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 3⅞fc~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,..., yfc~ι) ∈ Sfffc~1 ∣ yj≥0, ∑yj<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,... ,xn 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,... ,xn, 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},