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Part 4 b Forward-Backward Algorithm & Viterbi Algorithm CSE717, SPRING 2008 CUBS, Univ at Buffalo

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Production Probability Computation of : Summing over all possible state sequence s Time complexity

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Forward Variable Define can be computed by Forward Algorithm

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Initial Values of Forward Variables

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Recursive Computation of Forward Variables O1O1 s1s1 stst OtOt s t+1 O t+1

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Forward Algorithm Let 1. Initialization 2. Recursion for all times t, t=1,…,T–1 3. Termination Complexity

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Backward Variable Define Posterior probability of state i at time t

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Backward Algorithm Let 1. Initialization 2. Recursion for all times t, t=T–1,…,1 3. Termination

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Decoding Problem (Find a state sequence s* that produces the observations with maximal posterior probability) 12 N 312 N 312 N 312 N 3

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Partially Optimal Path Probability Define s* can be obtained from partially optimal paths given by the computation of

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Viterbi Algorithm Define s* can be obtained from partially optimal paths given by the computation of

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Viterbi Algorithm Let 1. Initialization 2. Recursion for all times t, t=1,…,T–1 3. Termination 4. Back-Tracking of Optimal Path for all times t, t= T–1,…,1

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Configuration of Forward-Backward Algorithm and Viterbi Algorithm Configuration Forward algorithm computes Viterbi Algorithm computes Parallel Implementation

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Discussion Problem in Part 3 Question: given N samples x 1, …, x N of a mixture density of two normal distributions can you get maximum likelihood estimators of parameters ?

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Single Gaussian The log likelihood is a polynomial and can be maximized by means of taking differentials. Mixture of 2-Gaussian Exponential cannot be removed by taking log () It’s hard to find expressions of ML estimators

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Discussion Problem in Part 3 Question: given N samples x 1, …, x N of a mixture density of two normal distributions can you get maximum likelihood estimators of parameters ? Conclusion: No analytic solution, but numerical solutions may exist

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