The name is absent

43

the vector of random probabilities (F(Bι),..., F(F⅛)) follows a Dirichlet distribution
as defined in the previous subsection:

(F(β₁),..., F(B_k)) ~ Dirichlet(αF₀(B₁),..., aF₀(B_k)).

We denote this by F ~ DP(α,F₀). The DP prior is indexed with two parameters:
the precision parameter a and the base measure F₀. The base measure F₀ defines
the expectation F(F(B)) — F₀(F), and a is the precision parameter that defines the
variance. For details of the role of these parameters see Walker et al. (1999).

We say that the collection of random elements X₁,..., X_n is a random sample of
size n of the Dirichlet Process DP if given, F ~ DP, X₁,..., X_n ∣ F is a random
sample from F. One of the main characteristics of the DP is that it is conjugate,
implying a simple posterior updating rule. Let δ_x denote a point mass at x. If

X₁,...,X_n⅛^d∙F and F~DP(α,F₀),

then the posterior distribution of F is, again, a DP given by

F∣X₁,...,X_n~DP(α + n,F₁),                (3.4)

with Fi E(F ∣ X₁,... ,X_n) oc F₀ + ɪɪɪ δχ_i a compromise between the empirical
distribution and the base measure F₀ (See Figure 3.1).

Sethuraman (1994) shows that any F ~ DP(α, F₀) can be represented as

∞

F(-) = ∑w_hδ_μh, μ_h^-^d-F₀                  (3.5)

h=l

where

w_h = U_h ɪɪ(l — U_j) with U_h ~^{l d}'Beta(l, a),
j<h

That is, a realization of a DP is a discrete random measure with countable support.
Its support is a sequence of values independently sampled from F₀ with respective
jump sizes generated by a “stick breaking” procedure. In particular F is a discrete
measure, that is, the DP generates, almost surely, discrete random measures. Figure

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