The name is absent

103

Observing that

p{yi I θ_i)G₀(θ_i) = ^Р^^Уг I                = p{θ_i I y_i) /p(y_i I θ)dθ,

P∖Vi)                       J

we get that with probability <⅛ := ot∕Q'¹ × f р(у_г ∣ θ) dθ, θt is a new observation from
p⅛i I θ_i).

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(θ_i∖y_i). In contrast, we will set θ_i equal to θ_k~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 = θ_k ∣ 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 θ_k 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.

<<   112   113   114   115   116   117   118   >>

More intriguing information