. Computational Genomics Lecture 10 Hidden Markov Models (HMMs) © Ydo Wexler & Dan Geiger (Technion) and by Nir Friedman (HU) Modified by Benny Chor (TAU)

2 Outline u Finite, or Discrete, Markov Models u Hidden Markov Models u Three major questions: u Q1.: Computing the probability of a given observation. A1.: Forward – Backward (Baum Welch) dynamic programming algorithm. u Q2.: Computing the most probable sequence, given an observation. A2.: Viterbi’s dynamic programming Algorithm u Q3.: Learn best model, given an observation,. A3.: Expectation Maximization (EM): A Heuristic.

3 Markov Models u A discrete (finite) system: l N distinct states. l Begins (at time t=1) in some initial state(s). l At each time step (t=1,2,…) the system moves from current to next state (possibly the same as the current state) according to transition probabilities associated with current state. u This kind of system is called a finite, or discrete Markov model u After Andrei Andreyevich Markov ( )

4 Outline u Markov Chains (Markov Models) u Hidden Markov Chains (HMMs) u Algorithmic Questions u Biological Relevance

5 Discrete Markov Model: Example u Discrete Markov Model with 5 states. u Each a ij represents the probability of moving from state i to state j u The a ij are given in a matrix A = {a ij }  The probability to start in a given state i is  i, The vector  repre- sents these start probabilities.

6 Markov Property Markov Property: The state of the system at time t+1 depends only on the state of the system at time t X t=1 X t=2 X t=3 X t=4 X t=5

7 Markov Chains Stationarity Assumption Probabilities independent of t when process is “stationary” So, This means that if system is in state i, the probability that the system will next move to state j is p ij, no matter what the value of t is

8 raining today rain tomorrow p rr = 0.4 raining today no rain tomorrow p rn = 0.6 no raining today rain tomorrow p nr = 0.2 no raining today no rain tomorrow p rr = 0.8 Simple Minded Weather Example

9 Transition matrix for our example Note that rows sum to 1 Such a matrix is called a Stochastic Matrix If the rows of a matrix and the columns of a matrix all sum to 1, we have a Doubly Stochastic Matrix

10 Coke vs. Pepsi Given that a person’s last cola purchase was Coke ™, there is a 90% chance that her next cola purchase will also be Coke ™. If that person’s last cola purchase was Pepsi™, there is an 80% chance that her next cola purchase will also be Pepsi™. coke pepsi

11 Coke vs. Pepsi Given that a person is currently a Pepsi purchaser, what is the probability that she will purchase Coke two purchases from now? The transition matrices are: (corresponding to one purchase ahead)

12 Coke vs. Pepsi Given that a person is currently a Coke drinker, what is the probability that she will purchase Pepsi three purchases from now?

13 Coke vs. Pepsi Assume each person makes one cola purchase per week. Suppose 60% of all people now drink Coke, and 40% drink Pepsi. What fraction of people will be drinking Coke three weeks from now? Let (Q 0,Q 1 )=(0.6,0.4) be the initial probabilities. We will regard Coke as 0 and Pepsi as 1 We want to find P(X 3 =0) P 00

14 Equilibrium (Stationary) Distribution coke pepsi u Suppose 60% of all people now drink Coke, and 40% drink Pepsi. What fraction will be drinking Coke 10,100,1000,10000 … weeks from now? u For each week, probability is well defined. But does it converge to some equilibrium distribution [p 0,p 1 ]? u If it does, then eqs. : 0.9 p p 1 = p 0, 0.8 p p 0 = p 1 must hold, yielding p 0 = 2/3, p 1 =1/3.

15 Simulation: Markov Process Coke vs. Pepsi Example (cont) week - i Pr[X i = Coke] 2/3 stationary distribution coke pepsi

16 Equilibrium (Stationary) Distribution Whether or not there is a stationary distribution, and whether or not it is unique if it does exist, are determined by certain properties of the process. Irreducible means that every state is accessible from every other state. Aperiodic means that there exists at least one state for which the transition from that state to itself is possible. Positive recurrent means that the expected return time is finite for every state. coke pepsi

17 Equilibrium (Stationary) Distribution u If the Markov chain is positive recurrent, there exists a stationary distribution. If it is positive recurrent and irreducible, there exists a unique stationary distribution, and furthermore the process constructed by taking the stationary distribution as the initial distribution is ergodic. Then the average of a function f over samples of the Markov chain is equal to the average with respect to the stationary distribution,ergodic

18 Equilibrium (Stationary) Distribution u Writing P for the transition matrix, a stationary distribution is a vector π which satisfies the equation l Pπ = π. u In this case, the stationary distribution π is an eigenvector of the transition matrix, associated with the eigenvalue 1. eigenvectoreigenvalue

19 Discrete Markov Model - Example u States – Rainy:1, Cloudy:2, Sunny:3 u Matrix A – u Problem – given that the weather on day 1 (t=1) is sunny(3), what is the probability for the observation O:

20 Discrete Markov Model – Example (cont.) u The answer is -

21 Types of Models u Ergodic model Strongly connected - directed path w/ positive probabilities from each state i to state j (but not necessarily complete directed graph)

22 Third Example: A Friendly Gambler Game starts with 10$ in gambler’s pocket – At each round we have the following: Gambler wins 1$ with probability p Gambler loses 1$ with probability 1-p – Game ends when gambler goes broke (no sister in bank), or accumulates a capital of 100$ (including initial capital) – Both 0$ and 100$ are absorbing states 01 2 N-1 N p p p p 1-p Start (10$) or

23 Fourth Example: A Friendly Gambler 01 2 N-1 N p p p p 1-p Start (10$) Irreducible means that every state is accessible from every other state. Aperiodic means that there exists at least one state for which the transition from that state to itself is possible. Positive recurrent means that the expected return time is finite for every state. If the Markov chain is positive recurrent, there exists a stationary distribution. Is the gambler’s chain positive recurrent? Does it have a stationary distribution (independent upon initial distribution)?

24 Let Us Change Gear u Enough with these simple Markov chains. u Our next destination: Hidden Markov chains. 0.9 Fair loaded head tail /2 1/4 3/4 1/2 Start 1/2

25 Hidden Markov Models (probabilistic finite state automata) Often we face scenarios where states cannot be directly observed. We need an extension: Hidden Markov Models a 11 a 22 a 33 a 44 a 12 a 23 a 34 b 11 b 14 b 12 b Observed phenomenon a ij are state transition probabilities. b ik are observation (output) probabilities. b 11 + b 12 + b 13 + b 14 = 1, b 21 + b 22 + b 23 + b 24 = 1, etc.

26 Hidden Markov Models - HMM H1H1 H2H2 H L-1 HLHL X1X1 X2X2 X L-1 XLXL HiHi XiXi Hidden variables Observed data

27 Example: Dishonest Casino Actually, what is hidden in this model?

fair loaded H H T T /2 1/4 3/41/2 Hidden Markov Models - HMM Coin-Tossing Example Fair/Loaded Head/Tail X1X1 X2X2 X L-1 XLXL XiXi H1H1 H2H2 H L-1 HLHL HiHi transition probabilities emission probabilities Q1.: What is the probability of the sequence of observed outcome (e.g. HHHTHTTHHT), given the model?

29 H1 H2HL-1HL X1 X2XL-1XL Hi Xi L tosses Fair/Load ed Head/Tail 0.9 Fair loaded head tail /2 1/4 3/4 1/2 Start 1/2 Loaded Coin Example (cont.) Q1.: What is the probability of the sequence of observed outcome (e.g. HHHTHTTHHT), given the model?

30 HMMs – Question I  Given an observation sequence O = ( O 1 O 2 O 3 … O L ), and a model M = {A, B,   }  how do we efficiently compute P( O | M ), the probability that the given model M produces the observation O in a run of length L ? u This probability can be viewed as a measure of the quality of the model M. Viewed this way, it enables discrimination/selection among alternative models M 1, M 2, M 3 …

31 HMM Recognition (question I) u For a given model M = { A, B, p} and a given state sequence Q 1 Q 2 Q 3 … Q L,, the probability of an observation sequence O 1 O 2 O 3 … O L is P(O|Q,M) = b Q1O1 b Q2O2 b Q3O3 … b QTOT u For a given hidden Markov model M = { A, B, p} the probability of the state sequence Q 1 Q 2 Q 3 … Q L is (the initial probability of Q 1 is taken to be  Q1 ) P(Q|M) = p Q1 a Q1Q2 a Q2Q3 a Q3Q4 … a QL-1QL u So, for a given HMM, M the probability of an observation sequence O 1 O 2 O 3 … O T is obtained by summing over all possible state sequences

32 HMM – Recognition (cont.) P(O| M) =  Q  P(O|Q) P(Q|M) =  Q  Q 1 b Q 1 O 1 a Q 1 Q 2 b Q 2 O 2 a Q 2 Q 3 b Q 2 O 2 … u Requires summing over exponentially many paths u Can this be made more efficient?

33 HMM – Recognition (cont.) u Why isn’t it efficient? – O(2LQ L ) l For a given state sequence of length L we have about 2L calculations  P(Q|M) =  Q 1 a Q 1 Q 2 a Q 2 Q 3 a Q 3 Q 4 … a Q T-1 Q T H P(O|Q) = b Q 1 O 1 b Q 2 O 2 b Q 3 O 3 … b Q T O T l There are Q L possible state sequence l So, if Q=5, and L=100, then the algorithm requires 200x5 100 computations l Instead, we will use the forward-backward (F-B) algorithm of Baum (68) to do things more efficiently.

34 1. Compute the posteriori belief in H i (specific i) given the evidence {x 1,…,x L } for each of H i ’s values h i, namely, compute p(h i | x 1,…,x L ). 2. Do the same computation for every H i but without repeating the first task L times. Coin-Tossing Example Seeing the set of outcomes {x 1,…,x L }, compute p(loaded | x 1,…,x L ) for each coin toss Q.: what is the most likely sequence of values in the H-nodes to generate the observed data?