- α 1 | 0  = ρ( 21 )
- k = 1 : K
- α k | k  = p ( y k  =  ŷ k | z k | k  –1
- α k +1|k  =  f z k p( z k +1|z k )α k |k The forward recursion is given by
- α 1 | 0 = [ P − 1 , P − 1 , − 2 log | 2 π P | − ½ µ T P − 1 µ ]
- k = 1 . . . K
= α k | k  = [ h k | k , K k | k : 9 k | k ]
hk | k  = CT R-1 ˆ yk  + hk | k - 1 Kk | k  = CT R-1C + K k | k - 1 9k | k  = 9k | k –1  = ½ log | 2π R |  = ½ ˆ y T R –1 ˆ y k
= α k +1|k  = [h k +1|k ; K k +1|k ; 9 k +1|k ]
Mk = (ATQ-1A + Kk |k)-1 hk +1|k  = Q-1AMk hk |k Kk +1|k  = Q-1 - Q-1AMkATQ-1 9k +1 k  = 9k |k  = 2 log |2π Q|  + 2 log |2π M k | + 2 h k |k M k hk |k B.1.2  BACKWARD  MESSAGE  PASSING
We can compute the likelihood also by starting from y K .

$$p\big{(}y_{1:K}\big{)}=\int_{z_{1}}p\big{(}z_{1}\,|p(y_{1}|z_{1})\int_{z_{2}}p(z_{2}|z_{1})p(y_{2}|z_{2})\ldots\int_{z_{K}}p\big{(}z_{K}|z_{K-1}\big{)}p\big{(}y_{K}|z_{K}\big{)}\,.$$

In this case the backward propagation can be summarized as
- βκ | κ +1 = 1
- k = K . . . 1
− βk βk = p(yk = ˆ γk|zk)βk|k +1
= β k − 1 | k  =  ∫ z k p ( z k | z k − 1 )β k | k The recursion is given by
- [h*K|K+1: K*|K+1: 9*K|K+1] = [0,0,0]
- k = K . . . 1
= β k | k  = [ h * | k  : K * | k  : 9 * | k ]
h x | k  = CT R-1 yk  + h x | k +1 K k | k  = CT R-1C + K k | k +1