MACHINE LEARNING TECHNIQUES - LASA

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Figure 7-2: Schematic illustrating the concept of emission probabilities associated to each state in a HMM.
3
o
t is the observable at time t. It can take any value v∈° .
7.2.2 Estimating a HMM
Designing an HMM consists in determining the hidden process that explains at best the
observations. The unkown variables in a HMM are the number of states, the transitions and initial
probabilities and the emission probabilities. Since the matrix A is quite large, sometimes, often
people chose a sparse matrix, i.e. they set to zero most of the probabilities hence allowing only
some transitions across some states. The most popular model is the so-called left-right model
that allows transitions solely from state 1, to state 2, and so forth until reaching the final state.
Estimating the parameters of the HMM is done through a variant of Expectation-Maximization
called the Baum-Welch procedure. For a fixed topology (i.e. number of states), one estimates the
set of parameters λ = ( AB , , π)
by maximizing the likelihood of the observations given the
model:
where q { q q }
PO ( | λ) = ∑ PO ( | q, λ) Pq ( | λ)
(7.2)
q
= ,...., is one particular set of expected state transitions during the T observation
1 T
steps. In the example of Figure 7-1, the set q is{ q = s, q = s , q = s, q = s , q = s }.
1 1 2 2 3 1 4 3 5 3
Computing all possible combinations of states in Equation (7.2) is prohibitive. To simplify
computation, one uses dynamic programming through the so-called forward-backward
computation. The principle is illustrated in Figure 7-3 left. It consists of propagating forward in
time the estimate of the probability of being in a particular state given the set of observations. At
each time step, the estimate of being in state i is given by:
α () i = P( o... o, q = s | λ)
(7.3)
t 1 t t i
© A.G.Billard 2004 – Last Update March 2011