Hidden Markov Models 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

CS262 Lecture 5, Win06, Batzoglou Hidden Markov Models FAIRLOADED FAIRLOADEDFAIR Alphabet  = { b1, b2, …, bM } Set of states Q = { 1,..., K } Transition probabilities  a ij = transition prob from state i to state j Emission probabilities  e i (b) = P( x i = b |  i = k) Markov Property: P(  t+1 = k | “whatever happened so far”) = P(  t+1 = k |  1,  2, …,  t, x 1, x 2, …, x t )= P(  t+1 = k |  t ) Given a sequence x = x 1 ……x N, A parse of x is a sequence of states  =  1, ……,  N

CS262 Lecture 5, Win06, Batzoglou Likelihood of a Parse Simply, multiply all the orange arrows! (transition probs and emission probs) 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

CS262 Lecture 5, Win06, Batzoglou Likelihood of a parse Given a sequence x = x 1 ……x N and a parse  =  1, ……,  N, To find how likely is the parse: (given our HMM) P(x,  ) = P(x 1, …, x N,  1, ……,  N ) = a 0  1 a  1  2 ……a  N-1  N e  1 (x 1 )……e  N (x N ) 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2 A compact way to write a 0  1 a  1  2 ……a  N-1  N e  1 (x 1 )……e  N (x N ) Number all parameters a ij and e i (b); n params Example: a 0Fair :  1 ; a 0Loaded :  2 ; … e Loaded (6) =  18 Then, count in x and  the # of times each parameter j = 1, …, n occurs F(j, x,  ) = # parameter  j occurs in (x,  ) (call F(.,.,.) the feature counts) Then, P(x,  ) =  j=1…n  j F(j, x,  ) = = exp [  j=1…n log(  j )  F(j, x,  ) ]

CS262 Lecture 5, Win06, Batzoglou Example: the dishonest casino Let the sequence of rolls be: x = 1, 2, 1, 5, 6, 2, 1, 5, 2, 4 Then, what is the likelihood of  = F, F, …, F? (say initial probs a 0Fair = ½, a oLoaded = ½) ½  P(1 | Fair) P(Fair | Fair) P(2 | Fair) P(Fair | Fair) … P(4 | Fair) = ½  (1/6) 10  (0.95) 9 = ~= 0.5  10 -9

CS262 Lecture 5, Win06, Batzoglou Example: the dishonest casino So, the likelihood the die is fair in this run is just  OK, but what is the likelihood of  = L, L, …, L ? ½  P(1 | Loaded) P(Loaded, Loaded) … P(4 | Loaded) = ½  (1/10) 9  (1/2) 1 (0.95) 9 = ~= 0.16  Therefore, it somewhat more likely that all the rolls are done with the fair die, than that they are all done with the loaded die

CS262 Lecture 5, Win06, Batzoglou Example: the dishonest casino Let the sequence of rolls be: x = 1, 6, 6, 5, 6, 2, 6, 6, 3, 6 Now, what is the likelihood  = F, F, …, F? ½  (1/6) 10  (0.95) 9 = 0.5  10 -9, same as before What is the likelihood  = L, L, …, L? ½  (1/10) 4  (1/2) 6 (0.95) 9 = ~= 0.5  So, it is 100 times more likely the die is loaded

CS262 Lecture 5, Win06, Batzoglou The three main questions on HMMs 1.Evaluation GIVEN a HMM M, and a sequence x, FIND Prob[ x | M ] 2.Decoding GIVENa HMM M, and a sequence x, FINDthe sequence  of states that maximizes P[ x,  | M ] 3.Learning GIVENa HMM M, with unspecified transition/emission probs., and a sequence x, FINDparameters  = (e i (.), a ij ) that maximize P[ x |  ]

CS262 Lecture 5, Win06, Batzoglou Notational Conventions The model M is: architecture (#states, etc), parameters  = a ij, e i (.) P[x | M] is the same with P[ x |  ], and P[ x ], when the architecture, and the parameters, respectively, are implied P[ x,  | M ], P[ x,  |  ] and P[ x,  ] are the same when the architecture, and the parameters, are implied LEARNING: we write P[ x |  ] to emphasize that we are seeking the  * that maximizes P[ x |  ]

CS262 Lecture 5, Win06, Batzoglou Problem 1: Decoding Find the most likely parse of a sequence

CS262 Lecture 5, Win06, Batzoglou Decoding GIVEN x = x 1 x 2 ……x N Find  =  1, ……,  N, to maximize P[ x,  ]  * = argmax  P[ x,  ] Maximizes a 0  1 e  1 (x 1 ) a  1  2 ……a  N-1  N e  N (x N ) Dynamic Programming! V k (i) = max {  1…  i-1} P[x 1 …x i-1,  1, …,  i-1, x i,  i = k] = Prob. of most likely sequence of states ending at state  i = k 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2 Given that we end up in state k at step i, maximize product to the left and right

CS262 Lecture 5, Win06, Batzoglou Decoding – main idea Inductive assumption: Given that for all states k, and for a fixed position i, V k (i) = max {  1…  i-1} P[x 1 …x i-1,  1, …,  i-1, x i,  i = k] What is V l (i+1)? From definition, V l (i+1) = max {  1…  i} P[ x 1 …x i,  1, …,  i, x i+1,  i+1 = l ] = max {  1…  i} P(x i+1,  i+1 = l | x 1 …x i,  1,…,  i ) P[x 1 …x i,  1,…,  i ] = max {  1…  i} P(x i+1,  i+1 = l |  i ) P[x 1 …x i-1,  1, …,  i-1, x i,  i ] = max k [ P(x i+1,  i+1 = l |  i =k) max {  1…  i-1} P[x 1 …x i-1,  1,…,  i-1, x i,  i =k] ] = max k [ P(x i+1 |  i+1 = l ) P(  i+1 = l |  i =k) V k (i) ] = e l (x i+1 ) max k a kl V k (i)

CS262 Lecture 5, Win06, Batzoglou The Viterbi Algorithm Input: x = x 1 ……x N Initialization: V 0 (0) = 1(0 is the imaginary first position) V k (0) = 0, for all k > 0 Iteration: V j (i) = e j (x i )  max k a kj V k (i – 1) Ptr j (i) = argmax k a kj V k (i – 1) Termination: P(x,  *) = max k V k (N) Traceback:  N * = argmax k V k (N)  i-1 * = Ptr  i (i)

CS262 Lecture 5, Win06, Batzoglou The Viterbi Algorithm Similar to “aligning” a set of states to a sequence Time: O(K 2 N) Space: O(KN) x 1 x 2 x 3 ………………………………………..x N State 1 2 K V j (i)

CS262 Lecture 5, Win06, Batzoglou Viterbi Algorithm – a practical detail Underflows are a significant problem P[ x 1,…., x i,  1, …,  i ] = a 0  1 a  1  2 ……a  i e  1 (x 1 )……e  i (x i ) These numbers become extremely small – underflow Solution: Take the logs of all values V l (i) = log e k (x i ) + max k [ V k (i-1) + log a kl ]

CS262 Lecture 5, Win06, Batzoglou Example Let x be a long sequence with a portion of ~ 1/6 6’s, followed by a portion of ~ ½ 6’s… x = … … Then, it is not hard to show that optimal parse is (exercise): FFF…………………...F LLL………………………...L 6 characters “123456” parsed as F, contribute.95 6  (1/6) 6 = 1.6  parsed as L, contribute.95 6  (1/2) 1  (1/10) 5 = 0.4  “162636” parsed as F, contribute.95 6  (1/6) 6 = 1.6  parsed as L, contribute.95 6  (1/2) 3  (1/10) 3 = 9.0  10 -5

CS262 Lecture 5, Win06, Batzoglou Problem 2: Evaluation Find the likelihood a sequence is generated by the model

CS262 Lecture 5, Win06, Batzoglou Generating a sequence by the model Given a HMM, we can generate a sequence of length n as follows: 1.Start at state  1 according to prob a 0  1 2.Emit letter x 1 according to prob e  1 (x 1 ) 3.Go to state  2 according to prob a  1  2 4.… until emitting x n 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xnxn 2 1 K 2 0 e 2 (x 1 ) a 02

CS262 Lecture 5, Win06, Batzoglou A couple of questions Given a sequence x, What is the probability that x was generated by the model? Given a position i, what is the most likely state that emitted x i ? Example: the dishonest casino Say x = 12341… …21341 Most likely path:  = FF……F However: marked letters more likely to be L than unmarked letters P(box: FFFFFFFFFFF) = (1/6) 11 * = * 0.54 = P(box: LLLLLLLLLLL) = [ (1/2) 6 * (1/10) 5 ] * * = 1.56*10 -7 * = FF

CS262 Lecture 5, Win06, Batzoglou Evaluation We will develop algorithms that allow us to compute: P(x)Probability of x given the model P(x i …x j )Probability of a substring of x given the model P(  I = k | x)Probability that the i th state is k, given x A more refined measure of which states x may be in

CS262 Lecture 5, Win06, Batzoglou The Forward Algorithm We want to calculate P(x) = probability of x, given the HMM Sum over all possible ways of generating x: P(x) =   P(x,  ) =   P(x |  ) P(  ) To avoid summing over an exponential number of paths , define f k (i) = P(x 1 …x i,  i = k) (the forward probability)

CS262 Lecture 5, Win06, Batzoglou The Forward Algorithm – derivation Define the forward probability: f k (i) = P(x 1 …x i,  i = k) =   1…  i-1 P(x 1 …x i-1,  1,…,  i-1,  i = k) e k (x i ) =  l   1…  i-2 P(x 1 …x i-1,  1,…,  i-2,  i-1 = l) a lk e k (x i ) =  l P(x 1 …x i-1,  i-1 = l) a lk e k (x i ) = e k (x i )  l f l (i-1) a lk

CS262 Lecture 5, Win06, Batzoglou The Forward Algorithm We can compute f k (i) for all k, i, using dynamic programming! Initialization: f 0 (0) = 1 f k (0) = 0, for all k > 0 Iteration: f k (i) = e k (x i )  l f l (i-1) a lk Termination: P(x) =  k f k (N) a k0 Where, a k0 is the probability that the terminating state is k (usually = a 0k )

CS262 Lecture 5, Win06, Batzoglou Relation between Forward and Viterbi VITERBI Initialization: V 0 (0) = 1 V k (0) = 0, for all k > 0 Iteration: V j (i) = e j (x i ) max k V k (i-1) a kj Termination: P(x,  *) = max k V k (N) FORWARD Initialization: f 0 (0) = 1 f k (0) = 0, for all k > 0 Iteration: f l (i) = e l (x i )  k f k (i-1) a kl Termination: P(x) =  k f k (N) a k0

CS262 Lecture 5, Win06, Batzoglou Motivation for the Backward Algorithm We want to compute P(  i = k | x), the probability distribution on the i th position, given x We start by computing P(  i = k, x) = P(x 1 …x i,  i = k, x i+1 …x N ) = P(x 1 …x i,  i = k) P(x i+1 …x N | x 1 …x i,  i = k) = P(x 1 …x i,  i = k) P(x i+1 …x N |  i = k) Then, P(  i = k | x) = P(  i = k, x) / P(x) Forward, f k (i)Backward, b k (i)

CS262 Lecture 5, Win06, Batzoglou The Backward Algorithm – derivation Define the backward probability: b k (i) = P(x i+1 …x N |  i = k) =   i+1…  N P(x i+1,x i+2, …, x N,  i+1, …,  N |  i = k) =  l   i+1…  N P(x i+1,x i+2, …, x N,  i+1 = l,  i+2, …,  N |  i = k) =  l e l (x i+1 ) a kl   i+1…  N P(x i+2, …, x N,  i+2, …,  N |  i+1 = l) =  l e l (x i+1 ) a kl b l (i+1)

CS262 Lecture 5, Win06, Batzoglou The Backward Algorithm We can compute b k (i) for all k, i, using dynamic programming Initialization: b k (N) = a k0, for all k Iteration: b k (i) =  l e l (x i+1 ) a kl b l (i+1) Termination: P(x) =  l a 0l e l (x 1 ) b l (1)

CS262 Lecture 5, Win06, Batzoglou Computational Complexity What is the running time, and space required, for Forward, and Backward? Time: O(K 2 N) Space: O(KN) Useful implementation technique to avoid underflows Viterbi: sum of logs Forward/Backward: rescaling at each position by multiplying by a constant

CS262 Lecture 5, Win06, Batzoglou Posterior Decoding We can now calculate f k (i) b k (i) P(  i = k | x) = ––––––– P(x) Then, we can ask What is the most likely state at position i of sequence x: Define  ^ by Posterior Decoding:  ^ i = argmax k P(  i = k | x)

CS262 Lecture 5, Win06, Batzoglou Posterior Decoding For each state,  Posterior Decoding gives us a curve of likelihood of state for each position  That is sometimes more informative than Viterbi path  * Posterior Decoding may give an invalid sequence of states  Why?

CS262 Lecture 5, Win06, Batzoglou Posterior Decoding P(  i = k | x) =   P(  | x) 1(  i = k) =  {  :  [i] = k} P(  | x) x 1 x 2 x 3 …………………………………………… x N State 1 l P(  i =l|x) k 1(  ) = 1, if  is true 0, otherwise

CS262 Lecture 5, Win06, Batzoglou Viterbi, Forward, Backward VITERBI Initialization: V 0 (0) = 1 V k (0) = 0, for all k > 0 Iteration: V l (i) = e l (x i ) max k V k (i-1) a kl Termination: P(x,  *) = max k V k (N) FORWARD Initialization: f 0 (0) = 1 f k (0) = 0, for all k > 0 Iteration: f l (i) = e l (x i )  k f k (i-1) a kl Termination: P(x) =  k f k (N) a k0 BACKWARD Initialization: b k (N) = a k0, for all k Iteration: b l (i) =  k e l (x i +1) a kl b k (i+1) Termination: P(x) =  k a 0k e k (x 1 ) b k (1)

CS262 Lecture 5, Win06, Batzoglou A+C+G+T+ A-C-G-T- A modeling Example CpG islands in DNA sequences

CS262 Lecture 5, Win06, Batzoglou Methylation & Silencing One way cells differentiate is methylation  Addition of CH 3 in C-nucleotides  Silences genes in region CG (denoted CpG) often mutates to TG, when methylated In each cell, one copy of X is silenced, methylation plays role Methylation is inherited during cell division

CS262 Lecture 5, Win06, Batzoglou Example: CpG Islands CpG nucleotides in the genome are frequently methylated (Write CpG not to confuse with CG base pair) C  methyl-C  T Methylation often suppressed around genes, promoters  CpG islands

CS262 Lecture 5, Win06, Batzoglou Example: CpG Islands In CpG islands,  CG is more frequent  Other pairs (AA, AG, AT…) have different frequencies Question: Detect CpG islands computationally

CS262 Lecture 5, Win06, Batzoglou A model of CpG Islands – (1) Architecture A+C+G+T+ A-C-G-T- CpG Island Not CpG Island

CS262 Lecture 5, Win06, Batzoglou A model of CpG Islands – (2) Transitions How do we estimate the parameters of the model? Emission probabilities: 1/0 1.Transition probabilities within CpG islands Established from many known (experimentally verified) CpG islands (Training Set) 2.Transition probabilities within other regions Established from many known non-CpG islands + ACGT A C G T ACGT A C G T

CS262 Lecture 5, Win06, Batzoglou Log Likehoods— Telling “Prediction” from “Random” ACGT A C G T Another way to see effects of transitions: Log likelihoods L(u, v) = log[ P(uv | + ) / P(uv | -) ] Given a region x = x 1 …x N A quick-&-dirty way to decide whether entire x is CpG P(x is CpG) > P(x is not CpG)   i L(x i, x i+1 ) > 0

CS262 Lecture 5, Win06, Batzoglou A model of CpG Islands – (2) Transitions What about transitions between (+) and (-) states? They affect  Avg. length of CpG island  Avg. separation between two CpG islands XY 1-p 1-q pq Length distribution of region X: P[l X = 1] = 1-p P[l X = 2] = p(1-p) … P[l X = k] = p k (1-p) E[l X ] = 1/(1-p) Geometric distribution, with mean 1/(1-p)

CS262 Lecture 5, Win06, Batzoglou A+C+G+T+ A-C-G-T- 1–p A model of CpG Islands – (2) Transitions No reason to favor exiting/entering (+) and (-) regions at a particular nucleotide To determine transition probabilities between (+) and (-) states 1.Estimate average length of a CpG island: l CPG = 1/(1-p)  p = 1 – 1/l CPG 2.For each pair of (+) states k, l, let a kl  p × a kl 3.For each (+) state k, (-) state l, let a kl = (1-p)/4 (better: take frequency of l in the (-) regions into account) 4.Do the same for (-) states A problem with this model: CpG islands don’t have exponential length distribution This is a defect of HMMs – compensated with ease of analysis & computation

CS262 Lecture 5, Win06, Batzoglou Applications of the model Given a DNA region x, The Viterbi algorithm predicts locations of CpG islands Given a nucleotide x i, (say x i = A) The Viterbi parse tells whether x i is in a CpG island in the most likely general scenario The Forward/Backward algorithms can calculate P(x i is in CpG island) = P(  i = A + | x) Posterior Decoding can assign locally optimal predictions of CpG islands  ^ i = argmax k P(  i = k | x)

CS262 Lecture 5, Win06, Batzoglou What if a new genome comes? We just sequenced the porcupine genome We know CpG islands play the same role in this genome However, we have no known CpG islands for porcupines We suspect the frequency and characteristics of CpG islands are quite different in porcupines How do we adjust the parameters in our model? LEARNING