43
the vector of random probabilities (F(Bι),..., F(F⅛)) follows a Dirichlet distribution
as defined in the previous subsection:
(F(β1),..., F(Bk)) ~ Dirichlet(αF0(B1),..., aF0(Bk)).
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,..., Xn is a random sample of
size n of the Dirichlet Process DP if given, F ~ DP, X1,..., Xn ∣ 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 δx denote a point mass at x. If
X1,...,Xn⅛d∙F and F~DP(α,F0),
then the posterior distribution of F is, again, a DP given by
F∣X1,...,Xn~DP(α + n,F1), (3.4)
with Fi E(F ∣ X1,... ,Xn) oc F0 + ɪɪɪ δχi 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(-) = ∑whδμh, μh^-d-F0 (3.5)
h=l
where
wh = Uh ɪɪ(l — Uj) with Uh ~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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