13
discussion below. The logistic model allows a nice interpretation of the parameters.
Posterior simulation in the described model is rather complicated. To facilitate
posterior simulation, Albert and Chib (1993) propose an equivalent augmented model.
They introduce a vector of continuous latent random variables v = (v1,..., υn), define
the binary response as zi = l(ι⅛ > 0) and assume a linear relationship between zi and
the vector xi, that is:
zi = xtiβ + ei for i = 1,..., n.
The distribution of the random errors ei plays an important role in their model. In
particular when the errors are standard normal (logistic) distributed we are working
with a probit (logistic) model. The variance is set to 1 for identifiability reasons. In
the augmented probit model, the joint pdf of the data and the parameters is given
by:
n
p(∕3,v,z) ocp(∕3) × JJ{l(r⅛ > 0)l(¾ = 1) + l(υi < 0)l(zi = 0)} × N{vi ∣ xtiβ, 1),
i=l
where N{y ∣ m, s) denotes the normal cdf with mean m and variance s. This implies
the complete conditional posterior distribution
n
P<β I v,z) (xp{β) × ∩⅜ I ≈∙∕3,1). (2.1)
2=1
And p(vl I z, β) is a truncated (at zero) normal distribution. It is truncated from the
right when zi — 0 and truncated from the left when zi = 1. The expression (2.1) is
the posterior distribution of β when considering the Bayesian linear regression model
zi = xtiβ + ei with ei ~ A(0,1). Denote by Np(m, Σ) the р-dimensional normal
distribution with mean vector m and covariance matrix Σ. Considering a prior for
β, p(∕?) = Np(r∏β,∑β), yields the posterior distribution p(β ∣ z) = Np(mι, ∑i) where
∑ι = (∑01 + XtX')~1 and mɪ — ∑ι(∑^1mιa + XtZ~), where X is the design matrix
with г—th row equal to xi. A Gibbs sampler defined by iterative draws from the two
conditional posterior distributions above is used to generate a Monte Carlo posterior