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8.8 Generalized State-Space Models 303
where αt+1|t and λt+1|t are, for the moment, unspecified functions of y (t) . (The subscript
t +1|t on the parameters is a shorthand way to indicate dependence on the conditional
distribution of Xt+1 given Y (t) .) With this specification of the state densities, the
parameters αt+1|t are related to the best one-step predictor of Yt through the formula
αt+1|t/λt+1|t ˆYt+1 : E Yt+1|y (t) . (8.8.38)
Proof We have from (8.8.7) and (8.8.33) that
E(Yt+1|y (t) )
∞
∞
yt+10
∞
−∞
yt+1p(yt+1|xt+1)p
−∞
xt+1|y (t) dxt+1
b ′ (xt+1)p xt+1|y (t) dxt+1.
Addition and subtraction of αt+1|t/λt+1|t then gives
E(Yt+1|y (t) )
∞
−∞
∞
−∞
b ′ (xt+1) − αt+1|t
λt+1|t
p xt+1|y (t) dxt+1 + αt+1|t
−λ −1
t+1|t p′ xt+1|y (t) dxt+1 + αt+1|t
λt+1|t
−λ −1
t+1|tp xt+1|y (t)xt+1∞ αt+1|t
+
xt+1−∞
λt+1|t
αt+1|t
.
λt+1|t
λt+1|t
Letting At|t−1 A(αt|t−1,λt|t−1), we can write the posterior density of Xt given
Y (t) as
p xt|y (t) exp{ytxt − b(xt) + c(yt)} exp{αt|t−1xt − λt|t−1b(xt)
+ At|t−1}/p yt|y (t−1)
exp{λt|t αt|txt − b(xt) − At|t},
f(xt; αt,λt),
where we find, by equating coefficients of xt and b(xt), that the coefficients λt and αt
are determined by
λt 1 + λt|t−1, (8.8.39)
αt yt + αt|t−1. (8.8.40)
The family of prior densities in (8.8.37) is called a conjugate family of priors for
the observation equation (8.8.35), since the resulting posterior densities are again
members of the same family.