The name is absent

20

For reasons of identifiability we fix the variance of the normal kernels in (2.4). We
recommend σ∣ = 1 and σ% = 4. See the argument below.

For later reference we state the joint probability model. Let z = (¾ : i =
1,... ,n,j — 1,..., J) denote the data. Let fa denote the prior distribution for β_g.
Assume that μ_jg for j = 1,..., J and g = 1,..., G are a priori independent, that for
each j, given the imputed hyperparameter φj, μj₁,..., μ₃c is a random sample from
f_μ(∙∖φβ and that φ_i,... ,φj ~ f_φ. Let β= (β₁,... ,βj) denote all probit regression
coefficients, and similarly for μ, r, v, w and φ. Let N(x∖m, s) indicate a normal pdf
with mean m and variance s evaluated at x. The joint distribution of z and all the
model parameters is

p(z, v, r, β, μ, w, φ) =

ΠΓ=ι Π∫=1Φ¾j < ^vH < θ_zij+ιj] ∏Γ=ι ∏∫=ι M⅝' I ^xT⅛ + ^ri + Mj,wy> σ∣)

× ∏iι∕⅛(∕3_j)∏r=ιWdO,σ_r²)

× ∏ix {n^pT^Wi3=9}∏₉⅛r¹} nil {fM ∏iιΛ⅛ I ⅛)}

(2.9)

To show that the cutpoints 0₇∙⅛ in (2.2) can be fixed, consider the following
simplified version of the right side in (2.4). Let υ ~ ∑G=₁p_gN(μ_g,σ²) and define
π⅛ ≡ Pr(z = k) = Pr{θ_k < V < 0fc₊ι), к = 0,1,..., к. We show by a constructive
argument that an appropriate choice of (G, p_g,μ_g,g — 1,..., G) can approximate an
arbitrary set of desired cell probabilities (πθ,πj,... ,π*_κ). In particular, the parallel
regression assumptions of the probit model is not required. A similar argument was
used in Kottas et al. (2005) for infinite Dirichlet process mixtures of normal distribu-
tions. Consider a mixture of normal distributions with G ≥ K components. Place one
component of the mixture into each interval [0_fc, 0fc₊ι) by choosing μ_k — ∣(0fc + 0⅛+ι)>
and set p⅛ = 7Γ⅛. Specify σ such that 1 — e of the probabilities of each kernel is be-
tween the adjacent cutpoints. This trivially achieves ∣π⅛ — π⅛∣ < e for к = 0,1,..., К.
Therefore, the cutpoints θ_jk in (2.2) can be fixed without loss of generality.

We recommend as a default choice of cutpoint parameters 0⅛: Θq — —∞, θκ+ι =

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