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The Expectation Maximization (EM) Algorithm

The Expectation Maximization (EM) Algorithm. … continued!. Repeat until convergence!. General Idea. Start by devising a noisy channel Any model that predicts the corpus observations via some hidden structure (tags, parses, …) Initially guess the parameters of the model!

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The Expectation Maximization (EM) Algorithm

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  1. The Expectation Maximization (EM) Algorithm … continued! 600.465 - Intro to NLP - J. Eisner

  2. Repeat until convergence! General Idea • Start by devising a noisy channel • Any model that predicts the corpus observations via some hidden structure (tags, parses, …) • Initially guess the parameters of the model! • Educated guess is best, but random can work • Expectation step: Use current parameters (and observations) to reconstruct hidden structure • Maximization step: Use that hidden structure (and observations) to reestimate parameters 600.465 - Intro to NLP - J. Eisner

  3. initialguess E step Guess of unknown parameters (probabilities) M step General Idea Guess of unknown hidden structure (tags, parses, weather) Observed structure(words, ice cream) 600.465 - Intro to NLP - J. Eisner

  4. initialguess M step For Hidden Markov Models E step Guess of unknown parameters (probabilities) Guess of unknown hidden structure (tags, parses, weather) Observed structure(words, ice cream) 600.465 - Intro to NLP - J. Eisner

  5. initialguess M step For Hidden Markov Models E step Guess of unknown parameters (probabilities) Guess of unknown hidden structure (tags, parses, weather) Observed structure(words, ice cream) 600.465 - Intro to NLP - J. Eisner

  6. initialguess M step For Hidden Markov Models E step Guess of unknown parameters (probabilities) Guess of unknown hidden structure (tags, parses, weather) Observed structure(words, ice cream) 600.465 - Intro to NLP - J. Eisner

  7. correct test trees accuracy expensive and/or wrong sublanguage cheap, plentifuland appropriate Grammar LEARNER training trees Grammar Reestimation E step P A R S E R s c o r e r test sentences M step 600.465 - Intro to NLP - J. Eisner

  8. EM by Dynamic Programming: Two Versions • The Viterbi approximation • Expectation: pick the best parse of each sentence • Maximization: retrain on this best-parsed corpus • Advantage: Speed! • Real EM • Expectation: find all parses of each sentence • Maximization: retrain on all parses in proportion to their probability (as if we observed fractional count) • Advantage: p(training corpus) guaranteed to increase • Exponentially many parses, so don’t extract them from chart – need some kind of clever counting why slower? 600.465 - Intro to NLP - J. Eisner

  9. compose these? Examples of EM • Finite-State case: Hidden Markov Models • “forward-backward” or “Baum-Welch” algorithm • Applications: • explain ice cream in terms of underlying weather sequence • explain words in terms of underlying tag sequence • explain phoneme sequence in terms of underlying word • explain sound sequence in terms of underlying phoneme • Context-Free case: Probabilistic CFGs • “inside-outside” algorithm: unsupervised grammar learning! • Explain raw text in terms of underlying cx-free parse • In practice, local maximum problem gets in the way • But canimprove a good starting grammar via raw text • Clustering case: explain points via clusters 600.465 - Intro to NLP - J. Eisner

  10. Our old friend PCFG S NP time VP | S) = p(S  NP VP | S) * p(NP  time | NP) p( PP V flies * p(VP  V PP | VP) P like NP * p(V  flies | V) * … Det an N arrow 600.465 - Intro to NLP - J. Eisner

  11. S S AdvP # copies of this sentencein the corpus NP VP Today stocks V PRT were up Viterbi reestimation for parsing • Start with a “pretty good” grammar • E.g., it was trained on supervised data (a treebank) that is small, imperfectly annotated, or has sentences in a different style from what you want to parse. • Parse a corpus of unparsed sentences: • Reestimate: • Collect counts: …; c(S  NP VP) += 12; c(S) += 2*12; … • Divide: p(S  NP VP | S) = c(S  NP VP) / c(S) • May be wise to smooth … 12Today stocks were up … 12 600.465 - Intro to NLP - J. Eisner

  12. Similar, but now we consider all parses of each sentence Parse our corpus of unparsed sentences: Collect counts fractionally: …; c(S  NP VP) += 10.8; c(S) += 2*10.8; … …; c(S  NP VP) += 1.2; c(S) += 1*1.2; … True EM for parsing NP S S AdvP # copies of this sentencein the corpus NP VP Today S stocks V PRT were up VP V PRT NP NP up stocks Today were … 12Today stocks were up … 10.8 1.2 600.465 - Intro to NLP - J. Eisner

  13. Where are the constituents? p=0.5 coal energy witness expert

  14. Where are the constituents? p=0.1 coal energy witness expert

  15. Where are the constituents? p=0.1 coal energy witness expert

  16. Where are the constituents? p=0.1 coal energy witness expert

  17. Where are the constituents? p=0.2 coal energy witness expert

  18. Where are the constituents? coal energy witness expert +0.1 +0.1 +0.2 = 1 0.5 +0.1

  19. Where are NPs, VPs, … ? NP locations VP locations S VP PP NP NP V P Det N Time Time flies flies like like an an arrow arrow

  20. Where are NPs, VPs, … ? NP locations VP locations Time Time flies flies like like an an arrow arrow (S (NP Time) (VP flies (PP like (NP an arrow)))) p=0.5

  21. Where are NPs, VPs, … ? NP locations VP locations Time Time flies flies like like an an arrow arrow (S (NP Time flies) (VP like (NP an arrow))) p=0.3

  22. Where are NPs, VPs, … ? NP locations VP locations Time Time flies flies like like an an arrow arrow (S (VP Time (NP (NP flies) (PP like (NP an arrow))))) p=0.1

  23. Where are NPs, VPs, … ? NP locations VP locations Time Time flies flies like like an an arrow arrow (S (VP (VP Time (NP flies)) (PP like (NP an arrow)))) p=0.1

  24. Where are NPs, VPs, … ? NP locations VP locations Time Time flies flies like like an an arrow arrow +0.3 +0.1 = 1 0.5 +0.1

  25. How many NPs, VPs, … in all? NP locations VP locations Time Time flies flies like like an an arrow arrow +0.3 +0.1 = 1 0.5 +0.1

  26. How many NPs, VPs, … in all? NP locations VP locations Time Time flies flies like like an an arrow arrow 2.1 NPs(expected) 1.1 VPs(expected)

  27. Where did the rules apply? S  NP VP locations NP  Det N locations Time Time flies flies like like an an arrow arrow

  28. Where did the rules apply? S  NP VP locations NP  Det N locations Time Time flies flies like like an an arrow arrow (S (NP Time) (VP flies (PP like (NP an arrow)))) p=0.5

  29. Where is S  NP VP substructure? S  NP VP locations NP  Det N locations Time Time flies flies like like an an arrow arrow (S (NP Time flies) (VP like (NP an arrow))) p=0.3

  30. Where is S  NP VP substructure? S  NP VP locations NP  Det N locations Time Time flies flies like like an an arrow arrow (S (VP Time (NP (NP flies) (PP like (NP an arrow))))) p=0.1

  31. Where is S  NP VP substructure? S  NP VP locations NP  Det N locations Time Time flies flies like like an an arrow arrow (S (VP (VP Time (NP flies)) (PP like (NP an arrow)))) p=0.1

  32. Why do we want this info? • Grammar reestimation by EM method • E step collects those expected counts • M step sets • Minimum Bayes Risk decoding • Find a tree that maximizes expected reward,e.g., expected total # of correct constituents • CKY-like dynamic programming algorithm • The input specifies the probability of correctness for each possible constituent (e.g., VP from 1 to 5)

  33. Why do we want this info? • Soft features of a sentence for other tasks • NER system asks: “Is there an NP from 0 to 2?” • True answer is 1 (true) or 0 (false) • But we return 0.3, averaging over all parses • That’s a perfectly good feature value – can be fed as to a CRF or a neural network as an input feature • Writing tutor system asks: “How many times did the student use S  NP[singular] VP[plural]?” • True answer is in {0, 1, 2, …} • But we return 1.8, averaging over all parses

  34. Similar, but now we consider all parses of each sentence Parse our corpus of unparsed sentences: Collect counts fractionally: …; c(S  NP VP) += 10.8; c(S) += 2*10.8; … …; c(S  NP VP) += 1.2; c(S) += 1*1.2; … But there may be exponentiallymany parses of a length-n sentence! How can we stay fast? Similar to taggings… True EM for parsing NP S S AdvP # copies of this sentencein the corpus NP VP Today S stocks V PRT were up VP V PRT NP NP up stocks Today were … 12Today stocks were up … 10.8 1.2 600.465 - Intro to NLP - J. Eisner

  35. The dynamic programming computation of a. (b is similar but works back from Stop.) Day 1: 2 cones Day 2: 3 cones Day 3: 3 cones a=0.1*0.08+0.1*0.01 =0.009 a=0.009*0.08+0.063*0.01 =0.00135 a=0.1 p(C|C)*p(3|C) p(C|C)*p(3|C) C C C p(C|Start)*p(2|C) 0.8*0.1=0.08 0.8*0.1=0.08 0.5*0.2=0.1 Start p(H|Start)*p(2|H) p(H|H)*p(3|H) 0.5*0.2=0.1 p(H|H)*p(3|H) H H H 0.8*0.7=0.56 0.8*0.7=0.56 p(C|H)*p(3|C) p(C|H)*p(3|C) a=0.1 0.1*0.1=0.01 0.1*0.1=0.01 p(H|C)*p(3|H) p(H|C)*p(3|H) a=0.1*0.07+0.1*0.56 =0.063 a=0.009*0.07+0.063*0.56 =0.03591 0.1*0.7=0.07 0.1*0.7=0.07 Analogies to a, b in PCFG? Call these aH(2) and bH(2) aH(3) and bH(3) 600.465 - Intro to NLP - J. Eisner

  36. “Inside Probabilities” S NP time VP | S) = p(S  NP VP | S) * p(NP  time | NP) p( PP V flies * p(VP  V PP | VP) P like NP * p(V  flies | V) * … Det an N arrow • Sum over all VP parses of “flies like an arrow”: VP(1,5) = p(flies like an arrow | VP) • Sum over all S parses of “time flies like an arrow”: S(0,5) = p(time flies like an arrow | S) 600.465 - Intro to NLP - J. Eisner

  37. Compute  Bottom-Up by CKY S NP time VP | S) = p(S  NP VP | S) * p(NP  time | NP) p( PP V flies * p(VP  V PP | VP) P like NP * p(V  flies | V) * … Det an N arrow VP(1,5) = p(flies like an arrow | VP) = … S(0,5) = p(time flies like an arrow | S) = NP(0,1) * VP(1,5) * p(S  NP VP|S) + … 600.465 - Intro to NLP - J. Eisner

  38. Compute  Bottom-Up by CKY 1 S  NP VP 6 S  Vst NP 2 S  S PP 1 VP  V NP 2 VP  VP PP 1 NP  Det N 2 NP  NP PP 3 NP  NP NP 0 PP  P NP

  39. Compute  Bottom-Up by CKY 2-1 S  NP VP 2-6 S  Vst NP 2-2 S  S PP 2-1 VP  V NP 2-2 VP  VP PP 2-1 NP  Det N 2-2 NP  NP PP 2-3 NP  NP NP 2-0 PP  P NP S 2-22

  40. Compute  Bottom-Up by CKY 2-1 S  NP VP 2-6 S  Vst NP 2-2 S  S PP 2-1 VP  V NP 2-2 VP  VP PP 2-1 NP  Det N 2-2 NP  NP PP 2-3 NP  NP NP 2-0 PP  P NP S 2-27

  41. The Efficient Version: Add as we go 2-1 S  NP VP 2-6 S  Vst NP 2-2 S  S PP 2-1 VP  V NP 2-2 VP  VP PP 2-1 NP  Det N 2-2 NP  NP PP 2-3 NP  NP NP 2-0 PP  P NP

  42. s(0,5) S(0,2) s(0,2)* PP(2,5)*p(S  S PP | S) PP(2,5) The Efficient Version: Add as we go 2-1 S  NP VP 2-6 S  Vst NP 2-2 S  S PP 2-1 VP  V NP 2-2 VP  VP PP 2-1 NP  Det N 2-2 NP  NP PP 2-3 NP  NP NP 2-0 PP  P NP +2-22 +2-27

  43. X Y Z what if you changed + to max? what if you replaced all rule probabilities by 1? j k i Compute  probs bottom-up (CKY) need some initialization up here for the width-1 case for width := 2 to n (* build smallest first *) for i := 0 to n-width (* start *) let k := i + width (* end *) for j := i+1 to k-1 (* middle *) for all grammar rules X  Y Z X(i,k) +=p(X Y Z | X) * Y(i,j) *Z(j,k) 600.465 - Intro to NLP - J. Eisner

  44. S “outside” the VP NP time VP VP(1,5) = p(time VP today | S) VP NP today VP(1,5) = p(flies like an arrow | VP) “inside” the VP Inside & Outside Probabilities PP V flies P like NP VP(1,5) *VP(1,5) = p(time [VPflies like an arrow] today| S) Det an N arrow 600.465 - Intro to NLP - J. Eisner

  45. S NP time VP VP(1,5) = p(time VP today | S) VP NP today / S(0,6) p(time flies like an arrow today | S) Inside & Outside Probabilities PP V flies VP(1,5) = p(flies like an arrow | VP) P like NP VP(1,5) *VP(1,5) = p(timeflies like an arrowtoday & VP(1,5) | S) Det an N arrow = p(VP(1,5) | time flies like an arrow today, S) 600.465 - Intro to NLP - J. Eisner

  46. S NP time VP VP(1,5) = p(time VP today | S) C C C VP NP today Start H H H Inside & Outside Probabilities PP V flies VP(1,5) = p(flies like an arrow | VP) P like NP strictly analogousto forward-backward in the finite-state case! Det an N arrow So VP(1,5) *VP(1,5) / s(0,6) is probability that there is a VP here, given all of the observed data (words) 600.465 - Intro to NLP - J. Eisner

  47. S NP time VP VP(1,5) = p(time VP today | S) VP NP today Inside & Outside Probabilities PP V flies V(1,2) = p(flies | V) P like NP PP(2,5) = p(like an arrow | PP) Det an N arrow So VP(1,5) * V(1,2) *PP(2,5) / s(0,6) is probability that there is VP  V PP here, given all of the observed data (words) … or is it? 600.465 - Intro to NLP - J. Eisner

  48. S NP time VP VP(1,5) = p(time VP today | S) C C C VP NP today Start H H H sum prob over all position triples like (1,2,5) to get expected c(VP V PP); reestimate PCFG! Inside & Outside Probabilities strictly analogousto forward-backward in the finite-state case! PP V flies V(1,2) = p(flies | V) P like NP PP(2,5) = p(like an arrow | PP) Det an N arrow So VP(1,5) * p(VP V PP) *V(1,2) *PP(2,5) / s(0,6) is probability that there is VP V PP here (at 1-2-5), given all of the observed data (words) 600.465 - Intro to NLP - J. Eisner

  49. Summary: VP(1,5) += p(V PP |VP) * V(1,2) * PP(2,5) inside 1,5 inside 1,2 inside 2,5 VP(1,5) p(flies like an arrow |VP) VP += p(V PP |VP) * p(flies |V) p(like an arrow |PP) * Compute  probs bottom-up(gradually build up larger blue “inside” regions) PP(2,5) PP V flies V(1,2) P like NP Det an N arrow 600.465 - Intro to NLP - J. Eisner

  50. Summary: V(1,2) += p(V PP |VP) * VP(1,5) * PP(2,5) S outside 1,2 outside 1,5 inside 2,5 p(time Vlike an arrow today | S) V(1,2) Compute  probs top-down (uses  probs as well) (gradually build up larger pink “outside” regions) NP time VP NP today VP(1,5) VP PP V flies PP(2,5) +=p(time VP today | S) * p(V PP |VP) * p(like an arrow | PP) P like NP Det an N arrow 600.465 - Intro to NLP - J. Eisner

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