Discrimination of Rival Models by Posterior Probability

For model Mj. let p(Y Mj. a) denote the predictive probability density of prospective data Y predicted in the manner of Eq. (6.1-10). with a known. Bayes theorem then gives the posterior probability of model Mj conditional on given Y and cr as 

Introducing a vector 6je of estimable parameters for model Mj, and integrating Eq. (6.6-22) over the permitted range of Oj, gives the posterior probability of model Mj in the form 

We treat the prior density p 6je Mj) as uniform over the region of appreciable likelihood. This approximation, consistent with Eq. (6.1-13), reduces Eq. (6.6-24) to 

The posterior probability p Mj Y, a), being based on data, has a sampling distribution over conceptual replications of the experimental design. We require this distribution to have expectation p(Mj) whenever model Mj is true. With this sensible requirement, Eq. (6.6-25) yields 

The model with the largest posterior probability is normally preferred, but before acceptance it should be tested for goodness of fit, as in Section 6.5. GREGPLUS automatically performs this test and summarizes the results for all the candidate models if the variance cr been specified. 

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