Applied Bayesian Modelling - Free

90 REGRESSION MODELSTable 3.1 Probit models for nodal involvement, regression coefficientsand marginal likelihood approximationsModel 1 with 3predictorsMean St. devn. 2.50% Median 97.50%b 0 0.74 0.40 1.50 0.74 0.07b 1 1.42 0.67 0.17 1.40 2.79b 2 1.30 0.48 0.40 1.29 2.22b 3 1.08 0.43 0.25 1.08 1.92Model 2 with 4 predictorsb 0 0.79 0.42 1.64 0.78 0.01b 1 1.63 0.70 0.30 1.63 3.07b 2 1.26 0.49 0.32 1.26 2.21b 3 0.96 0.45 0.10 0.96 1.84b 4 0.55 0.45 0.33 0.54 1.45Marginal likelihood estimates by iteration batchHarmonic MeanCPO MethodIterations Model 1 Model 2 Model 1 Model 21001±3000 29.86 28.85 29.20 29.853001±5000 28.41 30.83 29.18 30.035001±7000 28.89 28.47 28.90 29.887001±9000 28.55 30.23 29.07 29.879001:11000 28.47 28.60 29.15 29.5011000:13000 28.33 29.35 28.91 29.7813000±15000 29.25 29.58 29.28 29.86Program 3.1(B) follows Albert and Chib (1993), and takes the latent data z from thedistribution zjy. Then b is evaluated against N p (^b, C 1 ) for the samples z (t) by substitutingin Equation (3.10). This gives a marginal likelihood estimate of 34.04 for Model1 and 35.28 for Model 2, a Bayes factor of 3.46; see Model B in Program 3.1(B).Model A in Program 3.1(B) illustrates the basic truncated Normal sampling needed toimplement the Albert±Chib algorithm for probit regression.The CPO estimate (Equation (2.13) in Chapter 2) is obtained by taking minus logs ofthe posterior means of the inverse likelihoods (the quantities G[] in Program 3.1A,Model A), and then totalling over all cases. This estimator leads to B 12 ˆ 2:06, and ismore stable over batches than the harmonic mean estimate.A Pseudo Bayes Factor (PsBF) is also provided by the Geisser±Eddy cross-validationmethod based on training samples of n 1 cases and prediction of the remaining case.Program 3.1(C) (Nodal Involvement, Cross-Validation) evaluates 53 predictive likelihoodsof cases 1, 2, 3, : : , 53 based on models evaluated on cases {2, : : , 53}, {1, 3, 4, : : , 53},{1, 2, 4, : : , 53), . . . :{1, 2, . . . , 52}, respectively. Each omitted case y i provides a predictivelikelihood and Bayes factor under model j (where here j ˆ 1, 2), based on thecomponentsf j ( y i jb), p j (bjy (i) )