3174657. PMID 21876135.

[9] H. Jeffreys (1961). The Theory of Probability (3rd ed.). Oxford. p. 432

[10] 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 0548210.

[11] Sharpening Ockham's Razor On a Bayesian Strop