The name is absent

45

The predictive distribution (3.4) can be shown to be

1 ⁱ~¹

=       +                 forz = 2,...,n.    (3.7)

^{г l} j=ι

Multiplying p{Xι) and p(X_i ∣ Xi,..., X_i~ι), i — 2,... ,n, defines the joint marginal
distribution p(Xι,... ,X_n)∙ When F₀ has finite support and a = 1, the two expres-
sions above imply that sampling (X↑,... ,X_n) can be interpreted as drawing from an
urn whose initial proportion of balls of color x is F₀({τ}). We draw the г-th ball from
the urn, its color X_i is registered, the ball is put back into the urn and a new ball of
the same color as the one just extracted is added to the urn.

Mostly due to its computational advantages and the easy interpretation of the
parameters, the DP is the most popular nonparametric Bayesian prior. Nevertheless,
an almost surely discrete prior is not desirable in many applications. A simple solution
is to consider a convolution of G ~ DP with a continuous kernel. Such a (hierarchical)
model is known as Dirichlet process mixtures (DPM) (MacEachern, 1994, Escobar
and West, 1995):

x_i I θ_i ~ F(x_i I θ_i) and independent for i = 1,..., n
θι,...,θ_n I Gⁱ⅛^dG                                          (3.8)

G~DP(o,G₀),

i.e. x_i ~ ʃ F(x_i I θ) dG(θ) and G ~ DP(α, G₀).

The model above is easily extended to consider non identically distributed samples
such as the regression model: x_i ∣ θ_i ~ N(zfθ_i, σ) where z_i is a vector of covariates of
the same dimension as θ_i and σ the common precision. In general, suppose that the
distributions of each x_i are (possibly different) known distributions F_i, indexed with
Q_i, and possibly additional parameters σ that are common across all i, i.e.,

X_i∖θ_i,σ~ F_i(X_i∖θ_i,σ).

Moreover, we can include a hyperprior distribution for the parameter a and, besides,
consider that the base measure G₀ depends on parameters 7 with prior distribution

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