Hidden Markov Model (HMM) Tagging

1. Hidden Markov Model (HMM) Tagging • Using an HMM to do POS tagging • HMM is a special case of Bayesian inference

2. Hidden Markov Model (HMM) Taggers • Goal: maximize P(word|tag) x P(tag|previous n tags) • P(word|tag) • word/lexical likelihood • probability that given this tag, we have this word • NOT probability that this word has this tag • modeled through language model (word-tag matrix)‏ • P(tag|previous n tags)‏ • tag sequence likelihood • probability that this tag follows these previous tags • modeled through language model (tag-tag matrix)‏ Lexical information Syntagmatic information

3. POS tagging as a sequence classification task • We are given a sentence (an “observation” or “sequence of observations”)‏ • Secretariat is expected to race tomorrow. • sequence of n words w1…wn. • What is the best sequence of tags which corresponds to this sequence of observations? • Probabilistic/Bayesian view: • Consider all possible sequences of tags • Out of this universe of sequences, choose the tag sequence which is most probable given the observation sequence of n words w1…wn.

4. Getting to HMM • Let T = t1,t2,…,tn • Let W = w1,w2,…,wn • Goal: Out of all sequences of tags t1…tn, get the the most probable sequence of POS tags T underlying the observed sequence of words w1,w2,…,wn • Hat ^ means “our estimate of the best = the most probable tag sequence” • Argmaxxf(x) means “the x such that f(x) is maximized” it maximizes our estimate of the best tag sequence

5. Bayes Rule We can drop the denominator: it does not change for each tag sequence; we are looking for the best tag sequence for the same observation, for the same fixed set of words

6. Bayes Rule

7. Likelihood and prior

8. Likelihood and prior Further Simplifications 1. the probability of a word appearing depends only on its own POS tag, i.e, independent of other words around it n 2. BIGRAM assumption: the probability of a tag appearing depends only on the previous tag 3. The most probable tag sequence estimated by the bigram tagger

9. Likelihood and prior Further Simplifications 2. BIGRAM assumption: the probability of a tag appearing depends only on the previous tag. Bigrams are groups of two written letters, two syllables, or two words; they are a special case of N-gram. Bigrams are used as the basis for simple statistical analysis of text The bigram assumption is related to the first-order Markov assumption

10. Likelihood and prior Further Simplifications 3. The most probable tag sequence estimated by the bigram tagger --------------------------------------------------------------------------------------------------------------- n biagram assumption

11. Probability estimates • Tag transition probabilities p(ti|ti-1)‏ • Determiners likely to precede adjectives and nouns • That/DT flight/NN • The/DT yellow/JJ hat/NN • So we expect P(NN|DT) and P(JJ|DT) to be high

12. Estimating probability • Tag transition probabilities p(ti|ti-1)‏ • Compute P(NN|DT) by counting in a labeled corpus: # of times DT is followed by NN

13. Two kinds of probabilities • Word likelihood probabilities p(wi|ti)‏ • P(is|VBZ) = probability of VBZ (3sg Pres verb) being “is” • Compute P(is|VBZ) by counting in a labeled corpus: If we were expecting a third person singular verb, how likely is it that this verb would be is?

14. An Example: the verb “race” Two possible tags: • Secretariat/NNP is/VBZ expected/VBN to/TO race/VB tomorrow/NR • People/NNS continue/VB to/TO inquire/VB the/DT reason/NN for/IN the/DTrace/NN for/IN outer/JJ space/NN How do we pick the right tag?

15. Disambiguating “race”

16. Disambiguating “race” • P(NN|TO) = .00047 • P(VB|TO) = .83 The tag transition probabilities P(NN|TO) and P(VB|TO) answer the question: ‘How likely are we to expect verb/noun given the previous tag TO?’ • P(race|NN) = .00057 • P(race|VB) = .00012 Lexical likelihoods from the Brown corpus for ‘race’ given a POS tag NN or VB.

17. Disambiguating “race” • P(NR|VB) = .0027 • P(NR|NN) = .0012 tag sequence probability for the likelihood of an adverb occurring given the previous tag verb or noun • P(VB|TO)P(NR|VB)P(race|VB) = .00000027 • P(NN|TO)P(NR|NN)P(race|NN)=.00000000032 Multiply the lexical likelihoods with the tag sequence probabilities: the verb wins

18. Hidden Markov Models • What we’ve described with these two kinds of probabilities is a Hidden Markov Model (HMM)‏ • Let’s just spend a bit of time tying this into the model • In order to define HMM, we will first introduce the Markov Chain, or observable Markov Model.

19. Definitions • A weighted finite-state automaton adds probabilities to the arcs • The sum of the probabilities leaving any arc must sum to one • A Markov chain is a special case of a WFST in which the input sequence uniquely determines which states the automaton will go through • Markov chains can’t represent inherently ambiguous problems • Useful for assigning probabilities to unambiguous sequences

20. Hidden Markov Models Formal definition • States Q = q1, q2…qN; • Observations O = o1, o2…oN; • Each observation is a symbol from a vocabulary V = {v1,v2,…vV} • Transition probabilities (prior)‏ • Transition probability matrix A = {aij} • Observation likelihoods (likelihood)‏ • Output probability matrix B={bi(ot)} a set of observation likelihoods, each expressing the probability of an observation ot being generated from a state i, emission probabilities • Special initial probability vector  i the probability that the HMM will start in state i, each i expresses the probability p(qi|START)‏

21. Assumptions • Markov assumption: the probability of a particular state depends only on the previous state • Output-independence assumption: the probability of an output observation depends only on the state that produced that observation

22. HMM Taggers • Two kinds of probabilities • A transition probabilities (PRIOR) • B observation likelihoods (LIKELIHOOD) • HMM Taggers choose the tag sequence which maximizes the product of word likelihood and tag sequence probability

23. Weighted FSM corresponding to hidden states of HMM

24. observation likelihoods for POS HMM

25. Transition matrix for the POS HMM

26. The output matrix for the POS HMM

27. HMM Taggers • The probabilities are trained on hand-labeled training corpora (training set)‏ • Combine different N-gram levels • Evaluated by comparing their output from a test set to human labels for that test set (Gold Standard)

28. The Viterbi Algorithm • best tag sequence for "John likes to fish in the sea"? • efficiently computes the most likely state sequence given a particular output sequence • based on dynamic programming

29. A smaller example a b b a 0.2 0.8 0.4 • What is the best sequence of states for the input string “bbba”? • Computing all possible paths and finding the one with the max probability is exponential 0.6 0.7 end start r q 1 1 0.5 0.3 0.5

30. Possible improvements • in bigram POS tagging, we condition a tag only on the preceding tag • why not... • use more context (ex. use trigram model) • more precise: • “is clearly marked” --> verb, past participle • “he clearly marked” --> verb, past tense • combine trigram, bigram, unigram models • condition on words too • but with an n-gram approach, this is too costly (too many parameters to model)‏

31. Further issues with Markov Model tagging • Unknown words are a problem since we don’t have the required probabilities. Possible solutions: • Assign the word probabilities based on corpus-wide distribution of POS ??? • Use morphological cues (capitalization, suffix) to assign a more calculated guess. • Using higher order Markov models: • Using a trigram model captures more context • However, data sparseness is much more of a problem.