14
sample of β: β1,... ,βfvl. These values lead to Monte Carlo estimation of π for a
future observation:
1 M
Pr[zf = 1 ∣ ɪʃ,z] ≈ — ∑ φ 1^τfβm),
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 zi falls in the random interval [0fc, θk+ι), where O = (θi,..., 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,. ≤ υi < θzi+1)} × N(υi ∣ xtiβ, 1),
i=l
The complete conditional posterior distribution for β is exactly the same as in the
binary data model given above; p(yi ∣ z, /3) is a truncated normal distribution taking
values in the interval [θzi,θzi+β). Assume a noninformative prior for θ∣ς, p(0⅛) <× 1.
The complete conditional distribution of 0⅛ is uniform in the interval [maxi{t⅛ : zi —
k},mmi{υi : zi = 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).