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

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Outline for our next topic Hidden Markov models – the theory Probabilistic interpretation of alignments using HMMs Later in the course: Applications of HMMs to biological sequence modeling and discovery of features such as genes

Example: The Dishonest Casino A casino has two dice: Fair die P(1) = P(2) = P(3) = P(5) = P(6) = 1/6 Loaded die P(1) = P(2) = P(3) = P(5) = 1/10 P(6) = 1/2 Casino player switches back-&-forth between fair and loaded die once every 20 turns Game: 1.You bet $1 2.You roll (always with a fair die) 3.Casino player rolls (maybe with fair die, maybe with loaded die) 4.Highest number wins $2

Question # 1 – Evaluation GIVEN A sequence of rolls by the casino player QUESTION How likely is this sequence, given our model of how the casino works? This is the EVALUATION problem in HMMs

Question # 2 – Decoding GIVEN A sequence of rolls by the casino player QUESTION What portion of the sequence was generated with the fair die, and what portion with the loaded die? This is the DECODING question in HMMs

Question # 3 – Learning GIVEN A sequence of rolls by the casino player QUESTION How “loaded” is the loaded die? How “fair” is the fair die? How often does the casino player change from fair to loaded, and back? This is the LEARNING question in HMMs

The dishonest casino model FAIRLOADED P(1|F) = 1/6 P(2|F) = 1/6 P(3|F) = 1/6 P(4|F) = 1/6 P(5|F) = 1/6 P(6|F) = 1/6 P(1|L) = 1/10 P(2|L) = 1/10 P(3|L) = 1/10 P(4|L) = 1/10 P(5|L) = 1/10 P(6|L) = 1/2

Definition of a hidden Markov model Definition: A hidden Markov model (HMM) Alphabet  = { b 1, b 2, …, b M } Set of states Q = { 1,..., K } Transition probabilities between any two states a ij = transition prob from state i to state j a i1 + … + a iK = 1, for all states i = 1…K Start probabilities a 0i a 01 + … + a 0K = 1 Emission probabilities within each state e i (b) = P( x i = b |  i = k) e i (b 1 ) + … + e i (b M ) = 1, for all states i = 1…K K 1 … 2

A HMM is memory-less At each time step t, the only thing that affects future states is the current state  t 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 ) K 1 … 2

A parse of a sequence Given a sequence x = x 1 ……x N, A parse of x is a sequence of states  =  1, ……,  N 1 2 K … 1 2 K … 1 2 K … … … … 1 2 K … x1x1 x2x2 x3x3 xKxK 2 1 K 2

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 ) = P(x N,  N | x 1 …x N-1,  1, ……,  N-1 ) P(x 1 …x N-1,  1, ……,  N-1 ) = P(x N,  N |  N-1 ) P(x 1 …x N-1,  1, ……,  N-1 ) = … = P(x N,  N |  N-1 ) P(x N-1,  N-1 |  N-2 )……P(x 2,  2 |  1 ) P(x 1,  1 ) = P(x N |  N ) P(  N |  N-1 ) ……P(x 2 |  2 ) P(  2 |  1 ) P(x 1 |  1 ) P(  1 ) = 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

Example: the dishonest casino Let the sequence of rolls be: x = 1, 2, 1, 5, 6, 2, 1, 6, 2, 4 Then, what is the likelihood of  = Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair, Fair? (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

Example: the dishonest casino So, the likelihood the die is fair in all this run is just  OK, but what is the likelihood of  = Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded, Loaded? ½  P(1 | Loaded) P(Loaded, Loaded) … P(4 | Loaded) = ½  (1/10) 8  (1/2) 2 (0.95) 9 = = 0.79  Therefore, it somewhat more likely that the die is fair all the way, than that it is loaded all the way

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

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 |  ]

Let’s not be confused by notation P[ x | M ]: The probability that sequence x was generated by the model The model is: architecture (#states, etc) + parameters  = a ij, e i (.) So, P[x | M] is the same with P[ x |  ], and P[ x ], when the architecture, and the parameters, respectively, are implied Similarly, P[ x,  | M ], P[ x,  |  ] and P[ x,  ] are the same when the architecture, and the parameters, are implied In the LEARNING problem we always write P[ x |  ] to emphasize that we are seeking the  * that maximizes P[ x |  ]

Problem 1: Decoding Find the best parse of a sequence

Decoding GIVEN x = x 1 x 2 ……x N We want to find  =  1, ……,  N, such that P[ x,  ] is maximized  * = argmax  P[ x,  ] We can use dynamic programming! Let V k (i) = max {  1,…,i-1} P[x 1 …x i-1,  1, …,  i-1, x i,  i = k] = Probability 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

Decoding – main idea 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 Vl(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] ] = e l (x i+1 ) max k a kl V k (i)

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)

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)

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 ]

Example Let x be a 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

Problem 2: Evaluation Find the likelihood a sequence is generated by the model

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

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 = 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 * =

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

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)

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 ) = e k (x i )  l f l (i-1) a lk

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 )

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

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)

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)

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)

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

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)

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?

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