Bayesian analysis of ordinal survey data using the Dirichlet process ...

Dunson and Gelfand 2009, Gill and Casella 2009). The nonparametric prior specificationin our model and the associated clustering of subjects provides another essential differencebetween our approach and that of Rossi, Gilula and Allenby (2001).3 COMPUTATIONThe model described in section 2 is generally referred to as a Dirichlet process mixturemodel, and various Markov chain methodologies have been developed to facilitatesampling-based analyses (Neal 2000).sophistication on the part of the programmer.However, these algorithms require considerableA goal in this paper is to simplify the programming aspect of the analysis by carryingout computations in WinBUGS. The basic idea behind WinBUGS is that the programmerneed only specify the statistical model, the prior and the data. The Markov chaincalculations are done in the background whereby the user is then supplied with Markovchain output. Markov chain output is then conveniently averaged to give approximationsof posterior means.To implement the analysis of our model in WinBUGS, we make use of the constructivedefinition of the Dirichlet process given by Sethuraman (1994). The definition is knownas the stick breaking representation, and in the context of our problem, it is given asfollows: Generate a set of iid atoms (a ∗ i , b ∗ i ) from tr-Normal 2 (µ G , Σ G ) and generate a setof weights w i = y i∏ i−1j=1(1 − y j ) where the y i are iid with y i ∼ Beta(1, α) for i = 1, . . . , ∞.Thenwhere δ (a ∗i ,b ∗ i ) is the point mass at (a ∗ i , b ∗ i ).∞∑G = w i δ (a ∗i ,b ∗ i ) (6)i=110