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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 P A R S E R s c o r e r test sentences 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. Viterbi for tagging • Naïve idea: Give each word its highest-prob tag according to forward-backward. • Do this independently of other words. • Det Adj 0.35 • Det N 0.2 • N V 0.45 • Oops! We get Det V. • This is defensible, but not a coherent sequence. • Might screw up subsequent processing (parsing, chunking …) that depends on coherence. 600.465 - Intro to NLP - J. Eisner

  10. Viterbi for tagging • Naïve idea: Give each word its highest-prob tag according to forward-backward. • Do this independently of other words. • Det Adj 0.35 • Det N 0.2 • N V 0.45 • Better idea: Use Viterbi to pick single best tag sequence (best path). • Can still use full EM to train the parameters, if time permits, but use Viterbi to tag once trained. 600.465 - Intro to NLP - J. Eisner

  11. compose these? Examples • 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 PCFG • 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

  12. Inside-Outside Algorithm • How would we do the Viterbi approximation? • How would we do full EM? 600.465 - Intro to NLP - J. Eisner

  13. 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

  14. 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

  15. “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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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 PP(2,5) PP V flies V(1,2) P like NP Det an N arrow 600.465 - Intro to NLP - J. Eisner

  29. 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) 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

  30. Summary: PP(2,5) += p(V PP |VP) * VP(1,5) * V(1,2) outside 2,5 outside 1,5 inside 1,2 p(time fliesPP today | S) PP(2,5) Compute  probs top-down (uses  probs as well) S NP time VP NP today VP(1,5) VP PP V flies +=p(time VP today | S) * p(V PP |VP) * p(flies| V) V(1,2) P like NP Det an N arrow 600.465 - Intro to NLP - J. Eisner

  31. VP(1,5) VP Compute  probs bottom-up When you build VP(1,5), from VP(1,2) and VP(2,5)during CKY,increment VP(1,5) by p(VP VP PP) * VP(1,2) *PP(2,5) Why? VP(1,5) is total probability of all derivations p(flies like an arrow | VP)and we just found another. (See earlier slide of CKY chart.) PP(2,5) VP(1,2) PP VP flies P like NP Det an N arrow 600.465 - Intro to NLP - J. Eisner

  32. X Y Z j k i Compute  probs bottom-up (CKY) 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) * Y(i,j) *Z(j,k) 600.465 - Intro to NLP - J. Eisner

  33. Compute  probs top-down (reverse CKY) n downto 2 “unbuild” biggest first 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 Y(i,j) += ??? Z(j,k) += ??? X Y Z j k i 600.465 - Intro to NLP - J. Eisner

  34. PP(2,5) VP(1,2) PP VP VP(1,2) PP(2,5) already computed on bottom-up pass Compute  probs top-down S After computing  during CKY, revisit constits in reverse order (i.e., bigger constituents first).When you “unbuild” VP(1,5) from VP(1,2) and VP(2,5),increment VP(1,2) by VP(1,5)* p(VP  VP PP) *PP(2,5) and increment PP(2,5) by VP(1,5)* p(VP  VP PP) *VP(1,2) NP time VP VP(1,5) VP NP today VP(1,2) is total prob of all ways to gen VP(1,2) and all outside words. 600.465 - Intro to NLP - J. Eisner

  35. Compute  probs top-down (reverse CKY) n downto 2 “unbuild” biggest first 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 Y(i,j) +=X(i,k)* p(X  Y Z) *Z(j,k) Z(j,k) +=X(i,k)* p(X  Y Z) *Y(i,j) X Y Z j k i 600.465 - Intro to NLP - J. Eisner

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