
526 Current Trends in Bayesian Methodology with Applications
the complete posterior density π
γ
(ψ, w|y) we get the target posterior density
π
γ
(ψ|y), that is,
Z
R
n
π
γ
(ψ, w|y)dw = π
γ
(ψ|y).
So if we can generate a Markov chain {ψ
(i)
, w
(i)
}
N
i=1
with stationary den-
sity π
γ
(ψ, w|y), then the marginal chain {ψ
(i)
}
N
i=1
has the stationary density
π
γ
(ψ|y) defined in (25 .7). This is the standard technique of data augmenta-
tion and here w is playing the r ole of “latent” va riables (or “missing data”)
[21].
Since we are using conjugate priors for (β, σ
2
) in (25.6), integ rating
π
γ
(ψ, w|y) with respect β we have
π
γ
(σ
2
, τ
2
, w|y) ∝ (στ)
−n
exp{−
1
2τ
2
n
X
i=1
(y
i
− h
λ
(w
i
))
2
}
τ
−