The name is absent

63

3.4.2 Posterior Simulation

The model defined in (3.26)-(3.30) is a DPM. It includes a conjugate Poisson sample
model and gamma baseline distribution of the DP, and conditionally conjugate prior
specifications for other parameters. This greatly facilitates posterior simulation by
MCMC simulation.

The implementation of posterior MCMC follows the standard posterior simulation
method for DPM models given in, for example, Neal (2000) and MacEachern and
Müller (1998). This method was described in the Section 3.2. In particular, we use
the model augmentation with a latent Beta random variable proposed by West (1992)
(step (d) of the Section 3.2), to implement inference for the total mass parameter a.

The remaining parameters μi,t_μ,tβ and have closed form conditional posterior
distributions conditional on currently imputed values for all other parameters and
latent variables. Let N denote the data. When stating a complete conditional pos-
terior distribution on this section, we make explicit only the relevant quantities (for
example, since i_μ∣N,μι,... ,μ_n,tβ,tg does not depend on N, tβ or t§ we just write
tμ |Д1 > ∙ ∙ ■ ) ∕⅛)∙

We apply the Gibbs sampling scheme consisting of steps (a)-(e) in Section 3.2 to
simulate from the posterior distribution of the model proposed model. We notice that
the parameters τ_i, F_i(∙ ∣ θ_i), and 7 of the general DPM model (3.9) are equivalent to
N_i, (3.26), (tβ,tg) in our specific proposed model. There is no common parameter ,σ,
for all N_i^,s and, then, step (c) is not necessary. The rest of the parameters in (3.9) and
its correspondents in our specific application have the same names; an extra step (f)
is included in the Gibbs sampling scheme to account for the parameters μ_i∙, and, the
following configuration - a set of variables (K, (β*, δ*^y), φ) equivalent to (∕¾, δ₁),...,
(βn,δ_n), see Section 3.2- is required:

• (∕3*, <P) = (∕3∣, jɪ),..., (∕⅞-, ¾∙) represents the K unique values of (βι,δβ),...,

(β_n,δ_n).

<<   72   73   74   75   76   77   78   >>

More intriguing information