49
where θn+ι is a new sample from G0 and Λι+ι ɪs the cdf of x∏+ι∙ The Gibbs sampling
scheme to simulate parameters from the posterior distribution is based on the above
discussion and described next.
Gibbs Sampling Scheme: MacEachern (1994) introduced the following Gibbs sam-
pling scheme to simulate random samples θ1,... ,θn from the posterior under model
(3.9). The posterior is easiest described in terms of the configuration parameters θ*,
K and φ. We are still assuming that σ, 7 and a are known, the simulation of these
parameters can be added, as we will see later, straightforward to the Gibbs sampling
scheme, so that, we keep these variables out of the notation. Call (3.13),
κ~i
p(θi I xn, θ~i, φ~i, K~i) = qi0gi0(θi) + ∑ qikδθ→(θi). (3.18)
fc=l
Under model (3.9) the weights дгк are given by,
{cahi(xi), к = 0,
(3.19)
cnklfi(xi I ¾), 1 ≤ к ≤ K~i.
Here fi is the pdf corresponding to Fi, glo is the posterior pdf of θl, obtained by
updating g0 with the likelihood fi(xi ∣ θi), that is,
gιθ(θi) oc fi(xi I θi)go(θi), (3.20)
whose normalization constant, hi(xi), is the marginal density of Xi,
hi(xi) = J fi(xi I θi)g0(θi) dθi,
and c is a normalization constant (across к = 0,..., K~l).
Equation (3.18) implies the posterior distribution for the indicator variables in
the configuration,
Pr{φi = k∖xn, θ~i, φ~i, K~i) = qik. . (3.21)
We can simulate samples of θ*, K and φ by iterating over the following transition
probabilities:
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