The name is absent

50

(a) Given the previous values for θ*, K and φ, we generate a new configuration by
sampling new values for the indicator variables using (3.21), replacing θ↑,θ⅛,....
When sampling φ_i = 0 we associate the observation i, x_i, with a new draw from
g_i0 given in (3.20) and update the configuration accordingly.

Note that using a base measure G₀ conjugate to f_i is mathematically convenient.
It makes sampling from g_i0 and computation of h_i easier. For computational reasons
conjugate families should be considered when appropriate. In the model introduced
in Section 3.4 this is the case.

Repeated use of step (a) defines a Markov chain with limiting distribution equal
to the posterior distribution p(θ*,K,φ ∣ rrⁿ). But, this Markov chain mixes poorly.
Since it shifts one value of θ_i at a time, rarely does a value of in a clusters change.
This change needs the chain to pass trough a middle state of low probability in which
all indices in the cluster Sk are moved to other clusters. Convergence is accelerated
by avoiding this phenomenon. Once the values of θ*, K and φ have been obtained in
step (a), the vector θ* is resampled conditional on the values of K and φ. That is, a
second step is added to the Gibbs sampling scheme (after each step (a)),

(b) Given K and φ, we draw a new set of parameters Θ* by sampling new values
from the posterior distribution (3.14) .

Sequential iterations over (a) and (b) defines a Markov chain that converges to
the posterior distribution p(β*,K,φ ∣ xⁿ). For details about this convergence see
MacEachern and Müller (1998) . The posterior distribution of any function of the
parameters can be estimated based on the posterior sample. For example the aver-
age of (3.17), with respect to the simulated values of flɪ,..., θ*_κ, and θ_n+1 estimates
the predictive distribution p(x_n+ι ∣ ʃ") = G. This average is used for the density
estimation. See the example given in the next subsection.

Now we extend the Gibbs sampling scheme to include the common parameter
σ. Let p(σ) denote the prior for σ. We draw this parameter from the appropriate

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