The name is absent

14

sample of β: β¹,... ,β^fvl. These values lead to Monte Carlo estimation of π for a
future observation:

1 ^M

Pr[z_f = 1 ∣ ɪʃ,z] ≈ — ∑ ^{φ 1}^^τ_fβ^m),
m=l

where Φ denotes the standard normal cdf.

2.2.3 Multiresponse Categorical Data Model

Albert and Chib (1993) generalized the model given above to consider ordinal re-
sponses ¾ ∈ {0,...,K- 1} with K > 2. The response ¾ is equal to к if the latent
variable z_i falls in the random interval [0_fc, θk+ι), where O = (θ_i,..., ¾~ι) is an im-
puted latent random vector of cutpoints, Θq = —∞ and θχ = ∞. For identifiability
reasons, θ₁ is fixed to O. See Figure 2.1 for a graphical representation in an example.
The joint distribution of the parameters and the data is:

Tl

p(β,N,∙z.,θ') xp(β,θ) × ∏{1(0₂,. ≤ υ_i < θ_zi+1)} × N(υ_i ∣ x^t_iβ, 1),
i=l

The complete conditional posterior distribution for β is exactly the same as in the
binary data model given above; p(y_i ∣ z, /3) is a truncated normal distribution taking
values in the interval [θ_zi,θ_zi+β). Assume a noninformative prior for θ∣_ς, p(0⅛) <× 1.
The complete conditional distribution of 0⅛ is uniform in the interval [max_i{t⅛ : z_i —
k},mm_i{υ_i : z_i = к + 1}).

Straightforward implementation of the Gibbs sampling scheme using these com-
plete conditional distributions yields a poorly mixing Markov chain. The cutpoints
and the latent variables move too slowly. Cowles (1996) algorithm accelerates con-
vergence by replacing alternate sampling from p(θ ∣ z,v,∕3) and p(y ∣ z, β,θβ by
instead sampling from the joint distribution of the latent variables and the cutpoints
conditional in the data and the rest of the parameters,

p(v, O I β, z) oc p(v I z, β, θ) × p(θ I z, /3).

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