equal prior probability, so that , the Bayes factor is equal to the ratio of the 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. 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, with the caveat that approximate-Bayesian estimates of Bayes factors are often biased. 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). to use minimum description length (MDL). 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: style="text-align: center; margin-left: auto; margin-right: auto; border: none;" ! K !! dHart !! bits !! Strength of evidence |- | < 100 || < 0 || < 0 || Negative (supports M2) |- | 100 to 101/2 || 0 to 5 || 0 to 1.6 || Barely worth mentioning |- | 101/2 to 101 || 5 to 10 || 1.6 to 3.3 || Substantial |- | 101 to 103/2 || 10 to 15 || 3.3 to 5.0 || Strong |- | 103/2 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