The name is absent

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,..., υ_n), define
the binary response as z_i = l(ι⅛ > 0) and assume a linear relationship between z_i and
the vector x_i, that is:

z_i = x^t_iβ + e_i for i = 1,..., n.

The distribution of the random errors e_i 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(υ_i < 0)l(z_i = 0)} × N{v_i ∣ x^t_iβ, 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(v_l I z, β) is a truncated (at zero) normal distribution. It is truncated from the
right when z_i — 0 and truncated from the left when z_i = 1. The expression (2.1) is
the posterior distribution of β when considering the Bayesian linear regression model
z_i = x^t_iβ + e_i with e_i ~ A(0,1). Denote by N_p(m, Σ) the р-dimensional normal
distribution with mean vector m and covariance matrix Σ. Considering a prior for
β, p(∕?) = N_p(r∏β,∑β), yields the posterior distribution p(β ∣ z) = N_p(mι, ∑_i) where
∑ι = (∑0¹ + X^tX')~¹ and mɪ — ∑ι(∑^¹m_ιa + X^tZ~), where X is the design matrix
with г—th row equal to x_i. A Gibbs sampler defined by iterative draws from the two
conditional posterior distributions above is used to generate a Monte Carlo posterior

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