Pattern Recognition and Machine Learning

13.3. Linear Dynamical Systems 641

Figure 13.22 An illustration of a linear dy-

namical system being used to

track a moving object. The blue

points indicate the true positions

of the object in a two-dimensional

space at successive time steps,

the green points denote noisy

measurements of the positions,

and the red crosses indicate the

means of the inferred posterior

distributions of the positions ob-

tained by running the Kalman fil-

tering equations. The covari-

ances of the inferred positions

are indicated by the red ellipses,

which correspond to contours

having one standard deviation.

̂β(zn), which, for continuous latent variables, can be written in the form

cn+1̂β(zn)=

∫

β̂(zn+1)p(xn+1|zn+1)p(zn+1|zn)dzn+1. (13.99)

We now multiply both sides of (13.99) bŷα(zn)and substitute forp(xn+1|zn+1)
andp(zn+1|zn)using (13.75) and (13.76). Then we make use of (13.89), (13.90)
Exercise 13.29 and (13.91), together with (13.98), and after some manipulation we obtain

μ̂n = μn+Jn

(

μ̂n+1−AμN

)

(13.100)

V̂n = Vn+Jn

(

V̂n+1−Pn

)

JTn (13.101)

where we have defined

Jn=VnAT(Pn)−^1 (13.102)

and we have made use ofAVn=PnJTn. Note that these recursions require that the

forward pass be completed first so that the quantitiesμnandVnwill be available

for the backward pass.

For the EM algorithm, we also require the pairwise posterior marginals, which

can be obtained from (13.65) in the form

ξ(zn− 1 ,zn)=(cn)−^1 ̂α(zn− 1 )p(xn|zn)p(zn|z− 1 )β̂(zn)

=

N(zn− 1 |μn− 1 ,Vn− 1 )N(zn|Azn− 1 ,Γ)N(xn|Czn,Σ)N(zn|μ̂n,V̂n)

cn̂α(zn)

.

(13.103)

Substituting for̂α(zn)using (13.84) and rearranging, we see thatξ(zn− 1 ,zn)is a
Gaussian with mean given with componentsγ(zn− 1 )andγ(zn), and a covariance
Exercise 13.31 betweenznandzn− 1 given by

cov[zn,zn− 1 ]=Jn− 1 V̂n. (13.104)