552 Current Trends in Bayesian Methodology with Applications
26.5.4 Partially observed finite HMM
Consider the following model described probabilistically as
r
k
∼ Π(r
k
|r
k−1
), y
k
∼ p(y
k
|r
k
), z
k
∼ p(z
k
|y
k
), (26.34)
where the dynamics of r
k
is given by a homogeneous first order finite state
Markov cha in. Here r
k
and y
k
together describe a partially observed finite
HMM (POfHMM). However instead of y
k
, z
k
is actually observed. The graph-
ical representation of the model is shown in Figure 2 6.3(b).
Similar to (26.5.3), the joint density can be decomposed as
p(r
k
, y
0:k
|z
0:k
) = p(r
k
|y
0:k
, z
0:k
) p(y
0:k
|z
0:k
). (26.35)
Again, the density p(y
0:k
|z
0:k
) can be targeted using a PF in the form of N
weighted particles as p(y
0:k
|z
0:k
) ≈
P
N
i=1
ω
(i)
k
δ(y
0:k
− y
(i)
0:k
). Then for each ...