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8.8 Generalized State-Space Models 309
Table 8.4 Prediction density of Y53 given Y (52) for data in
Figure 8.7.
Number of goals
0 1 2 3 4 5
p(y53|y (52) ) .472 .326 .138 .046 .013 .004
Example 8.8.8 The exponential case
Figure 8-10
One-step predictors
of the goal data.
Suppose Yt given Xt has an exponential density with mean −1/Xt (Xt < 0). The
observation density is given by
p(yt|xt) exp{ytxt + ln(−xt)}, yt > 0,
which has the form (8.8.31) with b(x) −ln(−x) and c(y) 0. The state densities
corresponding to the family of conjugate priors (see (8.8.37)) are given by
Goals
0 1 2 3 4 5
o
p xt+1|y (t) exp{αt+1|txt+1 − λt+1|tb(xt+1) + At+1|t}, −∞