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8.8 Generalized State-Space Models 293
and that Xt+1 given Xt, X (t−1) , Y (t) is independent of X (t−1) , Y (t) with conditional
density function
p(xt+1|xt) : p xt+1|xt, x (t−1) , y (t)
t 1, 2,.... (8.8.2)
We shall also assume that the initial state X1 has probability density p1. The joint
density of the observation and state variables can be computed directly from (8.8.1)–
(8.8.2) as
p(y1,...,yn,x1,...,xn) p yn|xn, x (n−1) , y (n−1) p xn, x (n−1) , y (n−1)
p(yn|xn)p xn|x (n−1) , y (n−1) p y (n−1) , x (n−1)
p(yn|xn)p(xn|xn−1)p y (n−1) , x (n−1)
···
n
n
p(yj|xj) p(xj|xj−1) p1(x1),
j1
and since (8.8.2) implies that {Xt} is Markov (see Problem 8.22),
n
p(y1,...,yn|x1,...,xn) p(yj|xj) . (8.8.3)
j1
We conclude that Y1,...,Yn are conditionally independent given the state variables
X1,...,Xn, so that the dependence structure of {Yt} is inherited from that of the state
process {Xt}. The sequence of state variables {Xt} is often referred to as the hidden
or latent generating process associated with the observed process.
In order to solve the filtering and prediction problems in this setting, we shall
determine the conditional densities p xt|y (t) of Xt given Y (t) , and p xt|y (t−1) of Xt
given Y (t−1) , respectively. The minimum mean squared error estimates of Xt based
on Y (t) and Y (t−1) can then be computed as the conditional expectations, E Xt|Y (t)
and E Xt|Y (t−1) .
An application of Bayes’s theorem, using the assumption that the distribution of
Yt given Xt, X (t−1) , Y (t−1) does not depend on X (t−1) , Y (t−1) , yields
and
p xt|y (t) p(yt|xt)p xt|y (t−1) /p yt|y (t−1)
p xt+1|y (t)
j2
(8.8.4)
p xt|y (t) p(xt+1|xt)dµ(xt). (8.8.5)
(The integral relative to dµ(xt) in (8.8.4) is interpreted as the integral relative to dxt
in the continuous case and as the sum over all values of xt in the discrete case.) The
initial condition needed to solve these recursions is
p x1|y (0) : p1(x1). (8.8.6)