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. An alternative table, widely cited, is provided by Kass and Raftery (1995): style="text-align: center; margin-left: auto; margin-right: auto; border: none;" ! log10 K !! K !! Strength of evidence |- | 0 to 1/2 || 1 to 3.2 || Not worth more than a bare mention |- | 1/2 to 1 || 3.2 to 10 || Substantial |- | 1 to 2 || 10 to 100 || Strong |- | > 2 || > 100 || Decisive |- |} 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 = , 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: Thus we have for M1 whereas for M2 we have The ratio is then 1.2, 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 = is 0.02, and as a two-tailed test of getting a figure as extreme as or more extreme than 115 is 0.04. 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 , whence (rather than averaging over all possible q). That gives a likelihood ratio of 0.1 and points towards M2. M2 is