Since we do not know the endpoint of this word, we can obtain the probability p(z t = 1, z t+1 = 0) by marginalizing over the endpoint j (• • • means values all 0): Case 0-1: Similarly, as shown in Figure 8(c) this case means that a word of length at least 2 begins before time t and ends at time t. Therefore, we can marginalize over the start point of a possible word to obtain the marginal probability: Case 0-0: In principle, this means that a word begins before time t and ends later than (and including) time t + 1.
therefore, we can marginalize over both the start and end time of a possible word spanning [t, t+1] to obtain: in fact we can avoid this nested computation because the probability of p(z t , z t+1 ) over the possible values of z t and z t+1 must sum to 1.