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Exploring Hidden Markov Models: Calculation, Inference, and Learning Techniques

This document provides a comprehensive overview of Hidden Markov Models (HMMs), focusing on their mathematical foundations and practical applications. It includes detailed explanations of state transition and emission probabilities, inference methods, and model fit calculations. Key questions addressed include how to compute the likelihood of observations, infer hidden states, and estimate model probabilities. The document references the approaches detailed in Manning and Schütze (2000), emphasizing efficient algorithms for practical computation in natural language processing contexts.

joseph-brewer
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Exploring Hidden Markov Models: Calculation, Inference, and Learning Techniques

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  1. Hidden Markov Models Richard Golden (following approach of Chapter 9 of Manning and Schutze, 2000) REVISION DATE:April 15 (Tuesday), 2003

  2. a11=0.7 1 S0 S1 a12=0.3 a21=0.5 2 S2 a22=0.5 VMM (Visible Markov Model)

  3. HMM Notation • State Sequence Variables: X1, …, XT+1 • Output Sequence Variables: O1, …, OT • Set of Hidden States (S1, …, SN) • Output Alphabet (K1, …, KM) • Initial State Probabilities (1, .., N)i=p(X1=Si), i=1,…,N • State Transition Probabilities (aij) i,j{1,…,N}aij =p(Xt+1|Xt), t=1,…,T • Emission Probabilities (bij) i{1,…,N},j {1,…,M}bij=p(Xt+1=Si|Xt=Sj), t=1,…,T

  4. S0 S1 K3 K2 K1 HMM State-Emission Representation • Note that sometimes a Hidden Markov Model is represented by having the emission arrows come off the arcs • In this situation you would have a lot more emission arrows because there’s a lot more arcs… • But the transition and emission probabilities are the same…it just takes longer to draw on your powerpoint presentation (self-conscious presentation) a11=0.7 b11=0.6 b12=0.1 b13=0.3 1=1 a12=0.3 b22=0.7 b23=0.2 2=0 a21=0.5 S2 b21=0.1 a22=0.5

  5. Arc-Emission Representation • Note that sometimes a Hidden Markov Model is represented by having the emission arrows come off the arcs • In this situation you would have a lot more emission arrows because there’s a lot more arcs… • But the transition and emission probabilities are the same…it just takes longer to draw on your powerpoint presentation (self-conscious presentation)

  6. Fundamental Questions for HMMs • MODEL FIT • How can we compute likelihood of observations and hidden statesgiven known emission and transition probabilities? Compute:p(“Dog”/NOUN,”is”/VERB,”Good”/ADJ | {aij},{bkm}) • How can we compute likelihood of observationsgiven known emission and transition probabilities? p(“Dog”,”is”,”Good” | {aij},{bkm})

  7. Fundamental Questions for HMMs • INFERENCE • How can we infer the sequence of hidden statesgiven the observations and the known emission and transition probabilities? • Maximize: • p(“Dog”/?,”is”/?, “Good”/? | {aij},{bkm})with respect to the unknown labels

  8. Fundamental Questions for HMMs • LEARNING • How can we estimate the emission and transition probabilitiesgiven observations and assuming that hidden states are observable during learning process? • How can we estimate emission and transition probabilitiesgivenobservations only?

  9. Direct Calculation of Model Fit(note use of “Markov” Assumptions) Part 1 Follows directly from the definition of a conditional probability: p(o,x)=p(o|x)p(x) EXAMPLE:P(“Dog”/NOUN,”is”/VERB,”Good”/ADJ | {aij},{bij}) = p(“Dog”,”is”,”Good”|NOUN,VERB,ADJ {aij},{bij}) X p(NOUN,VERB,ADJ | aij},{bij})

  10. Direct Calculation of Likelihood of Labeled Observations(note use of “Markov” Assumptions)Part 2 EXAMPLE:Compute p(“DOG”/NOUN,”is”/VERB,”good”/ADJ|{aij},{bkm})

  11. a11=0.7 b11=0.6 b12=0.1 b13=0.3 1=1 a12=0.3 b22=0.7 S0 S1 K1 K2 K3 b23=0.2 a21=0.5 2=0 S2 b21=0.1 a22=0.5 Graphical Algorithm Representation of Direct Calculation of Likelihood of Observations and Hidden States (not hard!) Note that “good” is The name Of the dogj So it is a Noun! The likelihood of a particular “labeled” sequence of observations (e.g., p(“Dog”/NOUN,”is”/VERB,”Good”/NOUN|{aij},{bkm})) may be computed Using the “direct calculation” method using following simple graphical algorithm. Specifically, p(K3/S1, K2/S2, K1/S1 |{aij},{bkm}))= 1b13a12b22a21b11

  12. Extension to case where the likelihood of the observations given parameters is needed(e.g., p( “Dog”, ”is”, ”good” | {aij},{bij}) KILLER EQUATION!!!!!

  13. Efficiency of Calculations is Important (e.g., Model-Fit) • Assume 1 multiplication per microsecond • Assume N=1000 word vocabulary and T=7 word sentence. • (2T+1)NT+1 multiplications by “direct calculation” yields (2(7)+1)(1000)(7+1) is about475,000 million years of computer time!!! • 2N2T multiplications using “forward method”is about 14 seconds of computer time!!!

  14. Forward, Backward, and Viterbi Calculations • Forward calculation methods are thus very useful. • Forward, Backward, and Viterbi Calculations will now be discussed.

  15. S0 S1 Forward Calculations – Overview TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b11=0.6 b13=0.3 a11=0.7 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 b23=0.2 a22=0.5 b21=0.1 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  16. S0 S1 Forward Calculations – Time 2 (1 word example) TIME 2 NOTE: that 1 (2)+ 2 (2) is the likelihood of the observation/word “K3”in this “1 word example” K1 K2 K3 b13=0.3 a11=0.7 S1 1 a12=0.3 a21=0.5 2 S2 S2 a22=0.5 b23=0.2 K1 K2 K3

  17. S0 S1 Forward Calculations – Time 3 (2 word example) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 b12=0.1 1(3) b11=0.6 a11=0.7 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 a22=0.5 b21=0.1 b22=0.1 K1 K2 K3 K1 K2 K3

  18. S0 S1 Forward Calculations – Time 4 (3 word example) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 b11=0.6 a11=0.7 S1 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 S2 a22=0.5 b21=0.1 b23=0.2 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  19. Forward Calculation of Likelihood Function (“emit and jump”)

  20. S0 S1 Backward Calculations – Overview TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b11=0.6 b13=0.3 a11=0.7 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 b23=0.2 a22=0.5 b21=0.1 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  21. Backward Calculations – Time 4 TIME 4 K1 K2 K3 b11`=0.6 S1 S2 b21=0.1 K1 K2 K3

  22. Backward Calculations – Time 3 TIME 3 K1 K2 K3 b11`=0.6 S1 S2 b21=0.1 K1 K2 K3

  23. Backward Calculations – Time 2 TIME 3 TIME 4 TIME 2 NOTE: that 1 (2)+ 2 (2) is the likelihood the observation/word sequence “K2,K1”in this “2 word example” K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 a11=0.7 S1 S1 a12=0.3 a21=0.5 a22=0.5 S2 S2 b23=0.2 b22=0.7 K1 K2 K1 K2 K3 K3

  24. S0 S1 Backward Calculations – Time 1 TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b11=0.6 b13=0.3 a11=0.7 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 b23=0.2 a22=0.5 b21=0.1 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  25. Backward Calculation of Likelihood Function (“EMIT AND JUMP”)

  26. You get same answer going forward or backward!! Backward Forward

  27. The Forward-Backward Method • Note the forward method computes: • Note the backward method computes (t>1): • We can do the forward-backward methodwhich computes p(K1,…,KT) using formula (using any choice of t=1,…,T+1!):

  28. Example Forward-Backward Calculation! Backward Forward

  29. Solution to Problem 1 • The “hard part” of the 1st Problem was to find the likelihood of the observations for an HMM • We can now do this using either theforward, backward, or forward-backwardmethod.

  30. Solution to Problem 2: Viterbi Algorithm(Computing “Most Probable” Labeling) • Consider direct calculation of labeledobservations • Previously we summedthese likelihoods together across all possible labelings to solve the first problemwhich was to compute the likelihood of the observationsgiven the parameters (Hard part of HMM Question 1!). • We solved this problem using forward or backward method. • Now we want to compute all possible labelings and theirrespective likelihoods and pick the labeling which isthe largest! EXAMPLE:Compute p(“DOG”/NOUN,”is”/VERB,”good”/ADJ|{aij},{bkm})

  31. Efficiency of Calculations is Important (e.g., Most Likely Labeling Problem) • Just as in the forward-backward calculations wecan solve problem of computing likelihood of every possible one of the NT labelings efficiently • Instead of millions of years of computing time we can solve the problem in several seconds!!

  32. S0 S1 Viterbi Algorithm – Overview (same setup as forward algorithm) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b11=0.6 b13=0.3 a11=0.7 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 b23=0.2 a22=0.5 b21=0.1 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  33. S0 S1 Forward Calculations – Time 2 (1 word example) TIME 2 K1 K2 K3 b13=0.3 a11=0.7 S1 1=1 a12=0.3 a21=0.5 2=0 S2 S2 a22=0.5 b23=0.2 K1 K2 K3

  34. S0 S1 Backtracking – Time 2 (1 word example) TIME 2 K1 K2 K3 b13=0.3 a11=0.7 S1 1=1 a12=0.3 a21=0.5 2=0 S2 S2 a22=0.5 b23=0.2 K1 K2 K3

  35. S0 S1 Forward Calculations – (2 word example) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 S1 S1 a11=0.7 1 a21=0.5 a12=0.3 2 a22=0.5 S2 S2 S2 b23=0.2 b22=0.1 K1 K2 K3 K1 K2 K3

  36. S0 S1 BACKTRACKING – (2 word example) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 S1 S1 a11=0.7 1 a21=0.5 a12=0.3 2 a22=0.5 S2 S2 S2 b23=0.2 b22=0.1 K1 K2 K3 K1 K2 K3

  37. Formal Analysis of 2 word case

  38. S0 S1 Forward Calculations – Time 4 (3 word example) TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 b11=0.6 a11=0.7 S1 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 S2 a22=0.5 b21=0.1 b23=0.2 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  39. S0 S1 Backtracking to Obtain Labeling for 3 word case TIME 3 TIME 4 TIME 2 K1 K2 K3 K1 K2 K3 K1 K2 K3 b12=0.1 b13=0.3 b11=0.6 a11=0.7 S1 S1 S1 1 a12=0.3 a21=0.5 2 S2 S2 S2 S2 a22=0.5 b21=0.1 b23=0.2 b22=0.1 K1 K2 K3 K1 K2 K1 K2 K3 K3

  40. Formal Analysis of 3 word case

  41. Third Fundamental Question:Parameter Estimation • Make Initial Guess for {aij} and {bkm} • Compute probability one hidden state follows another given: {aij} and {bkm} and sequence of observations.(computed using forward-backward algorithm) • Compute probability of observed state given a hidden state given: {aij} and {bkm} and sequence of observations.(computed using forward-backward algorithm) • Use these computed probabilities tomake an improved guess for {aij} and {bkm} • Repeat this process until convergence • Can be shown that this algorithm does infact converge to correct choice for {aij} and {bkm}assuming that the initial guess was close enough..

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