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34is to use a likelihood ratio test (LRT). The test statistic is2 log( )pMLE (X|M 2 )∼ χ 2 Dp MLE (X|M 1 )2 −D 1(3.1)where p MLE (X|M) is the maximized likelihood of the data X given the modelM, and (D 2 − D 1 ) is the difference in degrees of freedom between the two models.The LRT requires that M 1 must be nested in M 2 ; the Bayes factor approach doesnot have this restriction.The Bayes factor B 21 for comparing the same two models has a form similarto the LRT.B 21 = p(X|M 2)p(X|M 1 )(3.2)Here, p(X|M) denotes the integrated likelihood rather than the maximizedlikelihood. If θ is the set of parameters in p(X|M) (so θ may be a vector), thenthe integrated likelihood is∫p(X|M i ) =p(X|θ, M i )p(θ|M i )dθ (3.3)where p(θ|M i ) is the prior density of θ given M i .Denote log(p(X|θ, M i )p(θ|M i )) by g(θ|M i ) , and rewrite the integrated likelihoodas∫p(X|M i ) =exp(g(θ|M i ))dθ (3.4)Suppose ˜θ is the posterior mode of θ, i.e. the value which is the mode of theposterior distribution of θ. The maximum likelihood estimate ˆθ and the posteriormode ˜θ converge to the same value as the sample size increases to infinity. Replacethe inner term in equation 3.4 by the first few terms of a Taylor series expansion