The name is absent

88

where x*_k is the set of the values of the covariate in the cluster A:, ξ_fc is a latent variable,
and g(∙ I ξ⅛) and q(β are auxiliary probability models. For categorical covariates x_i ∈
{1, ∙ ∙ ∙, Q} they introduce a vector ξ_k = (ξ⅛ι,... ,ξ_fe<ρ) and use q(x_i = q ∣ ξ_fcg) = ξ_kq
and g(ξ) as a Dirichlet distribution with parameters β₁,..., βq. We get,

_q<_x*∖          ∏⅞ ^γ<A ^{+ 77}⅛)                   /₄ η

Assuming that the parameters of this Dirichlet distribution are all equal to, say, β,
the expression above reduces to:

_z *_λ W)∏,Γ(∕⅛_g)             _f,₈,

5(¾) _v(_βγ_{3 V}ç_{#sk +} Q₅)                    ( ∙ )

and, when /3=1,

_{f υ} Π₉Γ(m_fcg + l) ∏_ρW_g!
mi°< r(#S_fe + Q) (#S_fc + Q-l)!

This is exactly the coefficient d(S⅛) in (4.4) of the definition of our NEPPM with
7 = 1.

The proposed model p(p_n ∣ xⁿ) defines a sequence of probability models across
sample sizes n. The question arises whether the model is coherent across n. Ideally
the model for n should arise from marginalization of the model under n + 1. Below
we show that, in general, this is not true. Let ρ_n = (Si,..., S_λ-) denote a partition
of the set {1,..., n}. Recall the model, using the notation in (4.8)

p(p_n = (S_l,...,S_κ)∖xⁿ) =

The normalizing constant, g_n(x"), is equal to ∑_pn ∏^L₁ cu(S_k)g(x_k).

Let {p_n,φ_n+1 — P) denote the partition of {1,... ,n + 1} after adding the index
n + 1 to the Ath cluster of p_n, for I — 1,..., K + 1. Then,

Pr(pⁿ, φ_n+l = I I xⁿ,x_n+1 = q)

Pr{pⁿ I xⁿ)

<<   97   98   99   100   101   102   103   >>

More intriguing information