Bioinformatics Hidden Markov Models

Markov Random Processes n A random sequence has the Markov property if its distribution is determined solely by its current state. Any random process having this property is called a Markov random process. n For observable state sequences (state is known from data), this leads to a Markov chain model. n For non-observable states, this leads to a Hidden Markov Model (HMM).

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

The casino models Honest casino Dishonest casino: Crooked casino:

The casino models (only one die) LOADEDFAIR P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 P(1) = 1/10 P(2) = 1/10 P(3) = 1/10 P(4) = 1/10 P(5) = 1/10 P(6) = 1/ L L L L F F F F Honest casino Crooked casino:

The casino models 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/ F L F L F L F L I 0.5 Dishonest casino:

The honest and crooked casino models P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 P(1) = 1/10 P(2) = 1/10 P(3) = 1/10 P(4) = 1/10 P(5) = 1/10 P(6) = 1/ F F F F n Then, what is the likelyhood of π = F, F, F, F, F, F, F, F, F, F? P(x|H)= P(1) P(2) … P(4) = (1/6)^10=1.7x D D D D Let the sequence of rolls be: x = 1, 2, 1, 5, 6, 2, 1, 6, 2, 4 n And the likelyhood of π = L, L, L, L, L, L, L, L, L, L? P(x|C)= P(1 ) … P(4) = (1/10) 8  (1/2) 2 = 2.5  Therefore, it is after all 6.8 times more likely that the model was honest all the way, than that it was crooked all the way.

The honest and crooked casino models P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 P(1) = 1/10 P(2) = 1/10 P(3) = 1/10 P(4) = 1/10 P(5) = 1/10 P(6) = 1/ F F F F n Then, what is the likelyhood of π = F, F, F, F, F, F, F, F, F, F? P(x|H)= P(1) P(2) … P(4) = (1/6)^10=1.7x D D D D Let the sequence of rolls be: x = 1, 6, 6, 5, 6, 2, 6, 6, 3, 6 n And the likelyhood of π = L, L, L, L, L, L, L, L, L, L? P(x|C)= P(1 ) … P(4) = (1/10) 4  (1/2) 6 = 1.6  Therefore, it is after all 98 times more likely that the model was crooked all the way, than that it was honest all the way.

Representation of a HMM Definition: A hidden Markov model (HMM) n Alphabet  = {a,b,c,…} = { b 1, b 2, …, b M } n Set of states Q = { 1,..., q } n Transition probabilities between any two states: p ij = transition prob from state i to state j p i1 + … + p iq = 1, for all states i = 1…q n Start probabilities p 0i such that p 01 + … + p 0q = 1 n Emission probabilities within each state e i (b) = P( x = b | q = i) e i (b 1 ) + … + e i (b M ) = 1, for all states i = 1…q q 1 … 2 a, b, c … 11 1 … … 2 q q … q q I...

General questions Evaluation problem: how likely is this sequence, given our model of how the casino works? GIVEN a HMM Mand a sequence x, FIND Prob[ x | M ] Decoding problem: what portion of the sequence was generated with the fair die, and what portion with the loaded die? GIVENa HMM M, and a sequence x, FINDthe sequence  of states that maximizes P[ x,  | M ] Learning problem: 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? Are there only two dies? GIVENa HMM M, with unspecified transition/emission probs. , and a sequence x, FINDparameters  that maximize P[ x |  ]

Evaluation problem: Forward Algorithm We want to calculate P(x | M) = probability of a sequence x, given the HMM M = Sum over all possible ways of generating x: Given x= 1, 4, 2, 3, 6, 6, 3…, how many ways generate x? only one way ? n Honest casino: n Crooked casino: n Dishonest casino: F F F F D D D D F L F L F L F L I 0.5

Evaluation problem: Forward Algorithm We want to calculate P(x | D) = probability of x, given the HMM D = Sum over all possible ways of generating x: F L F L F L F L I 0.5 Given x= 1, 4, 2, 3, 6, 6, 3…, how many ways generate x?2 |x| Naïve computation is very expensive: given |x| characters and N states, there are N |x| possible state sequences. Even small HMMs, |x|=10 and N=10, contain 10 billion different paths!

Evaluation problem: Forward Algorithm P(x ) = probability of x, given the HMM D = Sum over all possible ways of generating x: =   P(x,  ) =   P(x |  ) P(  ) Then, define f k (i) = P(x 1 …x i,  i = k) (the forward probability) x 1 x 2 x 3 x i a, b, c … 11 1 … … 2 q q … q q I The probability of prefix x 1 x 2 …x i f 1 (i) f 2 (i) f q (i)

Evaluation problem: Forward Algorithm The forward probability recurrence: f k (i) = P(x 1 …x i,  i = k) x 1 x 2 x 3 x i-1 x i a, b, c … 11 1 … … 2 q q … q q I 1 2 q k withf 0 (0) = 1 f k (0) = 0, for all k > 0 and cost space: O(Nq) time: O(Nq 2) =  h=1..q P(x 1 …x i-1,  i-1 = h)p hk e k (x i ) = e k (x i )  h=1..q P(x 1 …x i-1,  i-1 = h)p hk = e k (x i )  h=1..q f h (i-1) p hk f k (i)

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) FLFL f k (i) = e k (x i )  h=1..q f h (i-1) p hk 0 0 1/ / x =

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) FLFL f k (i) = e k (x i )  h=1..q f h (i-1) p hk / / x =

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) FLFL f k (i) = e k (x i )  h=1..q f h (i-1) p hk / / Then P(125) = x =

The dishonest casino model Honest casino Dishonest casino Prob (S | Honest casino model) = exp (-896) Prob (S | Dishonest casino model)= exp (-916) Prob (S | Honest casino model ) = exp (-896) Prob (S | Dishonest casino model )= exp (-847)

General questions Evaluation problem: how likely is this sequence, given our model of how the casino works? Decoding problem: what portion of the sequence was generated with the fair die, and what portion with the loaded die? Learning problem: 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? GIVEN a HMM Mand a sequence x, FIND Prob[ x | M ] GIVENa HMM M, and a sequence x, FINDthe sequence  of states that maximizes P[ x,  | M ] GIVENa HMM M, with unspecified transition/emission probs. , and a sequence x, FINDparameters  that maximize P[ x |  ]

Decoding problem We want to calculate path  * such that  * = argmax π P(x,  | M) = the sequence  of states that maximizes P(x,  | M) F L F L F L F L I 0.5 Naïve computation is very expensive: given |x| characters and N states, there are N |x| possible state sequences.

Evaluation problem: Viterbi algorithm Then, define v k (i) = argmax π P(x 1 …x i,  i = k) x 1 x 2 x 3 x i a, b, c … 11 1 … … 2 q q … q q I The sequence of states that maximizes x 1 x 2 …x i v 1 (i) v 2 (i) v q (i)  * = argmax π P(x,  | M) = the sequence  of states that maximizes P(x,  | M)

Evaluation problem: Viterbi algorithm The forward probability recurrence: v k (i) = argmax π P(x 1 …x i,  i = k) x 1 x 2 x 3 x i-1 x i a, b, c … 11 1 … … 2 q q … q q I 1 2 q k = max h [argmax π P(x 1 …x i-1,  i-1 = h)p hk e k (x i )] = e k (x i ) max h [p hk argmax π P(x 1 …x i-1,  i-1 = h)] = e k (x i ) max h [p hk v h (i-1)] v k (i)

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) = e k (x i ) max h=1..q v h (i-1) p hk 0 0 1/ / x = F 0.05 L f k (i) FLFL

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) = e k (x i ) max h=1..q v h (i-1) p hk / / max(0.013, ) max( ,0.0047) x = F 0.05 L FF LL f k (i) FLFL

The dishonest casino model 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/ F L F L F L F L I 0.5 f k (i) FLFL f k (i) = e k (x i ) max h=1..q v h (i-1) p hk / / max(0.013, ) max( ,0.0047) max(0.0022, ) max( ,00049) x = F 0.05 L FF LL FFF LLL Then, the most probable path is FFF !

The dishonest casino model Dishonest casino sequence of values: Dishonest casino sequence of states:

General questions Evaluation problem: how likely is this sequence, given our model of how the casino works? Decoding problem: what portion of the sequence was generated with the fair die, and what portion with the loaded die? Learning problem: 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? GIVEN a HMM Mand a sequence x, FIND Prob[ x | M ] GIVENa HMM M, and a sequence x, FINDthe sequence  of states that maximizes P[ x,  | M ] GIVENa HMM M, with unspecified transition/emission probs. , and a sequence x, FINDparameters  that maximize P[ x |  ]

Learning problem 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? GIVENa HMM M, with unspecified transition/emission probs. , and a sequence x, FINDparameters  that maximize P[ x |  ] We need a training data set. It could be: n A sequence of pairs (x,π) = (x 1,π 1 ), (x 2,π 2 ), …,(x n,π n ) where we know the set of values and the states. n A sequence of singles x = x 1,x 2, …,x n where we only know the set of values.

Learning problem: given (x,π) i=1..n From the training set we can define: n H ki as the number of times the transition from state k to state i appears in the training set. n J l (x) as the number of times the value x is emitted by state l. For instance, given the training set: Fair die, Loaded die H FF = 51 H FL = H LF = H LL =4426 J F (1) = 10 J F (2) = J F (3)= J F (4) = J F (5) = J F (6) = J L (1) = 0 J L (2) = J L (3)= J L (4) = J L (5) = J L (6) =

And we estimate the parameters of the HMM as n p kl = H ki / (H k1 + … + H kq ). n e l (r) = J l (r) /(J 1 (r) +…+ J q (r) ) Learning problem: given (x,π) i=1..n From the training set we have computed: n H ki as the number of times the transition from state k to state i appears in the training set. n J l (r) as the number of times the value r is emitted by state l. H FF = 51 H FL = 4 H LF = 4 H LL = 26 J F (1) = 10 e F (1)= J F (2) = 11 e F (2)= J F (3)= 9 e F (3)= J F (4) = 12 e F (4)= J F (5) = 8 e F (5)= J F (6) = 6 e F (6)= 2/563/ J L (1) = 0 e L (6)= J L (2) = 5 e L (6)= J L (3)=6 e L (6)= J L (4) = 3 e L (6)= J L (5) = 1 e L (6)= J L (6) = 14 e L (6)= 5/296/ p FF = 51/85=0.6 p FL = p LF = p LL = /29 10/56

Learning problem: given x i=1..n To choose the parameters of HMM that maximize P(x 1 ) x P(x 2 ) x …x P(x n ) that implies The use of standard (iterative) optimization algorithms: n Determine initial parameters values n Iterate until P(x 1 ) x P(x 2 ) x …x P(x n ) becomes smaller that some predeterminated threshold but the algorithm may converge to a point close to a local maximum, not to a global maximum.

Learning problem: algorithm From the training x i=1..n we estimate M 0 : n p ki as the probability of transitions. n e l (r) as the probability of emissions. Do (we have M s ) n Compute H ki as the expected number of times the transition from state k to state I is reached. n Compute J l (r) as the expected number of times the value r is emitted by state l. n Compute p ki =H ki / (H k1 + … + H kq ) and e l (r) = J l (r) /(J 1 (r) +…+ J q (r) ). n { we have M s+1 } Until some value smaller than the threshold { M is close to a local maximum}

Recall forward and backward algorithms The forward probability recurrence: f k (i) = P(x 1 …x i,  i = k) = e k (x i )  h=1..q f h (i-1) x 1 x i-1 x i x i+1 x i+2 x n a, b, c … 1 … 1 2 … 2 … q q I 1 2 q k 1 2 q l 1 … 1 2 … 2 … q q The backward probability recurrence: b l (i+1) = P(x i+1 …x n,  i+1 = l) =  h=1..q p lh e h (x i+2 ) b h (i+2) f k (i) b l (i+1)

Baum-Welch training algorithm J k (r) = the expected number of times the value r is emitted by state k. = ∑ all x ∑ all i Prob(state k emits r at step i in sequence x) = ∑ all x ∑ all i Prob(x 1 …x n ) = ∑ all x ∑ all i Prob(x 1 …x n ) Prob(x 1 …x n | state k emits x i ) δ(r = x i ) f k (i) b k (i) δ(r = x i ) a, b, c … 1 … 1 2 … 2 … q q I 1 2 q k 1 2 q … 1 … 2 … q

Baum-Welch training algorithm H kl (r) = as the expected number of times the transition from k to I is reached = ∑ all x ∑ all i Prob(transition from k to l is reached at step i in x) = ∑ all x ∑ all i Prob(x 1 …x n ) = ∑ all x ∑ all i Prob(x 1 …x n ) Prob(x 1 …x n | state k reaches state l ) f k (i) p kl e l (x i+1 ) b l (i+1) a, b, c … 1 … 1 2 … 2 … q q I 1 2 q k 1 2 q … 1 … 2 … q 1 2 q l

And we estimate the new parameters of the HMM as n p kl = H ki / (H k1 + … + H kq ). n e l (r) = J l (r) /(J 1 (r) +…+ J q (r) ) Baum-Welch training algorithm H ki as the expected number of times the transition from state k to state i appears. H kl (r) = ∑ all x ∑ all i Prob(x 1 …x n ) f k (i) p kl e l (x i+1 ) b l (i+1) J l (r) as the expected number of times the value r is emitted by state l. J l (r) = ∑ all x ∑ all i Prob(x 1 …x n ) f k (i) b k (i) δ(r = x i )

Baum-Welch training algorithm The algorithm has been applied to the sequences:. For |S|=500: n M= 6 N= 2 P0F= P0L= n PFF= PFL= PLL= PLF= n n n pi: n For |S|=50000: n M= 6 N= n n n n