jurafsky&martin_3rdEd_17 (1).pdf

Computation of the joint probability of being in

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Computation of the joint probability of being in state i at time t and state j at time t + 1. The figure shows the various probabilities that need to be combined to produce P ( q t = i , q t + 1 = j , O | l ) : the a and b probabilities, the transition probability a i j and the observation probability b j ( o t + 1 ) . After Rabiner (1989) which is c 1989 IEEE. Figure 9.14 shows the various probabilities that go into computing not-quite- x t : the transition probability for the arc in question, the a probability before the arc, the b probability after the arc, and the observation probability for the symbol just after the arc. These four are multiplied together to produce not-quite- x t as follows: not-quite- x t ( i , j ) = a t ( i ) a i j b j ( o t + 1 ) b t + 1 ( j ) (9.34) To compute x t from not-quite- x t , we follow the laws of probability and divide by P ( O | l ) , since P ( X | Y , Z ) = P ( X , Y | Z ) P ( Y | Z ) (9.35) The probability of the observation given the model is simply the forward proba- bility of the whole utterance (or alternatively, the backward probability of the whole utterance), which can thus be computed in a number of ways: P ( O | l ) = a T ( q F ) = b T ( q 0 ) = N X j = 1 a t ( j ) b t ( j ) (9.36) So, the final equation for x t is x t ( i , j ) = a t ( i ) a i j b j ( o t + 1 ) b t + 1 ( j ) a T ( q F ) (9.37) The expected number of transitions from state i to state j is then the sum over all t of x . For our estimate of a i j in Eq. 9.31 , we just need one more thing: the total
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138 C HAPTER 9 H IDDEN M ARKOV M ODELS expected number of transitions from state i . We can get this by summing over all transitions out of state i . Here’s the final formula for ˆ a i j : ˆ a i j = P T - 1 t = 1 x t ( i , j ) P T - 1 t = 1 P N k = 1 x t ( i , k ) (9.38) We also need a formula for recomputing the observation probability. This is the probability of a given symbol v k from the observation vocabulary V , given a state j : ˆ b j ( v k ) . We will do this by trying to compute ˆ b j ( v k ) = expected number of times in state j and observing symbol v k expected number of times in state j (9.39) For this, we will need to know the probability of being in state j at time t , which we will call g t ( j ) : g t ( j ) = P ( q t = j | O , l ) (9.40) Once again, we will compute this by including the observation sequence in the probability: g t ( j ) = P ( q t = j , O | l ) P ( O | l ) (9.41) o t+1 α t (j) o t-1 o t s j β t (j) Figure 9.15 The computation of g t ( j ) , the probability of being in state j at time t . Note that g is really a degenerate case of x and hence this figure is like a version of Fig. 9.14 with state i collapsed with state j . After Rabiner (1989) which is c 1989 IEEE. As Fig. 9.15 shows, the numerator of Eq. 9.41 is just the product of the forward probability and the backward probability: g t ( j ) = a t ( j ) b t ( j ) P ( O | l ) (9.42) We are ready to compute b . For the numerator, we sum g t ( j ) for all time steps t in which the observation o t is the symbol v k that we are interested in. For the denominator, we sum g t ( j ) over all time steps t . The result is the percentage of the
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9.5 HMM T RAINING : T HE F ORWARD -B ACKWARD A LGORITHM 139 times that we were in state j and saw symbol v k (the notation P T t = 1 s . t . O t = v k means “sum over all t for which the observation at time t was v k
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