Bayes factor
From Wikipedia, the free encyclopedia
It has been suggested that this article or section be merged with Bayesian model comparison. (Discuss) 
In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing.^{[1]}^{[2]}
Contents 
[edit] Definition
Given a model selection problem in which we have to choose between two models M_{1} and M_{2}, on the basis of a data vector x. The Bayes factor K is given by
where p(x  M_{i}) is called the marginal likelihood for model i. This is similar to a likelihoodratio test, but instead of maximizing the likelihood, Bayesians average it over the parameters. Generally, the models M_{1} and M_{2} will be parametrized by vectors of parameters θ_{1} and θ_{2}; thus K is given by
The logarithm of K is sometimes called the weight of evidence given by x for M_{1} over M_{2}, measured in bits, nats, or bans, according to whether the logarithm is taken to base 2, base e, or base 10.
[edit] Interpretation
A value of K > 1 means that the data indicate that M_{1} is more strongly supported by the data under consideration than M_{2}. 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:^{[3]}

K dB bits Strength of evidence < 1:1 < 0 Negative (supports M_{2}) 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.^{[4]}
The use of Bayes factors or classical hypothesis testing takes place in the context of inference rather than decisionmaking 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 decisionmaking under uncertainty in which the resulting action is to report a value. For decisionmaking, 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 KullbackLeibler divergence. If the logarithms are to the base 2 this is equivalent to Shannon information.
[edit] Example
Suppose we have a random variable which produces either a success or a failure. We want to compare a model M_{1} where the probability of success is q = ½, and another model M_{2} 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:
So we have
but
The ratio is then 1.197..., which is "barely worth mentioning" even if it points very slightly towards M_{1}.
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 M_{2}. 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 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 M_{1} 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 twotailed 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.
M_{2} is a more complex model than M_{1} 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.
[edit] See also
[edit] Statistical ratios
 Likelihoodratio test, frequentist analog
 Odds ratio
 Relative risk
[edit] References
 ^ Goodman S (1999). "Toward evidencebased medical statistics. 1: The P value fallacy". Ann Intern Med 130 (12): 995–1004. PMID 10383371.
 ^ Goodman S (1999). "Toward evidencebased medical statistics. 2: The Bayes factor". Ann Intern Med 130 (12): 1005–13. PMID 10383350.
 ^ 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: . MR82c:01049.