The name is absent

42

When aj = O, the corresponding Yj is a point mass at 0, i.e. Yj = 0 almost surely.
When Oj > 0 for all j, the pdf in 3⅞^fc~¹ of (У1,..., Yfc-ι) is

(∖ α⅛-l
l-∑t^⅛) ɪs(w.....№-i).

(3.3)

where S is the set,

.                                                                              fc-l

S = {(v∕ι,..., y_fc~ι) ∈ Sff^fc~¹ ∣ y_j≥0, ∑y_j<l}.

j=ι

When к = 2, (3.3) reduces to the Beta distribution, denoted by Beta(ou, o⅛).

3.2.2 Dirichlet Process and Dirichlet Process Mixture Model

Consider the complete separable metric space (Ferguson, 1974), X with its Borel
σ-algebra A. Let

ʃ = {F : F is a probability measure on X}.

Let A denote some suitable σ-algebra of subsets of F, for example the Borel sets
generated by the topology of weak convergence. We say that P is a random probability
measure (RPM) if P is a probability measure in (F, Λ). If F denotes a random
probability chosen according to P, then F(B) for B measurable set in X is a random
variable. Let ʃɪ,... ,x_n be a random sample from F. Consider a Bayesiari model,
with a prior distribution on F ~ P. Inference about F is based on the posterior
distribution F ∖ x₁,... ,x_n, using the information available in the sample.

With this framework in mind Ferguson (1973) stated two “desirable” characteristic
that the RPM, P, the prior for F, should satisfy: (i) Its support should be large and
(ii) the posterior distribution should be mathematically convenient. In the same
paper, he introduces and proves the existence of the Dirichlet Process (DP) RPM. A
measure F is generated by a DP if for any partition of the sample space, {Bι,..., B⅛},

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