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# hw5 - CSE 150 Assignment 5 Out Tue Feb 15 Due Tue Feb 22...

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CSE 150. Assignment 5 Out: Tue Feb 15 Due: Tue Feb 22 Recommended reading: Russell & Norvig, Chapter 15. L. R. Rabiner (1989). A tutorial on hidden Markov models and selected applications in speech recog- nition. Proceedings of the IEEE 77(2):257–286. 5.1 Inference in HMMs Consider a discrete HMM with hidden states S t , observations O t , transition matrix a ij = P ( S t +1 = j | S t = i ) and emission matrix b ik = P ( O t = k | S t = i ) . In class, we defined the forward-backward probabilities: α it = P ( o 1 , o 2 , . . . , o t , S t = i ) , β it = P ( o t +1 , o t +2 , . . . , o T | S t = i ) , for a particular observation sequence { o 1 , o 2 , . . . , o T } of length T . In terms of these probabilities, which you may assume to be given, as well as the transition and emission matrices of the HMM, show how to (efficiently) compute the posterior probability: P ( S t - 1 = i, S t +1 = k | S t = j, o 1 , o 2 , . . . , o T ) . In the above equation, you may assume that t > 1 and t < T ; in particular, you are not asked to consider the boundary cases.

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