1 / 22

Three Basic Problems

Three Basic Problems. Compute the probability of a text (observation) language modeling – evaluate alternative texts and models P m (W 1, N ) Compute maximum probability tag (state) sequence Tagging/classification arg max T 1, N P m (T 1, N | W 1, N )

aracely-trey
Download Presentation

Three Basic Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Three Basic Problems • Compute the probability of a text (observation) • language modeling – evaluate alternative texts and models Pm(W1,N) • Compute maximum probability tag (state) sequence • Tagging/classification arg maxT1,N Pm(T1,N | W1,N) • Compute maximum likelihood model • training / parameter estimation arg maxm Pm(W1,N)

  2. Compute Text Probability • Recall: P(W,T) = i P(ti-1ti) P(wi | ti) • Text probability: need to sum P(W,T) over all possible sequences – an exponential number • Dynamic programming approach – similar to the Viterbi algorithm • Will be used also for estimating model parameters from an untagged corpus

  3. Forward Algorithm Define: Ai(k) = P(w1,k, tk= ti); Nt – total num. of tags For i = 1 To Nt: Ai(1) = m(t0ti)m(w1 | ti) • For k = 2 To N; For j = 1 To Nt: • Aj(k) = [iAi(k-1)m(titj)]m(wk | tj) • Then: Pm(W1,N) = iAi(N) Complexity = O(Nt2 N) (like Viterbi,  instead of max)

  4. w1 w2 w3 A1(1) m(t1t1) m(t1t1) t1 t1 t1 m(t2t1) m(t2t1) A2(1) A2(2) A2(3) t2 t2 t2 m(t3t1) m(t3t1) m(t0ti) A3(1) A3(2) A3(3) t3 t3 t3 m(t4t1) m(t4t1) A4(1) A4(2) A4(3) t4 t4 t4 m(t5t1) m(t5t1) A5(1) A5(2) A5(3) t5 t5 t5 Pm(W1,3) Forward Algorithm A1(2) A1(3)

  5. Backward Algorithm Define Bi(k) = P(wk+1,N | tk=ti) • For i = 1 To Nt: Bi(N) = 1 • For k = N-1 To 1; For j = 1 To Nt: • Bj(k) = [i m(tjti)m(wk+1 | ti)Bi(k+1)] • Then: Pm(W1,N) = i m(t0ti)m(w1 | ti)Bi(1) Complexity = O(Nt2 N)

  6. w1 w2 w3 m(t1t1) m(t1t1) B1(3) t1 t1 t1 m(t0ti) m(t2t1) m(t2t1) B2(1) B2(2) B2(3) t2 t2 t2 m(t3t1) m(t3t1) B3(1) B3(2) B3(3) t3 t3 t3 m(t4t1) m(t4t1) B4(1) B4(2) B4(3) t4 t4 t4 m(t5t1) m(t5t1) B5(1) B5(2) B5(3) t5 t5 t5 Pm(W1,3) Backward Algorithm B1(1) B1(2)

  7. Estimation from Untagged Corpus: EM – Expectation-Maximization • Start with some initial model • Compute the probability of (virtually) each state sequence given the current model • Use this probabilistic tagging to produce probabilistic counts for all parameters, and use these probabilistic counts to estimate a revised model, which increases the likelihood of the observed output W in each iteration • Repeat until convergence Note: No labeled training required. Initialize by lexicon constraints regarding possible POS for each word (cf. “noisy counting” for PP’s)

  8. Notation • aij = Estimate of P(titj) • bjk = Estimate of P(wk | tj) • Ai(k) = P(w1,k, tk=ti) (from Forward algorithm) • Bi(k) = P(wk+1,N | tk=ti) (from Backwards algorithm)

  9. Estimating transition probabilities Define pk(i,j) as prob. of traversing arc titj at time k given the observations: pk(i,j) = P(tk = ti, tk+1 = tj | W) = P(tk = ti, tk+1 = tj,W) / P(W) = =

  10. Expected transitions • Define gi(k) = P(tk = ti | W), then: gi(k) = • Now note that: • Expected number of transitions from tag i = • Expected transitions from tag i to tag j =

  11. Re-estimation of Maximum Likelihood Parameters • a’ij = = • b’ik = =

  12. EM Algorithm • Choose initial model = <a,b,g(1)> • Repeat until results don’t improve (much): • Compute pk based on current model, using Forward & Backwards algorithms to compute A and B (Expectation for counts) • Compute new model <a’,b’,g’(1)> (Maximization of parameters) Note: Output likelihood is guaranteed to increase in each iteration, but might converge to a local maximum!

  13. Initialize Model by Dictionary Constraints • Training should be directed to correspond to the linguistic perception of POS (recall local max) • Achieved by a dictionary with possible POS for each word • Word-based initialization: • P(w|t) = 1 / #of listed POS for w, for the listed POS; and 0 for unlisted POS • Class-based initialization (Kupiec, 1992): • Group all words with the same possible POS into a ‘metaword’ • Estimate parameters and perform tagging for metawords • Frequent words are handled individually

  14. Some extensions for HMM POS tagging • Higher-order models: trigrams, possibly interpolated with bigrams • Incorporating text features: • Output prob = P(wi,fj| tk) where f is a vector of features (capitalized, ends in –d, etc.) • Features useful to handle unknown words • Combining labeled and unlabeled training (initialize with labeled then do EM)

  15. Transformational Based Learning (TBL) for Tagging • Introduced by Brill (1995) • Can exploit a wider range of lexical and syntactic regularities via transformation rules – triggering environment and rewrite rule • Tagger: • Construct initial tag sequence for input – most frequent tag for each word • Iteratively refine tag sequence by applying “transformation rules” in rank order • Learner: • Construct initial tag sequence for the training corpus • Loop until done: • Try all possible rules and compare to known tags, apply the best rule r* to the sequence and add it to the rule ranking

  16. Some examples 1. Change NN to VB if previous is TO • to/TO conflict/NN with  VB 2. Change VBP to VB if MD in previous three • might/MD vanish/VBP VB 3. Change NN to VB if MD in previous two • might/MD reply/NN VB 4. Change VB to NN if DT in previous two • the/DT reply/VB  NN

  17. Transformation Templates • Specify which transformations are possible For example: change tag A to tag B when: • The preceding (following) tag is Z • The tag two before (after) is Z • One of the two previous (following) tags is Z • One of the three previous (following) tags is Z • The preceding tag is Z and the following is W • The preceding (following) tag is Z and the tag two before (after) is W

  18. Lexicalization New templates to include dependency on surrounding words (not just tags): Change tag A to tag B when: • The preceding (following) word is w • The word two before (after) is w • One of the two preceding (following) words is w • The current word is w • The current word is w and the preceding (following) word is v • The current word is w and the preceding (following) tag is X (Notice: word-tag combination) • etc…

  19. Initializing Unseen Words • How to choose most likely tag for unseen words? Transformation based approach: • Start with NP for capitalized words, NN for others • Learn “morphological” transformations from: Change tag from X to Y if: • Deleting prefix (suffix) x results in a known word • The first (last) characters of the word are x • Adding x as a prefix (suffix) results in a known word • Word W ever appears immediately before (after) the word • Character Z appears in the word

  20. TBL Learning Scheme Unannotated Input Text Setting Initial State Ground Truth for Input Text Annotated Text Learning Algorithm Rules

  21. Greedy Learning Algorithm • Initial tagging of training corpus – most frequent tag per word • At each iteration: • Compute “error reduction” for each transformation rule: • #errors fixed - #errors introduced • Find best rule; If error reduction greater than a threshold (to avoid overfitting): • Apply best rule to training corpus • Append best rule to ordered list of transformations

  22. Morphological Richness • Parts of speech really include features: • NN2  Noun(type=common,num=plural) This is more visible in other languages with richer morphology: • Hebrew nouns: number, gender, possession • German nouns: number, gender, case, … • And so on…

More Related