640 13. SEQUENTIAL DATA
zn− 1 zn znFigure 13.21 The linear dynamical system can be viewed as a sequence of steps in which increasing un-
certainty in the state variable due to diffusion is compensated by the arrival of new data. In the left-hand plot,
the blue curve shows the distributionp(zn− 1 |x 1 ,...,xn− 1 ), which incorporates all the data up to stepn− 1.
The diffusion arising from the nonzero variance of the transition probabilityp(zn|zn− 1 )gives the distribution
p(zn|x 1 ,...,xn− 1 ), shown in red in the centre plot. Note that this is broader and shifted relative to the blue curve
(which is shown dashed in the centre plot for comparison). The next data observationxncontributes through the
emission densityp(xn|zn), which is shown as a function ofznin green on the right-hand plot. Note that this is not
a density with respect toznand so is not normalized to one. Inclusion of this new data point leads to a revised
distributionp(zn|x 1 ,...,xn)for the state density shown in blue. We see that observation of the data has shifted
and narrowed the distribution compared top(zn|x 1 ,...,xn− 1 )(which is shown in dashed in the right-hand plot
for comparison).
If we consider a situation in which the measurement noise is small compared
to the rate at which the latent variable is evolving, then we find that the posterior
Exercise 13.27 distribution forzndepends only on the current measurementxn, in accordance with
the intuition from our simple example at the start of the section. Similarly, if the
latent variable is evolving slowly relative to the observation noise level, we find that
the posterior mean forznis obtained by averaging all of the measurements obtained
Exercise 13.28 up to that time.
One of the most important applications of the Kalman filter is to tracking, and
this is illustrated using a simple example of an object moving in two dimensions in
Figure 13.22.
So far, we have solved the inference problem of finding the posterior marginal
for a nodezngiven observations fromx 1 up toxn. Next we turn to the problem of
finding the marginal for a nodezngiven all observationsx 1 toxN. For temporal
data, this corresponds to the inclusion of future as well as past observations. Al-
though this cannot be used for real-time prediction, it plays a key role in learning the
parameters of the model. By analogy with the hidden Markov model, this problem
can be solved by propagating messages from nodexNback to nodex 1 and com-
bining this information with that obtained during the forward message passing stage
used to compute theα̂(zn).
In the LDS literature, it is usual to formulate this backward recursion in terms
ofγ(zn)=̂α(zn)̂β(zn)rather than in terms of̂β(zn). Becauseγ(zn)must also be
Gaussian, we write it in the form
γ(zn)=α̂(zn)̂β(zn)=N(zn|μ̂n,V̂n). (13.98)To derive the required recursion, we start from the backward recursion (13.62) for