Hidden Markov Models

Transcription

1 0. Hidden Markov Models Based on Foundations of Statistical NLP by C. Manning & H. Schütze, ch. 9, MIT Press, 2002 Biological Sequence Analysis, R. Durbin et al., ch. 3 and 11.6, Cambridge University Press, 1998

2 PLAN 1. 1 Markov Models Markov assumptions 2 Hidden Markov Models 3 Fundamental questions for HMMs 3.1 Probability of an observation sequence: the Forward algorithm, the Backward algorithm 3.2 Finding the best sequence: the Viterbi algorithm 3.3 HMM parameter estimation: the Forward-Backward (EM) algorithm 4 HMM extensions 5 Applications

3 1 Markov Models (generally) 2. Markov Models are used to model a sequence of random variables in which each element depends on previous elements. X = X 1...X T X t S = {s 1,..., s N } X is also called a Markov Process or Markov Chain. S = set of states Π = initial state probabilities π i = P(X 1 = s i ); N i=1 π i = 1 A = transition probabilities: a ij = P(X t+1 = s j X t = s i ); N j=1 a ij = 1 i

4 3. Markov assumptions Limited Horizon: P(X t+1 = s i X 1...X t ) = P(X t+1 = s i X t ) (first-order Markov model) Time Invariance: P(X t+1 = s j X t = s i ) = p ij t Probability of a Markov Chain P(X 1...X T ) = P(X 1 )P(X 2 X 1 )P(X 3 X 1 X 2 )... P(X T X 1 X 2...X T 1 ) = P(X 1 )P(X 2 X 1 )P(X 3 X 2 )...P(X T X T 1 ) = π X1 Π T 1 t=1 a X t X t+1

5 A 1st Markov chain example: DNA (from [Durbin et al., 1998]) 4. A T Note: Here we leave transition probabilities unspecified. C G

6 A 2nd Markov chain example: CpG islands in DNA sequences Maximum Likelihood estimation of parameters using real data (+ and -) 5. a + st = c+ st st t c + st a st = c t c st + A C G T A C G T A C G T A C G T

7 6. Using log likelihoood (log-odds) ratios for discrimination S(x) = log 2 P(x model +) P(x model ) = L i=1 log 2 a + x i 1 x i a x i 1 x i = β A C G T A C G T L i=1 β xi 1 x i

8 2 Hidden Markov Models 7. K = output alphabet = {k 1,..., k M } B = output emission probabilities: b ijk = P(O t = k X t = s i, X t+1 = s j ) Notice that b ijk does not depend on t. In HMMs we only observe a probabilistic function of the state sequence: O 1...O T When the state sequence X 1...X T is also observable: Visible Markov Model (VMM) Remark: In all our subsequent examples b ijk is independent of j.

9 8. A program for a HMM t = 1; start in state s i with probability π i (i.e., X 1 = i); forever do move from state s i to state s j with prob. a ij (i.e., X t+1 = j); emit observation symbol O t = k with probability b ijk ; t = t + 1;

10 A 1st HMM example: CpG islands (from [Durbin et al., 1998]) 9. Notes: A + C + G + T+ 1. In addition to the transitions shown, there is also a complete set of transitions within each set (+ respectrively -). A C G T 2. Transition probabilities in this model are set so that within each group they are close to the transition probabilities of the original model, but there is also a small chance of switching into the other component. Overall, there is more chance of switching from + to - than viceversa.

11 A 2nd HMM example: The occasionally dishonest casino (from [Durbin et al., 1998]) 10. 1: 1/6 2: 1/6 3: 1/6 4: 1/6 5: 1/6 6: 1/ : 1/10 2: 1/10 3: 1/10 4: 1/10 5: 1/10 6: 1/ F L

12 A 2rd HMM example: The crazy soft drink machine (from [Manning & Schütze, 2000]) 11. P(Coke) = 0.6 Ice tea = 0.1 Lemon = Coke Preference Ice tea Preference 0.5 π CP =1 0.5 P(Coke) = 0.1 Ice tea = 0.7 Lemon = 0.2

13 A 4th example: A tiny HMM for 5 splice site recognition (from [Eddy, 2004]) 12.

14 3 Three fundamental questions for HMMs Probability of an Observation Sequence: Given a model µ = (A, B, Π) over S, K, how do we (efficiently) compute the likelihood of a particular sequence, P(O µ)? 2. Finding the Best State Sequence: Given an observation sequence and a model, how do we choose a state sequence (X 1,..., X T+1 ) to best explain the observation sequence? 3. HMM Parameter Estimation: Given an observation sequence (or corpus thereof), how do we acquire a model µ = (A, B, Π) that best explains the data?

15 Probability of an observation sequence P(O X, µ) = Π T t=1 P(O t X t, X t+1, µ) = b X1 X 2 O 1 b X2 X 3 O 2...b XT X T+1 O T P(O, µ) = P(O X, µ)p(x, µ) = π X1 Π T t=1a Xt X t+1 b Xt X t+1 O t X X 1...X T+1 Complexity : (2T + 1)N T+1, too inefficient better : use dynamic prog. to store partial results α i (t) = P(O 1 O 2...O t 1, X t = s i µ).

16 Probability of an observation sequence: The Forward algorithm 1. Initialization: α i (1) = π i, for 1 i N 2. Induction: α j (t + 1) = N i=1 α i(t)a ij b ijot, 1 t T, 1 j N 3. Total: P(O µ) = N i=1 α i(t + 1). Complexity: 2N 2 T

17 Proof of induction step: α j (t + 1) = P(O 1 O 2...O t 1 O t, X t+1 = j µ) N = P(O 1 O 2...O t 1 O t, X t = i, X t+1 = j µ) = = = = i=1 N P(O t, X t+1 = j O 1 O 2...O t 1, X t = i, µ)p(o 1 O 2...O t 1, X t = i µ) i=1 N P(O 1 O 2...O t 1, X t = i µ)p(o t, X t+1 = j O 1 O 2...O t 1, X t = i, µ) i=1 N α i (t)p(o t, X t+1 = j X t = i, µ) i=1 N α i (t)p(o t X t = i, X t+1 = j, µ)p(x t+1 = j X t = i, µ) = i=1 i=1 16. N α i (t)b ijot a ij

18 Closeup of the Forward update step 17. s 1 α 1 (t) a 1j b 1jO t s 2 α 2 (t) P(O... O, X = s µ ) 1 t t+1 j P(O... O, X = s µ ) 1 t 1 t i s N α N (t) a 2j b 2jO t a Nj b NjO t s j α j (t+1) t t+1

19 Trellis 18. s 1 Each node (s i, t) stores information about paths through s i at time t. s 2 s 3 State s N 1 2 Time t T+1

20 Probability of an observation sequence: The Backward algorithm β i (t) = P(O t...o T X t = i, µ) 1. Initialization: β i (T + 1) = 1, for 1 i N 2. Induction: β i (t) = N j=1 a ijb ijot β j (t + 1), 1 t T, 1 i N 3. Total: P(O µ) = N i=1 π iβ i (1) Complexity: 2N 2 T

21 Induction: The Backward algorithm: Proofs 20. β i (t) = P(O t O t+1...o T X t = i, µ) N = P(O t O t+1...o T, X t+1 = j X t = i, µ) j=1 N = P(O t O t+1...o T X t = i, X t+1 = j, µ)p(x t+1 = j X t = i, µ) j=1 N = P(O t+1...o T O t, X t = i, X t+1 = j, µ)p(o t X t = i, X t+1 = j, µ)a ij j=1 N = P(O t+1...o T X t+1 = j, µ)b ijot a ij = j=1 N β j (t + 1)b ijot a ij N Total: P(O µ) = P(O 1 O 2...O T X 1 = i, µ)p(x 1 = i µ) = j=1 i=1 i=1 N β i (1)π i

22 Combining Forward and Backward probabilities 21. Proofs: P(O, X t = i µ) = α i (t)β i (t) N P(O µ) = α i (t)β i (t) for 1 t T + 1 i=1 P(O, X t = i µ) = P(O 1...O T, X t = i µ) = P(O 1...O t 1, X t = i, O t...o T µ) = P(O 1...O t 1, X t = i µ)p(o t...o T O 1...O t 1, X t = i, µ) = α i (t)p(o t...o T X t = i, µ) = α i (t)β i (t) P(O µ) = N P(O, X t = i µ) = i=1 N α i (t)β i (t) i=1 Note: The total forward and backward formulae are special cases of the above one (for t = T + 1 and respectively t = 1).

23 3.2 Finding the best state sequence Posterior decoding 22. One way to find the most likely state sequence underlying the observation sequence: choose the states individually γ i (t) = P(X t = i O, µ) ˆX t Computing γ i (t): = argmax 1 i N γ i (t) for 1 t T + 1 γ i (t) = P(X t = i O, µ) = P(X t = i, O µ) P(O µ) = α i (t)β i (t) N j=1 α j(t)β j (t) Remark: ˆX maximizes the expected number of states that will be guessed correctly. However, it may yield a quite unlikely/unnatural state sequence.

24 Note Sometimes not the state itself is of interest, but some other property derived from it. 23. For instance, in the CpG islands example, let g be a function defined on the set of states: g takes the value 1 for A +, C +, G +, T + and 0 for A, C, G, T. Then P(π t = s j O)g(s j ) j designates the posterior probability that the symbol O t come from a state in the + set. Thus it is possible to find the most probable label of the state at each position in the output sequence O.

25 3.2.2 Finding the best state sequence The Viterbi algorithm 24. Compute the probability of the most likely path argmax P(X O, µ) = argmax X X through a node in the trellis P(X, O µ) δ i (t) = max X 1...X t 1 P(X 1...X t 1, O 1...O t 1, X t = s i µ) 1. Initialization: δ j (1) = π j, for 1 j N 2. Induction: (see the similarity with the Forward algorithm) δ j (t + 1) = max 1 i N δ i (t)a ij b ijot, 1 t T, 1 j N ψ j (t + 1) = argmax 1 i N δ i (t)a ij b ijot, 1 t T, 1 j N 3. Termination and readout of best path: P( ˆX, O µ) = max1 i N δ i (T + 1) ˆX T+1 = argmax 1 i N δ i (T + 1), ˆX t = ψ ˆXt+1 (t + 1)

26 Example: Variable calculations for the crazy soft drink machine HMM Output lemon ice tea coke t α CP (t) α IP (t) P(o 1...o t 1 ) β CP (t) β CP (t) P(o 1...o T ) γ CP (t) γ IP (t) ˆX t CP IP CP CP δ CP (t) δ IP (t) ψ CP (t) CP IP CP ψ IP (t) CP IP CP 25. ˆX t CP IP CP CP P( ˆX)

27 HMM parameter estimation Given a single observation sequence for training, we want to find the model (parameters) µ = (A, B, π) that best explains the observed data. Under Maximum Likelihood Estimation, this means: argmax µ P(O training µ) There is no known analytic method for doing this. However we can choose µ so as to locally maximize P(O training µ) by an iterative hill-climbing algorithm: Forward-Backward (or: Baum-Welch), which is a special case of the EM algorithm.

28 The Forward-Backward algorithm The idea Assume some (perhaps randomly chosen) model parameters. Calculate the probability of the observed data. Using the above calculation, we can see which transitions and signal emissions were probably used the most; by increasing the probabily of these, we will get a higher probability of the observed sequence. Iterate, hopefully arriving at an optimal parameter setting.

29 The Forward-Backward algorithm: Expectations Define the probability of traversing a certain arc at time t, given the observation sequence O 28. p t (i, j) = P(X t = i, X t+1 = j O, µ) p t (i, j) = P(X t = i, X t+1 = j, O µ) P(O µ) = N m=1 α i (t)a ij b ijot β j (t + 1) N n=1 α m(t)a mn b mnot β n (t + 1) = α i(t)a ij b ijot β j (t + 1) N m=1 α m(t)β m (t) Summing over t: T t=1 p t(i, j) = expected number of transitions from s i to s j in O N j=1 T t=1 p t(i, j) = expected number of transitions from s i in O

30 s i a ij b ijot s j... α i (t) β j (t+1) t 1 t t t+1

31 30. The Forward-Backward algorithm: Re-estimation From µ = (A, B, Π), derive ˆµ = (Â, ˆB, ˆΠ): ˆπ i = â ij = ˆbijk = N j=1 p 1(i, j) N N j=1 p 1(l, j) = p 1 (i, j) = γ i (1) N l=1 T t=1 p t(i, j) N T l=1 t=1 p t(i, l) t:o t =k, 1 t T p t(i, j) T t=1 p t(i, j) j=1

32 31. The Forward-Backward algorithm: Justification Theorem (Baum-Welch): P(O ˆµ) P(O µ) Note 1: However, it does not necessarily converge to a global optimum. Note 2: There is a straightforward extension of the algorithm that deals with multiple observation sequences (i.e., a corpus).

33 Example: Re-estimation of HMM parameters The crazy soft drink machine, after one EM iteration on the sequence O = (Lemon, Ice-tea, Coke) 32. P(Coke) = Ice tea = Lemon = Coke Ice tea Preference Preference π CP = P(Coke) = Ice tea = Lemon = 0 On this HMM, we obtained P(O) = , a significant improvement on the initial P(O) =

34 3.3.2 HMM parameter estimation: Viterbi version 33. Objective: maximize P(O Π (O), µ), where Π (O) is the Viterbi path for the sequence O Idea: Instead of estimating the parameters a ij, b ijk using the expected values of hidden variables (p t (i, j)), estimate them (as Maximum Likelihood), based on the computed Viterbi path. Note: In practice, this method performs poorer than the Forward-Backward (Baum-Welch) main version. However it is widely used, especially when the HMM used is primarily intended to produce Viterbi paths.

35 Proof of the Baum-Welch theorem In the general EM setup (not only that of HMM) Assume some statistical model determined by parameters θ the observed quantities x, and some missing data y that determines/influences the probability of x. The aim is to find the model (in fact, the value of the parameter θ) that maximises the log likelihood log P(x θ) = log y P(x, y θ) Given a valid model θ t, we want to estimate a new and better model θ t+1, i.e. one for which log P(x θ t+1 ) > log P(x θ t )

36 P(x, y θ) Chaining rule = P(y x, θ)p(x θ) log P(x θ) = log P(x, y θ) log P(y x, θ) By multiplying the last equality by P(y x, θ t ) and summing over y, it follows (since y P(y x, θt ) = 1): 35. log P(x θ) = y P(y x, θ t ) log P(x, y θ) y P(y x, θ t ) log P(y x, θ) The first sum will be denoted Q(θ θ t ). Since we want P(y x, θ) larger than P(y x, θ t ), the difference log P(x θ) log P(x θ t ) = Q(θ θ t ) Q(θ t θ t ) + y P(y x, θ t ) log P(y x, θt ) P(y x, θ) should be positive. Note that the last sum is the relative entropy of P(y x, θ t ) with respect to P(y x, θ), therefore it is non-negative. So, log P(x θ) log P(x θ t ) Q(θ θ t ) Q(θ t θ t ) with equality only if θ = θ t, or if P(x θ) = P(x θ t ) for some other θ θ t.

37 36. Taking θ t+1 = argmax θ Q(θ θ t ) will imply log P(x θ t+1 ) log P(x θ t ) 0. (If θ t+1 = θ t, the maximum has been reached.) Note: The function Q(θ θ t ) def. = y P(y x, θt ) log P(x, y θ) is an average of log P(x, y θ) over the distribution of y obtained with the current set of parameters θ t. This [LC: average] can be expressed as a function of θ in which the constants are expectation values in the old model. (See details in the sequel.) The (backbone of) EM algorithm: initialize θ to some arbitrary value θ 0 ; until a certain stop criterion is met, do: xxx E-step: compute the expectations E[y x, θ t ]; calculate the Q function; xxx M-step: compute θ t+1 = argmax θ Q(θ θ t ). Note: Since the likelihood increases at each iteration, the procedure will always reach a local (or maybe global) maximum asymptotically as t.

38 Note: For many models, such as HMM, both of these steps can be carried out analytically. If the second step cannot be carried out exactly, we can use some numerical optimisation technique to maximise Q. In fact, it is enough to make Q(θ t+1 θ t ) > Q(θ t θ t ), thus getting generalised EM algorithms. See [Dempster, Laird, Rubin, 1977], [Meng, Rubin, 1992], [Neal, Hinton, 1993]. 37.

39 Derivation of EM steps for HMM 38. In this case, the missing data are the state paths π. We want to maximize Q(θ θ t ) = π P(π x, θ t ) log P(x, π θ) For a given path, each parameter of the model will appear some number of times in P(x, π θ), computed as usual. We will note this number A kl (π) for transitions and E k (b, π) for emissions. Then, P(x, π θ) = Π M k=1 Π b[e k (b)] E k(b,π) Π M k=0 ΠM l=1 aa kl(π) kl By taking the logarithm in the above formula, it follows Q(θ θ t ) = [ M ] M M P(π x, θ t ) E k (b, π) log e k (b) + A kl (π) log a kl π k=1 b k=0 l=1

40 The expected values A kl and E k (b) can be written as expectations of A kl (π) and E k (b, π) with respect to P(π x, θ t ): 39. E k (b) = π P(π x, θ t )E k (b, π) and A kl = π P(π x, θ t )A kl (π) Therefore, Q(θ θ t ) = M E k (b) log e k (b) + k=1 b M k=0 M A kl log a kl To maximise, let us look first at the A term. The difference between this term for a 0 ij = A ij k A and for any other a ij is ik ( ) M M M A kl log a0 M kl = a 0 kl log a0 kl a kl a kl k=0 l=1 k=0 l A kl The last sum is a relative entropy, and thus it is larger than 0 unless a kl = a 0 kl. This proves that the maximum is at a0 kl. Exactly the same procedure can be used for the E term. l=1 l=1

41 40. For the HMM, the E-step of the EM algorithm consists of calculating the expectations A kl and E k (b). This is done by using the Forward and Backward probabilities. This completely determines the Q function, and the maximum is expressed directly in terms of these numbers. Therefore, the M-step just consists ofplugging A kl and E k (b) into the re-estimation formulae for a kl and e k (b). (See formulae (3.18) in the R. Durbin et al. BSA book.)

42 41. Null (epsilon) emissions 4 HMM extensions Initialization of parameters: improve chances of reaching global optimum Parameter tying: help coping with data sparseness Linear interpolation of HMMs Variable-Memory HMMs Acquiring HMM topologies from data

43 42. 5 Some applications of HMMs Speech Recognition Text Processing: Part Of Speech Tagging Probabilistic Information Retrieval Bioinformatics: genetic sequence analysis

44 Part Of Speech (POS) Tagging Sample POS tags for the Brown/Penn Corpora AT article BEZ is IN preposition JJ adjective JJR adjective: comparative MD modal NN noun: singular or mass NNP noun: singular proper PERIOD.:?! PN personal pronoun RB adverb RBR adverb: comparative TO to VB verb: base form VBD verb: past tense VBG verb: present participle, gerund VBN verb: past participle VBP verb: non-3rd singular present VBZ verb: 3rd singular present WDT wh-determiner (what, which)

45 44. POS Tagging: Methods [Charniak, 1993] Frequency-based: 90% accuracy now considered baseline performance [Schmid, 1994] [Brill, 1995] [Brants, 1998] [Chelba & Jelinek, 1998] Decision lists; artificial neural networks Transformation-based learning Hidden Markov Modelss lexicalized probabilistic parsing (the best!)

46 A fragment of a HMM for POS tagging (from [Charniak, 1997]) 45. P(large) = small = adj 0.45 =1 π det det noun P(a) = the = P(house) = stock = 0.001

47 46. Using HMMs for POS tagging argmax t 1...n P(w 1...n t 1...n )P(t 1...n ) P(t 1...n w 1...n ) = argmax t 1...n P(w 1...n ) = argmax t 1...n P(w 1...n t 1...n )P(t 1...n ) using the two Markov assumptions = argmax t 1...n Π n i=1 P(w i t i )Π n i=1 P(t i t i 1 ) Supervised POS Tagging: MLE estimations: P(w t) = C(w,t) C(t), P(t t ) = C(t,t ) C(t )

48 47. The Treatment of Unknown Words: use apriori uniform distribution over all tags: error rate 40% 20% feature-based estimation [ Weishedel et al., 1993 ]: P((w t) = 1 P(unknown word t)p(capitalized t)p(ending t) Z using both roots and suffixes [Charniak, 1993] Smoothing: P(t w) = C(t,w)+1 C(w)+k w [Church, 1988] where k w is the number of possible tags for w P(t t ) = (1 ǫ) C(t,t ) C(t ) + ǫ [Charniak et al., 1993]

49 48. Fine-tuning HMMs for POS tagging See [ Brants, 1998 ]

50 The Google PageRank Algorithm A Markov Chain worth no. 5 on Forbes list! ( billion USD, as of November 2007)

51 50. Sergey Brin and Lawrence Page introduced Google in 1998, a time when the pace at which the web was growing began to oustrip the ability of current search engines to yield usable results. In developing Google, they wanted to improve the design of search engines by moving it into a more open, academic environment. In addition, they felt that the usage of statistics for their search engine would provide an interesting data set for research. From David Austin, How Google finds your needle in the web s haystack, Monthly Essays on Mathematical Topics, 2006.

52 51. Notations Let n = the number of pages on Internet, and H and A two n n matrices defined by { 1 if page j points to page i (notation: Pj B h ij = i ) 0 otherwise { 1 if page i contains no outgoing links a ij = 0 otherwise α [0; 1] (this is a parameter that was initially set to 0.85) The transition matrix of the Google Markov Chain is G = α(h + A) + 1 α n where 1 is the n n matrix whose entries are all 1 1

53 52. The significance of G is derived from: the Random Surfer model the definition the (relative) importance of a page: combining votes from the pages that point to it I(P i ) = I(P j ) l j P j B i where l j is the number of links pointing out from Pj.

54 53. The PageRank algorithm [Brin & Page, 1998] G is a stochastic matrix (g ij [0; 1], n i=1 g ij = 1), therefore λ 1 the greatest eigenvalue of G is 1, and G has a stationary vector I (i.e., GI = I). G is also primitive ( λ 2 < 1, where λ 2 is the second eigenvalue of G) and irreducible (I > 0). From the matrix calculus it follows that I can be computed using the power method: if I 1 = GI 0, I 2 = GI 1,..., I k = GI k 1 then I k I. I gives the relative importance of pages.

55 54. Suggested readings Using Google s PageRank algorithm to identify important attributes of genes, G.M. Osmani, S.M. Rahman, 2006

56 ADDENDA 55. Formalisation of HMM algorithms in Biological Sequence Analysis [ Durbin et al, 1998 ] Note A begin state was introduced. The transition probability a 0k from this begin state to state k can be thought as the probability of starting in state k. An end state is assumed, which is the reason for a k0 in the termination step. If ends are not modelled, this a k0 will disappear. For convenience we label both begin and end states as 0. There is no conflict because you can only transit out of the begin state and only into the end state, so variables are not used more than once. The emission probabilities are considered independent of the origin state. (Thus te emission of (pairs of) symbols can be seen as being done when reaching the non-end states.) The begin and end states are silent.

57 56. Forward: 1. Initialization (i = 0): f 0 (0) = 1; f k (0) = 0, for k > 0 2. Induction (i = 1...L): f l (i) = e l (x i ) k f k(i 1)a kl 3. Total: P(x) = k f k(l)a k0. Backward: 1. Initialization (i = L): b k (L) = a k0, for all k 2. Induction (i = L 1,..., 1: b k (i) = l a kle l (x i+1 )b l (i + 1) 3. Total: P(x) = l a 0le l (x 1 )b l (1) Combining f and b: P(π k, x) = f k (i)b k (i)

58 57. Viterbi: 1. Initialization (i = 0): v 0 (0) = 1; v k (0) = 0, for k > 0 2. Induction (i = 1...L): v l (i) = e l (x i ) max k (v k (i 1)a kl ); ptr i (l) = argmax k v k (i 1)a kl ) 3. Termination and readout of best path: P(x, π ) = max k (v k (L)a k0 ); πl = argmax k v k (L)a k0, and πi 1 = ptr i(πi ), for i = L...1.

59 Baum-Welch: 1. Initialization: Pick arbitrary model parameters 2. Induction: For each sequence j = 1...n calculate f j k (i) and bj k (i) for sequence j using the forward and respectively backward algorithms. Calculate the expected number of times each transition of emission is used, given the training sequences: A kl = 1 P(x j f j k ) (i)a kle l (x j i+1 )bj l (i + 1) j i E kl = 1 P(x j f j k ) (i)bj k (i) j {i x j i =b} Calculate the new model parameters: a kl = A kl and e k (b) = E k(b) l A kl b E k (b ) Calculate the new log likelihood of the model. 3. Termination: Stop is the change in log likelihood is less than some predefined threshold or the maximum number of iterations is exceeded. 58.