53
the file). We apply the DPM model to estimate the distribution of the house prices
(disregarding the rest of the information in the file).
A particular case of model (3.9) is given in Escobar and West (1995), and imple-
mented in the R-package “DPpackage” (Alejandro Jara, 2007):
I. yi I μi,λi ~ N(μi,λi) for i = 1,... ,n,
II. (μι,λ1), ...,(μn,λn) ∣G~G
III. G I Q,fc0,μι,≠ι ~ DP(α,G0(∙ I ⅛o,μι,≠ι)),
G0(μi,λi I k0,μ1,ψ1') = N(μi ∣ m1, k0λi) Ga(λ ∣ r∕ι,≠ι)
IV. a ~ Ga(aa,bct'),mι ~ N(m2,s2β
ψι ~ Ga(η2,ψ2) and k0 ~ Gα(α⅛0∕2, ⅛0∕2),
where N(m,τ) denotes the normal distribution with mean m and precision τ, and
Ga(a,β') the gamma distribution with mean a∣β∖ ηι,aa,ba, s2,η2,ψ2, dk,bk > 0 and
m2 ∈ JL
(3.25)
Notice that the variables yi,(μi,λi) and (m1,⅛,ψι) of the model in (3.25) play
the role of the variables xt,θi and 7, respectively, in the general DPM (3.9). Let
Уп = {⅛∙∙∙ >y∏} represent the prices of n — 506 houses. Besides, we denote with
μ* = (μ*1,...,μ*κ) and λ* = (ʌɪ,..., AJf) the vectors of size K containing the unique
values of the μi and the λi respectively. We use the notation of the previous section
for the variables related with the configurations. When stating a complete conditional
posterior distribution in this section, we make explicit only the relevant quantities
(for example, since the complete conditional posterior distribution of k0 depends only
on m1,μ* and A*, we just write k0 ∣ m1,μ*, λ*).
We now describe steps (a)-(e) of the MCMC sampling scheme given in the previous
section for this specific example. There is no common parameter σ so that step (c)
is not necessary. Let t(y ∣ m, s, v) denote a student t distribution with location m,
scale s and degrees of freedom υ.
(a) Given the currently imputed values of (μ*, λ*), K and φ, generate a new config-
uration by simulating φi,..., φn from the complete conditional posterior distri-