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294 Chapter 8 State-Space Models
The factor p yt|y (t−1) appearing in the denominator of (8.8.4) is just a scale factor,
determined by the condition p xt|y (t) dµ(xt) 1. In the generalized state-space
setup, prediction of a future state variable is less important than forecasting a future
value of the observations. The relevant forecast density can be computed from (8.8.5)
as
p yt+1|y (t)
p(yt+1|xt+1)p xt+1|y (t) dµ(xt+1). (8.8.7)
Equations (8.8.1)–(8.8.2) can be regarded as a Bayesian model specification. A
classical Bayesian model has two key assumptions. The first is that the data Y1,...,Yt,
given an unobservable parameter (X (t) in our case), are independent with specified
conditional distribution. This corresponds to (8.8.3). The second specifies a prior
distribution for the parameter value. This corresponds to (8.8.2). The posterior
distribution is then the conditional distribution of the parameter given the data. In
the present setting the posterior distribution of the component Xt of X (t) is determined
by the solution (8.8.4) of the filtering problem.
Example 8.8.1 Consider the simplified version of the linear state-space model of Section 8.1,
Yt GXt + Wt, {Wt} ∼iid N(0, R), (8.8.8)
Xt+1 FXt + Vt, {Vt} ∼iid N(0, Q), (8.8.9)
where the noise sequences {Wt} and {Vt} are independent of each other. For this model
the probability densities in (8.8.1)–(8.8.2) become
p1(x1) n(x1; EX1, Var(X1)), (8.8.10)
p(yt|xt) n(yt; Gxt, R), (8.8.11)
p(xt+1|xt) n(xt+1; Fxt, Q), (8.8.12)
where n x; µ, σ 2 is the normal density with mean µ and variance σ 2 defined in
Example (a) of Section A.1.
To solve the filtering and prediction problems in this new framework, we first
observe that the filtering and prediction densities in (8.8.4) and (8.8.5) are both normal.
We shall write them, using the notation of Section 8.4, as
and
p xt|Y (t) n(xt; Xt|t,t|t) (8.8.13)
p xt+1|Y (t)
n xt+1; ˆXt+1,t+1 . (8.8.14)