Pattern Recognition and Machine Learning

13.2. Hidden Markov Models 611

Figure 13.6 Transition diagram showing a model whose la-

tent variables have three possible states corre-

sponding to the three boxes. The black lines

denote the elements of the transition matrix

Ajk.

A 12

A 23

A 31

A 21

A 32

A 13

A 11

A 22

A 33

k=1

k=2

k=3

hasK(K−1)independent parameters. We can then write the conditional distribution

explicitly in the form

p(zn|zn− 1 ,A)=

∏K

k=1

∏K

j=1

Ajkzn−^1 ,jznk. (13.7)

The initial latent nodez 1 is special in that it does not have a parent node, and so

it has a marginal distributionp(z 1 )represented by a vector of probabilitiesπwith

elementsπk≡p(z 1 k=1), so that

p(z 1 |π)=

∏K

k=1

πzk^1 k (13.8)

where

∑
kπk=1.
The transition matrix is sometimes illustrated diagrammatically by drawing the
states as nodes in a state transition diagram as shown in Figure 13.6 for the case of
K=3. Note that this does not represent a probabilistic graphical model, because
the nodes are not separate variables but rather states of a single variable, and so we
have shown the states as boxes rather than circles.
It is sometimes useful to take a state transition diagram, of the kind shown in
Figure 13.6, and unfold it over time. This gives an alternative representation of the
Section 8.4.5 transitions between latent states, known as alatticeortrellisdiagram, and which is
shown for the case of the hidden Markov model in Figure 13.7.
The specification of the probabilistic model is completed by defining the con-
ditional distributions of the observed variablesp(xn|zn,φ), whereφis a set of pa-
rameters governing the distribution. These are known asemission probabilities, and
might for example be given by Gaussians of the form (9.11) if the elements ofxare
continuous variables, or by conditional probability tables ifxis discrete. Because
xnis observed, the distributionp(xn|zn,φ)consists, for a given value ofφ,ofa
vector ofKnumbers corresponding to theKpossible states of the binary vectorzn.