Markov Models and Hidden Markov Models (HMMs)

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1 Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook.

2 Markov Model ( = Markov Chain A sequence of random variables visiing a se of saes Transiion probabiliy specifies he probabiliy of ransiing from one sae o he oher. Language Model! Markov Assumpion: nex sae depends only on he curren sae and independen of previous hisory. 2

3 Sample Markov Model for POS De 0.95 Noun sar PropNoun P(PropNoun Verb De Noun =? Verb sop

4 Sample Markov Model for POS sar 4 De Noun Verb PropNoun P(PropNoun Verb De Noun = 0.4*0.8*0.25*0.95*0.1= sop

5 Hidden Markov Model (HMM Probabilisic generaive model for sequences. HMM Definiion wih respec o POS agging: Saes = POS ags Observaion = a sequence of words Transiion probabiliy = bigram model for POS ags Observaion probabiliy = probabiliy of generaing each oken (word from a given POS ag Hidden means he exac sequence of saes (a sequence of POS ags ha generaed he observaion (a sequence of words are hidden.. 5

6 Figure 5.13 Hidden Markov Model (HMM represened as finie sae machine

7 Figure 5.14 Hidden Markov Model (HMM represened as finie sae machine Noe ha in his represenaion, he number of nodes (saes = he size of he se of POS ags

8 Figure 5.12 Hidden Markov Model (HMM represened as a graphical model Noe ha in his represenaion, he number of nodes (saes = he lengh of he word sequence.

9 Formal Definiion of an HMM 9 Wha are he parameers of HMM?

10 Three imporan problems in HMM Likelihood funcion L θ ; X Sricly speaking, likelihood is no a probabiliy. Likelihood is proporionae o P X θ 10

11 Three imporan problems in HMM Problem 1 (Likelihood Forward Algorihm Problem 2 (Decoding Vierbi Algorihm Problem 3 (Learning Forward-backward Algorihm 11

12 HMM Decoding: Vierbi Algorihm Decoding finds he mos likely sequence of saes ha produced he observed sequence. A sequence of saes = pos-ags A sequence of observaion = words Naïve soluion: brue force search by enumeraing all possible sequences of saes. problem? Dynamic Programming! Sandard procedure is called he Vierbi algorihm (Vierbi, 1967 and has O(N 2 T ime complexiy. 12

13 HMM Decoding: Vierbi Algorihm Inuiion:

14 HMM Decoding: Vierbi Algorihm Inuiion:

15 HMM Decoding: Vierbi Algorihm Inuiion:

16 HMM Decoding: Vierbi Algorihm Inuiion:

17 HMM Decoding: Vierbi Algorihm Inuiion:

18 HMM Decoding: Vierbi Algorihm Inuiion:

19 HMM Decoding: Vierbi Algorihm Inuiion:

20 HMM Decoding: Vierbi Algorihm Inuiion:

21 HMM Decoding: Vierbi Algorihm Inuiion:

22

23

24 HMM Likelihood of Observaion Given a sequence of observaions, O, and a model wih a se of parameers, λ, wha is he probabiliy ha his observaion was generaed by his model: P(O λ? 24

25 HMM Likelihood of Observaion Due o he Markov assumpion, he probabiliy of being in any sae a any given ime only relies on he probabiliy of being in each of he possible saes a ime 1. Forward Algorihm: Uses dynamic programming o exploi his fac o efficienly compue observaion likelihood in O(TN 2 ime. Compue a forward rellis ha compacly and implicily encodes informaion abou all possible sae pahs. 25

26 Forward Probabiliies Le ( be he probabiliy of being in sae afer seeing he firs observaions (by summing over all iniial pahs leading o. ( P( o, o2,... o, q s 1 26

27 Forward Sep s 1 s 2 s N -1 (i a 1 a 2 a 2 a N s (i Consider all possible ways of geing o s a ime by coming from all possible saes s i and deermine probabiliy of each. Sum hese o ge he oal probabiliy of being in sae s a ime while accouning for he firs 1 observaions. Then muliply by he probabiliy of acually observing o in s. 27

28

29 Forward Trellis s 1 s 0 s 2 s N s F T-1 T Coninue forward in ime unil reaching final ime poin and sum probabiliy of ending in final sae.

30

31 Forward Compuaional Complexiy Requires only O(TN 2 ime o compue he probabiliy of an observed sequence given a model. Explois he fac ha all sae sequences mus merge ino one of he N possible saes a any poin in ime and he Markov assumpion ha only he las sae effecs he nex one. 31

32 HMM Learning Supervised Learning: All raining sequences are compleely labeled (agged. Tha is, nohing is really hidden sricly speaking. Learning is very simple by MLE esimae Unsupervised Learning: All raining sequences are unlabeled (ags are unknown We do assume he number of ags, i.e. saes True HMM case. Forward-Backward Algorihm, (also known as Baum-Welch algorihm which is a special case of Expecaion Maximizaion (EM raining 32

33 HMM Learning: Supervised Esimae sae ransiion probabiliies based on ag bigram and unigram saisics in he labeled daa. Esimae he observaion probabiliies based on ag/word co-occurrence saisics in he labeled daa. Use appropriae smoohing if raining daa is sparse. 33 ( q, ( 1 i i i s q C s s q C a (, ( ( i k i i s q C v o s q C k b

34 HMM Learning: Unsupervised 34

35 Skech of Baum-Welch (EM Algorihm for Training HMMs Assume an HMM wih N saes. Randomly se is parameers λ=(a,b (making sure hey represen legal disribuions Unil converge (i.e. λ no longer changes do: E Sep: Use he forward/backward procedure o deermine he probabiliy of various possible sae sequences for generaing he raining daa M Sep: Use hese probabiliy esimaes o re-esimae values for all of he parameers λ 35

36 Backward Probabiliies Le (i be he probabiliy of observing he final se of observaions from ime +1 o T given ha one is in sae i a ime. ( i P( o, o 2,... ot q si, 1 36

37 Compuing he Backward Probabiliies Iniializaion Recursion Terminaion 37 N i a i if T 1 ( T N i o b a i N i 1, 1 ( ( ( ( ( ( ( ( o b a s s O P N F T

38 ( i, Esimaing Probabiliy of Sae Transiions Le (i, be he probabiliy of being in sae i a ime and sae a ime + 1 ( i, P( q s, q 1 s O, P( q s i, q 1 P( O s i, O ( i a i b ( o P( O 1 1 ( s 1 a 1i a 1 s 1 s 2 s N a 2i a 3i a Ni s i (i 1 ( a i b ( o 1 s a 2 a 3 a N s 2 s N

39 Re-esimaing A ˆ a i aˆ i expeced number of ransiions from saei o expeced number of ransiions from saei T 1 1 T 1 N 1 1 ( i, ( i,

40 Esimaing Observaion Probabiliies Le (i be he probabiliy of being in sae i a ime given he observaions and he model. ( ( ( (, (, ( ( O P O P O s q P O s q P

41 Re-esimaing B v v b k k imes in sae expeced number of observing imes in sae expeced number of ( ˆ T T v k v b k 1 1,s..o ( ( ( ˆ

42 Pseudocode for Baum-Welch (EM Algorihm for Training HMMs Assume an HMM wih N saes. Randomly se is parameers λ=(a,b (making sure hey represen legal disribuions Unil converge (i.e. λ no longer changes do: E Sep: Compue values for ( and (i, using curren values for parameers A and B. M Sep: Re-esimae parameers: a b ˆ i a i ˆ ( vk b ( vk 42

43

44

45

46

47 EM Properies Each ieraion changes he parameers in a way ha is guaraneed o increase he likelihood of he daa: P(O. Anyime algorihm: Can sop a any ime prior o convergence o ge approximae soluion. Converges o a local maximum.

48 Semi-Supervised Learning EM algorihms can be rained wih a mix of labeled and unlabeled daa. EM basically predics a probabilisic (sof labeling of he insances and hen ieraively rerains using supervised learning on hese prediced labels ( self raining. EM can also exploi supervised daa: 1 Use supervised learning on labeled daa o iniialize he parameers (insead of iniializing hem randomly. 2 Use known labels for supervised daa insead of predicing sof labels for hese examples during reraining ieraions.

49 Semi-Supervised Resuls Use of addiional unlabeled daa improves on supervised learning when amoun of labeled daa is very small and amoun of unlabeled daa is large. Can degrade performance when here is sufficien labeled daa o learn a decen model and when unsupervised learning ends o creae labels ha are incompaible wih he desired ones. There are negaive resuls for semi-supervised POS agging since unsupervised learning ends o learn semanic labels (e.g. eaing verbs, animae nouns ha are beer a predicing he daa han purely synacic labels (e.g. verb, noun.

50 Conclusions POS Tagging is he lowes level of synacic analysis. I is an insance of sequence labeling, a collecive classificaion ask ha also has applicaions in informaion exracion, phrase chunking, semanic role labeling, and bioinformaics. HMMs are a sandard generaive probabilisic model for sequence labeling ha allows for efficienly compuing he globally mos probable sequence of labels and suppors supervised, unsupervised and semi-supervised learning.

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