Intro
Admin Mid-Term Evaluation open remember to come by office hours once Recap From Rabiner, "A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition" Defining an HMM HMM is characterized by N, number of states in the model S, {S_1,S_2,S_3} state at time t is q_t M the number of distinct observation symbols per state V = {v_1,V_2,V_3} State transition probability A={a_ij} a_ij = P(q_t+1 = S_j | q_t = S_i) 1<= i, j <= N if all states can reach all states then a_ij > 0 for all i,m The Observation symbol probability distribution in state j B = {b_j(k)} b_j(k) = P(v_k at t | q_t = S_j) 1<= j <=N , 1<=k <=M The initial state distribution pi = {pi_i} pi_i = P(q_1=S_i) 1<= i <=N Example of how to use it generatively Model can be completely determined by M, N, A, B, and pi New Material Motivate with a house and motion sensor example 3 problems from Rabiner The 3 basic HMM problems |
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Problem 1 is the evaluation problem It is also considered the scoring problem Choosing among multiple models Problem 2 is an attempt to uncover the hidden variable or find the correct state sequence "correct" is not accurate -> some optimality criterion there are several possible Problem 3 is the learning problem "training" the HMM based on some observed data Solving Problem #1 |
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A more tractable version is called the Forward-Backward procedure |
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So this only describes the "forward variable" It is sufficient for problem 1 but we will want to use a backward variable for the other problems |
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Problem 2 Lots of possible solutions "optimal" choose the states which are individually most likely |
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Problems with this formulation State sequence may not be possible Other ways of optimizing are possible One well-known optimization is to optimize the single best state sequence Based on dynamic programming "Viterbi Algorithm" |
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Two points This is very similar to the solution to Problem 1 except that we are maximizing rather than summing and it should be clear that it can be done efficiently with a lattice structure Problem 3 Adusting the model parameters (A,B,pi) to maximize the probability of the observation sequence This is a global optimization problem No known analytical problem of maximize Use a local optimazation procedure called Baum-Welch algorithm iterative Expectation-Maximization procedure |
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