use a likelihood calculation and derive the posterior mean of the likelihood and deviance.

In Example 6.1 use a likelihood calculation and derive the posterior mean of the likelihood and deviance. Use the AIC and BIC criteria to compare solutions C = 1, 2, 3, 4. In Example 6.1 obtain the posterior probabilities under C = 3 that individual cases belong to different groups. These are averages over iterations of indicator variables.

Eye-tracking data Escobar and West (1998) present count data on eye tracking anomalies in 101 schizophrenic patients. The data are obviously highly overdispersed to be fit by a single Poisson, and solutions with C = 2, 3 and 4 groups are estimated here with an ordered means constraint. Assuming a Dirichlet prior for the group probabilities, a prior sample size of s0 = 4 is allocated equally between the C groups so that the prior Dirichlet weights are αj = 4/C, j = 1,…,C. Priors on the means are expressed as νj = log(μj), where the νj are normal with variance 1000 and subject to an ordering constraint. Iterations 1001–5000 of a two chain run show that the two-group solution has means μ1 = 0.7 and μ2 = 11.5 with respective subpopulation proportions 0.73 and 0.27. The three-group solution has means 0.48, 6.7 and 19.2 with respective proportions 0.66, 0.24 and 0.10.

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What conclusions might you draw from these data?

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