55
e Hyperparameters of G0, 7 : simulate from the complete conditional posterior
distributions
, I * X * X , , I I TΠ2S2 + A¾ Lt-1 /ʃfeʌfe . V—' X ɪ
p(m1 I μ , λ , r) = N mɪ ∣ --------- , S2 + k0 ∖ λfc ,
∖ S2 + fc0 Lfe=I λ⅛ fc=l /
(κ ∖
≠1 I η2 + Kηl,ψ2 + ∑λ*k I ,
fe=l /
and,
/, I * ∖ r< ( 1 I ÷ c¾ βk0 ÷ Σ2 τk(fj'k ~ ml)2λ
p(k0 I τ , m1, r) = Ga I k0 ∣----—-,------------------ ) .
Finally, we estimate the density of the house prices at a point yf based on the (ap-
proximated) sample of size M from the joint posterior distribution of the parameters:
(μ*m, λ*m, φm, Km, am, k%l, mψ, ηm), for m = 1,..., M, via (3.17) by,
1 M m к™ j
Λ/ I »”) = ʌʃ Σ ɪʌ'to ɪ Æ,ŋ + ∑^N(,y, I Λ∙"∙) ,
1 Vl LΛ IL Lt I h I
m=l k=l J
where (μ∏+ι,λζ+ι) is a new sample from gio given in (3.2.4) whose parameters are
specified by the corresponding parameters of the m — th simulated observation:
L,m γrim rim
⅝ î rn,l > rI ∙
We use the same values for the hyperparameters given in the set of hyperparame-
ters “prior4” in the example in the help of the function “DPdensity” in the R package
“DPpackage”. In the R example, the density of the velocities, relative to our own
galaxy, of 82 galaxies from six well-separated conic sections of the space is estimated.
The values are aa = 2,ba = l,m2 = 0,s2 = 10-5,α⅛o = l,∕⅞0 = 100,771 = η2 — 3,
and ψ2 — 0.5. After a 1000 iteration burn-in period we generated M = 10,000 ob-
servations from the posterior distribution of the parameters by taking an observation
every ten Gibbs iterations. All this using the R function “DPdensity”. The estimated
density is shown in Figure 3.2.