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HMM - Part 2

HMM - Part 2. The EM algorithm Continuous density HMM. The EM Algorithm. EM: Expectation Maximization Why EM? Simple optimization algorithms for likelihood functions rely on the intermediate variables, called latent data For HMM , the state sequence is the latent data

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HMM - Part 2

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  1. HMM - Part 2 • The EM algorithm • Continuous density HMM

  2. The EM Algorithm • EM: Expectation Maximization • Why EM? • Simple optimization algorithms for likelihood functions rely on the intermediate variables, called latent dataFor HMM, the state sequence is the latent data • Direct access to the data necessary to estimate the parameters is impossible or difficultFor HMM, it is almost impossible to estimate (A, B, ) without considering the state sequence • Two Major Steps : • E step: computes an expectation of the likelihood by including the latent variables as if they were observed • M step: computes the maximum likelihood estimates of the parameters by maximizing the expected likelihood found in the E step

  3. Three Steps for EM Step 1. Draw a lower bound • Use the Jensen’s inequality Step 2. Find the best lower bound  auxiliary function • Let the lower bound touch the objective function at the current guess Step 3. Maximize the auxiliary function • Obtain the new guess • Go to Step 2 until converge [Minka 1998]

  4. Given the current guess , the goal is to find a new guess such that Form an Initial Guess of =(A,B,) objective function current guess

  5. Step 1. Draw a Lower Bound objective function lower bound function

  6. auxiliary function Step 2. Find the Best Lower Bound objective function lower bound function

  7. Step 3. Maximize the Auxiliary Function objective function auxiliary function

  8. Update the Model objective function

  9. Step 2. Find the Best Lower Bound objective function auxiliary function

  10. Step 3. Maximize the Auxiliary Function objective function

  11. The lower bound function of Step 1. Draw a Lower Bound (cont’d) Objective function If f is a concave function, and X is a r.v., then E[f(X)]≤ f(E[X]) Apply Jensen’s Inequality

  12. Step 2. Find the Best Lower Bound (cont’d) • Find that makes the lower bound function touch the objective function at the current guess

  13. Take the derivative w.r.t Step 2. Find the Best Lower Bound (cont’d) Set it to zero

  14. Define We can check Q function Step 2. Find the Best Lower Bound (cont’d)

  15. Expectation EM for HMM Training • Basic idea • Assume we have and the probability that each Q occurred in the generation of O i.e., we have in fact observed a complete data pair (O,Q) with frequency proportional to the probability P(O,Q|) • We then find a new that maximizes • It can be guaranteed that • EM can discover parameters of model  to maximize the log-likelihood of the incomplete data, logP(O|), by iteratively maximizing the expectation of the log-likelihood of the complete data, logP(O,Q|)

  16. Solution to Problem 3 - The EM Algorithm • The auxiliary function where and can be expressed as

  17. example wi yi wj yj wk yk Solution to Problem 3 - The EM Algorithm (cont’d) • The auxiliary function can be rewritten as

  18. Solution to Problem 3 - The EM Algorithm (cont’d) • The auxiliary function is separated into three independent terms, each respectively corresponds to , , and • Maximization procedure on can be done by maximizing the individual terms separately subject to probability constraints • All these terms have the following form

  19. Solution to Problem 3 - The EM Algorithm (cont’d) • Proof: Apply Lagrange Multiplier Constraint

  20. wi yi Solution to Problem 3 - The EM Algorithm (cont’d)

  21. wj yj Solution to Problem 3 - The EM Algorithm (cont’d)

  22. wk yk Solution to Problem 3 - The EM Algorithm (cont’d)

  23. Solution to Problem 3 - The EM Algorithm (cont’d) • The new model parameter set can be expressed as:

  24. Discrete vs. Continuous Density HMMs • Two major types of HMMs according to the observations • Discrete and finite observation: • The observations that all distinct states generate are finite in number, i.e., V={v1, v2, v3, ……, vM}, vkRL • In this case, the observation probability distribution in state j, B={bj(k)}, is defined as bj(k)=P(ot=vk|qt=j), 1kM, 1jNot :observation at time t, qt: state at time t bj(k) consists of only M probability values • Continuous and infinite observation: • The observations that all distinct states generate are infinite and continuous, i.e., V={v| vRL} • In this case, the observation probability distribution in state j, B={bj(v)}, is defined as bj(v)=f(ot=v|qt=j), 1jNot :observation at time t, qt: state at time t bj(v) is a continuous probability density function (pdf) and is often a mixture of Multivariate Gaussian (Normal) Distributions

  25. Gaussian Distribution • A continuous random variable X is said to have a Gaussian distribution with mean μand variance σ2(σ>0) if X has a continuous pdf in the following form:

  26. Multivariate Gaussian Distribution • If X=(X1,X2,X3,…,XL) is an L-dimensional random vector with a multivariate Gaussian distribution with mean vectorand covariance matrix, then the pdf can be expressed as • If X1,X2,X3,…,XLare independent random variables, the covariance matrix is reduced to diagonal, i.e.,

  27. Observation vector Mean vector of the kth mixture of the jth state Covariance matrix of the kth mixture of the jth state Multivariate Mixture Gaussian Distribution • An L-dimensional random vector X=(X1,X2,X3,…,XL) is with a multivariate mixture Gaussian distribution if • In CDHMM,bj(v) is a continuous probability density function (pdf) and is often a mixture of multivariate Gaussian distributions

  28. Solution to Problem 3 – The Segmental K-means Algorithm • Assume that we have a training set of observations and an initial estimate of model parameters • Step 1 : Segment the training data The set of training observation sequences is segmented into states, based on the current model, by Viterbi Algorithm • Step 2 : Re-estimate the model parameters • Step 3: Evaluate the model If the difference between the new and current model scores exceeds a threshold, go back to Step 1; otherwise, return

  29. Solution to Problem 3 – The Segmental K-means Algorithm (cont’d) • 3 states and 4 Gaussian mixtures per state State s3 s3 s3 s3 s3 s3 s3 s3 s3 s2 s2 s2 s2 s2 s2 s2 s2 s2 s1 s1 s1 s1 s1 s1 s1 s1 s1 1 2 N O1 O2 ON {12,12,c12} {11,11,c11} K-means Global mean Cluster 1 mean Cluster 2mean {13,13,c13} {14,14,c14}

  30. Observation-independent assumption Solution to Problem 3 – The Intuitive View (CDHMM) • Define a new variable t(j,k) • probability of being in state j at time t with the k-th mixture component accounting for ot

  31. Solution to Problem 3 – The Intuitive View (CDHMM) (cont’d) • Re-estimation formulae for are

  32. A Simple Example The Forward/Backward Procedure S1 S1 S1 State S2 S2 S2 1 2 3 Time o1 o2 o3

  33. A Simple Example(cont’d) q: 1 1 1 q: 1 1 2 Total 8 paths

  34. A Simple Example(cont’d) back

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