103
Observing that
p{yi I θi)G0(θi) = Р^Уг I = p{θi I yi) /p(yi I θ)dθ,
P∖Vi) J
we get that with probability <⅛ := ot∕Q'1 × f р(уг ∣ θ) dθ, θt is a new observation from
p⅛i I θi).
Consider the DPM Gibbs sampling scheme (MacEachern and Müller, 1998 and
Neal, 2000), described in Subsection 3.2.3, with a Dirichlet process with total mass
parameter a and base measure Go- In this scheme, θi is set equal to θ*k~l with
probability := p(θi∖yi). In contrast, we will set θi equal to θk~i with probability
d ( z∏fc∕ + 1 ∖7
qik=qikX\#S^Q) ∙
The expression above has an important practical implication. Assume we have
code for posterior simulation under the DP prior with parameters a and Gq. Only a
slight modification in the predictive probability function, i.e., in Pr[θi = θk ∣ θ~l, yn],
of the Dirichlet process is necessary to implement posterior simulation under the
proposed
nonexchangeable product partition model.
In the implementation of the NEPPM in (4.1) the normal prior for θk in model
plays the role of G0 in the algorithm described above. Besides, we assume a gamma
distribution for the DP total mass parameter a. To do so, we use the model augmen-
tation with a latent Beta random variable proposed by West (1992) (step (d) of the
Section 3.2), to implement posterior inference for a.