Pattern Recognition and Machine Learning

13.3. Linear Dynamical Systems 639

where we have defined
Pn− 1 =AVn− 1 AT+Γ. (13.88)

We can now combine this result with the first factor on the right-hand side of (13.86)
by making use of (2.115) and (2.116) to give

μn = Aμn− 1 +Kn(xn−CAμn− 1 ) (13.89)

Vn =(I−KnC)Pn− 1 (13.90)

cn = N(xn|CAμn− 1 ,CPn− 1 CT+Σ). (13.91)

Here we have made use of the matrix inverse identities (C.5) and (C.7) and also
defined theKalman gain matrix

Kn=Pn− 1 CT

(

CPn− 1 CT+Σ

)− 1

. (13.92)

Thus, given the values ofμn− 1 andVn− 1 , together with the new observationxn,
we can evaluate the Gaussian marginal forznhaving meanμnand covarianceVn,
as well as the normalization coefficientcn.
The initial conditions for these recursion equations are obtained from

c 1 ̂α(z 1 )=p(z 1 )p(x 1 |z 1 ). (13.93)

Becausep(z 1 )is given by (13.77), andp(x 1 |z 1 )is given by (13.76), we can again
make use of (2.115) to calculatec 1 and (2.116) to calculateμ 1 andV 1 giving

μ 1 = μ 0 +K 1 (x 1 −Cμ 0 ) (13.94)

V 1 =(I−K 1 C)V 0 (13.95)

c 1 = N(x 1 |Cμ 0 ,CV 0 CT+Σ) (13.96)

where
K 1 =V 0 CT

(

CV 0 CT+Σ

)− 1

. (13.97)

Similarly, the likelihood function for the linear dynamical system is given by (13.63)
in which the factorscnare found using the Kalman filtering equations.
We can interpret the steps involved in going from the posterior marginal over
zn− 1 to the posterior marginal overznas follows. In (13.89), we can view the
quantityAμn− 1 as the prediction of the mean overznobtained by simply taking the
mean overzn− 1 and projecting it forward one step using the transition probability
matrixA. This predicted mean would give a predicted observation forxngiven by
CAzn− 1 obtained by applying the emission probability matrixCto the predicted
hidden state mean. We can view the update equation (13.89) for the mean of the
hidden variable distribution as taking the predicted meanAμn− 1 and then adding
a correction that is proportional to the errorxn−CAzn− 1 between the predicted
observation and the actual observation. The coefficient of this correction is given by
the Kalman gain matrix. Thus we can view the Kalman filter as a process of making
successive predictions and then correcting these predictions in the light of the new
observations. This is illustrated graphically in Figure 13.21.