Applied Bayesian Modelling - Free

112 REGRESSION MODELSseparation, and (b) that if cases do fall in the same group, they are expected to besimilar.An alternative to constrained priors involves re-analysis of the posterior MCMCsample, for example by random or constrained permutation sampling (Fruhwirtth-Schattner, 2001). Suppose unconstrained priors in model (3.20) are adopted, andparameter values u (t)j ˆ {b (t)j , t (t)j } are sampled for the nominal group j at iterationt. We may investigate first whether ± after accounting for the label switchingproblem ± there are patterns apparent on some of the parameter estimates whichsupport the presence of sub-populations in the data. Thus, if there is only p ˆ 1predictor and the model isy i S j N(m ij , t j )(3:21)m ij ˆ b 0j b ij x ithen a prior constraint which produces an identifiable mixture might be b 01 > b 02 , orb 11 > b 12 or t 1 > t 2 . (Sometimes more than one constraint may be relevant, such asb 01 > b 02 , and b 11 > b 12 ). Fruhwirtth-Schattner proposes random permutations of thenominal groups in the posterior sample from an unconstrained prior to assess whetherthere are any parameter restrictions apparent empirically in the output that may beassociated with sub-populations in the observations.From the output of an unconstrained prior run with J ˆ 2 groups, random permutationof the original sample labels means that the parameters nominally labelled as 1 atiteration t are relabelled as 2 with probability 0.5, and if this particular relabellingoccurs then the parameters at iteration t originally labelled as 2 are relabelled as 1.Otherwise, the original labelling holds. If J ˆ 3 then the nominal group samplesordered {1, 2, 3} keep the same label with probability 1/6, change to {1, 3, 2} withprobability 1/6, etc.Let ~u jk then denote the relabelled group j samples for parameters k ˆ 1, : : , K. (Asuffix for iteration t is understood.) The parameters relabelled as 1 (or any other singlelabel among the j ˆ 1, : : J) provide a complete exploration of the unconstrained parameterspace, and one may consider scatter plots involving ~u 1k against ~u 1m for all pairsk and m. If some or all the plots involving ~u 1k show separated clusters, then anidentifying constraint may be based on that parameter. To assess whether this isan effective constraint, the permutation method is applied based not on randomreassignment, but on the basis of reassignment to ensure the constraint is satisfied atall iterations.Example 3.10 Viral infections in potato plants Turner (2000) considers experiments inwhich viral infections in potato plants were assessed in relation to total aphid exposurecounts. The experiment was repeated 51 times. The data are in principle binomial,recording numbers of infected plants in a 9 9 grid with a single plant at each point.However, for reasons of transparency, a Normal approximation involving linear regressionwas taken. The outcome is then just the totals of plants infected y, which vary from0 to 24. A plot of the infected plant count against the number of aphids released (x)shows a clear bifurcation, with one set of ( y, x) pairs illustrating a positive impact ofaphid count on infections, while another set of pairs shows no relation.Here we consider two issues: the question of possible relabelling in the MCMCsample, so that samples for the nominal group 1 (say) in fact are a mix of parameters