Applied Bayesian Modelling - Free

54 HIERARCHICAL MIXTURE MODELSTable 2.6Student t parameters for fertilisation data, degrees of freedom unknownParameter Mean St. devn. 2.5% Median 97.5%k 0.11 0.12 0.01 0.06 0.47Mean 23.90 9.84 1.06 24.17 46.72n 29.57 52.66 2.07 10.17 289.30Variance 1225.00 709.20 237.10 1088.00 3655.00degrees of freedom is obtained (though the posterior mean for n is 30). The posteriormedian of s 2 is lower than in the Normal analysis, namely 1090, and the Normal meanis higher ± though not more precisely identified ± with an average of 24.This analysis illustrates how inferences about the second stage parameters may bedistorted by incorrectly assuming a Normal first stage density, but does not indicatewhich observations are the source of the heavy tails. In a third analysis (Model 3 inProgram 2.4), the scale mixture version of the Normal is applied, with n still anunknown. Taking the prior on the degrees of freedom as above we obtain ± fromiterations 50 000±100 000 of a three chain run ± factors l i for each sample memberthat provide a measure of outlier status. It may be noted that sampling convergence isimproved by this approach as compared to the standard Student t form above. Thelowest l i (i.e. most suspect points) are for corn plant pairs 2 and 15, the observations forwhich are both highly negative (Table 2.7). The high positive value for pair 13 is alsosuspect. The Normal mean parameter m is again estimated to have posterior mean 24,and the median variance is around 1050.Table 2.7Parameters for scale mixture density of fertilisation dataMean St. devn. 2.5% Median 97.5%Mean 24.1 9.8 3.9 24.4 42.8Degr. Freedom 23.7 40.8 2.2 8.7 140.8Variance 1199 706 304 1058 2950Weights for Individual CasesMean St. devn. 2.5% Median 97.5%l 1 1.03 0.53 0.22 0.97 2.37l 2 0.59 0.36 0.05 0.57 1.29l 3 1.06 0.56 0.24 0.99 2.47l 4 1.11 0.60 0.26 1.01 2.68l 5 1.05 0.55 0.23 0.98 2.43l 6 1.14 0.63 0.27 1.02 2.78l 7 1.14 0.62 0.28 1.02 2.78l 8 1.09 0.58 0.24 1.00 2.57l 9 1.10 0.60 0.25 1.01 2.64l 10 1.14 0.63 0.27 1.02 2.78l 11 0.97 0.49 0.19 0.93 2.17l 12 1.14 0.62 0.27 1.03 2.78l 13 0.82 0.42 0.12 0.83 1.74l 14 0.94 0.47 0.17 0.91 2.07l 15 0.68 0.38 0.07 0.68 1.45