Pattern Recognition and Machine Learning

Exercises 647

p(zn|Xn)

p(zn+1|Xn)

p(xn+1|zn+1)

p(zn+1|Xn+1) z

Figure 13.23 Schematic illustration of the operation of the particle filter for a one-dimensional latent
space. At time stepn, the posteriorp(zn|xn)is represented as a mixture distribution,
shown schematically as circles whose sizes are proportional to the weightsw(nl).Asetof
Lsamples is then drawn from this distribution and the new weightsw(nl+1) evaluated using
p(xn+1|z(nl)+1).

satisfies the conditional independence properties

p(xn|x 1 ,...,xn− 1 )=p(xn|xn− 1 ,xn− 2 ) (13.122)

forn=3,...,N.

13.2 ( ) Consider the joint probability distribution (13.2) corresponding to the directed

graph of Figure 13.3. Using the sum and product rules of probability, verify that

this joint distribution satisfies the conditional independence property (13.3) forn=

2 ,...,N. Similarly, show that the second-order Markov model described by the

joint distribution (13.4) satisfies the conditional independence property

p(xn|x 1 ,...,xn− 1 )=p(xn|xn− 1 ,xn− 2 ) (13.123)

forn=3,...,N.

13.3 ( ) By using d-separation, show that the distributionp(x 1 ,...,xN)of the observed

data for the state space model represented by the directed graph in Figure 13.5 does

not satisfy any conditional independence properties and hence does not exhibit the

Markov property at any finite order.

13.4 ( ) www Consider a hidden Markov model in which the emission densities are

represented by a parametric modelp(x|z,w), such as a linear regression model or

a neural network, in whichwis a vector of adaptive parameters. Describe how the

parameterswcan be learned from data using maximum likelihood.