The name is absent

62

measure. We discuss the probability model for G below. A parametric random effects
distribution for μ_i keeps computation simple.

(βi, δ_i∖G)~ G, i.i.d. , and μ_i ~ Ga(s_μ, s_μ ■ t_μ)             (3.27)

The parametrization of the Gamma distribution is chosen such that E(X) = а/b for
X ~ Ga(a,b). Choosing a prior for the random probability measure G requires a
nonparametric prior. The most commonly used model is the Dirichlet process (DP)
prior (Ferguson, 1973; Antoniak, 1974). We write G ~ DP(a,Go) for a DP prior on
the random probability measure G. The DP prior is indexed with two parameters,
a total mass parameter a and a base measure Go- The total mass parameter is a
precision parameter, and the base measure defines the prior expectation, E(G) — Gq.
See, for example MacEachern and Müller (1998), Walk et al. (1999), and Müller et
al. (2004) for recent reviews of the DP prior, including posterior inference for DP
mixtures similar to the model used here. We assume

G ~ DP(α, Gq) with Go(b, d) = Ga(b ∣ Sβ, Sβtβ) ■ Ga(d ∣ s⅛, s<5t⅛).       (3.28)

The base measure in the DP prior are independent gamma distributions. The model
is completed with a prior on the hyperparameters

a ~ Ga(a_a,b_a),t_β ~ Ga(t_β∖a_tβ, b_t0),                  (3.29)

t_δ ~ Ga(t_s∖a_tδ,b_ti), and,⅛_μ ~ Ga(t_μ∖a_tμ,b_tμ),               (3.30)

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