Introduction to Machine Learning CMU-10701 Hidden Markov Models Barnabás Póczos & Aarti Singh Slides courtesy: Eric Xing
i.i.d to sequential data So far we assumed independent, identically distributed data Sequential (non i.i.d.) data Time-series data E.g. Speech Characters in a sentence Base pairs along a DNA strand 2
Markov Models Joint distribution of n arbitrary random variables Chain rule Markov Assumption (m th order) Current observation only depends on past m observations 3
Markov Models Markov Assumption 1 st order 2 nd order 4
Markov Models Markov Assumption 1 st order # parameters in stationary model K-ary variables O(K 2 ) m th order O(K m+1 ) n-1 th order O(K n ) no assumptions complete (but directed) graph Homogeneous/stationary Markov model (probabilities don t depend on n) 5
Hidden Markov Models Distributions that characterize sequential data with few parameters but are not limited by strong Markov assumptions. S 1 S 2 S T-1 S T O 1 O 2 O T-1 O T Observation space O t ϵ {y 1, y 2,, y K } Hidden states S t ϵ {1,, I} 6
Hidden Markov Models S 1 S 2 S T-1 S T O 1 O 2 O T-1 O T 1 st order Markov assumption on hidden states {S t } t = 1,, T (can be extended to higher order). Note: O t depends on all previous observations {O t-1, O 1 } 7
Hidden Markov Models Parameters stationary/homogeneous markov model (independent of time t) Initial probabilities p(s 1 = i) = π i S 1 S 2 S T-1 S T O 1 O 2 O T-1 O T Transition probabilities p(s t = j S t-1 = i) = p ij Emission probabilities p(o t = y S t = i) = 8
HMM Example The Dishonest Casino A casino has two dices: Fair dice P(1) = P(2) = P(3) = P(5) = P(6) = 1/6 Loaded dice P(1) = P(2) = P(3) = P(5) = 1/10 P(6) = ½ Casino player switches back-&- forth between fair and loaded die with 5% probability 9
HMM Problems 10
HMM Example L F F F L L L F 11
State Space Representation Switch between F and L with 5% probability 0.05 0.95 L F 0.95 0.05 HMM Parameters Initial probs P(S 1 = L) = 0.5 = P(S 1 = F) Transition probs P(S t = L/F S t-1 = L/F) = 0.95 P(S t = F/L S t-1 = L/F) = 0.05 Emission probabilities P(O t = y S t = F) = 1/6 y = 1,2,3,4,5,6 P(O t = y S t = L) = 1/10 y = 1,2,3,4,5 = 1/2 y = 6 12
Three main problems in HMMs Evaluation Given HMM parameters & observation seqn find prob of observed sequence Decoding Given HMM parameters & observation seqn find most probable sequence of hidden states Learning Given HMM with unknown parameters and observation sequence find likelihood of observed data parameters that maximize 13
HMM Algorithms Evaluation What is the probability of the observed sequence? Forward Algorithm Decoding What is the probability that the third roll was loaded given the observed sequence? Forward-Backward Algorithm What is the most likely die sequence given the observed sequence? Viterbi Algorithm Learning Under what parameterization is the observed sequence most probable? Baum-Welch Algorithm (EM) 14
Evaluation Problem Given HMM parameters & observation sequence find probability of observed sequence S 1 S 2 S T-1 S T O 1 O 2 O T-1 O T requires summing over all possible hidden state values at all times K T exponential # terms! Instead: αk T Compute recursively 15
Forward Probability Compute forward probability αk t recursively over t S 1 S t-1 S t... Introduce S t-1 O t-1 O t Chain rule Markov assumption O 1 16
Forward Algorithm Can compute α t k for all k, t using dynamic programming: Initialize: α 1 k = p(o 1 S 1 = k) p(s 1 = k) for all k Iterate: for t = 2,, T α t k = p(o t S t = k) α t-1 p(s t = k S t-1 = i) i i for all k Termination: = α T k k 17
Decoding Problem 1 Given HMM parameters & observation sequence find probability that hidden state at time t was k Compute recursively αk t β k t S 1 S t-1 S t S t+1 S T-1 S T O t-1 O t O t+1 O 1 O T-1 O T 18
Backward Probability Compute forward probability β k t recursively over t S t S t+1 S t+2 S T... Introduce S t+1 O t O t+1 Chain rule Markov assumption O t+2 O T 19
Backward Algorithm Can compute β t k for all k, t using dynamic programming: Initialize: β T k = 1 for all k Iterate: for t = T-1,, 1 for all k Termination: 20
Most likely state vs. Most likely sequence Most likely state assignment at time t E.g. Which die was most likely used by the casino in the third roll given the observed sequence? Most likely assignment of state sequence E.g. What was the most likely sequence of die rolls used by the casino given the observed sequence? Not the same solution! MLA of x? MLA of (x,y)? 21
Decoding Problem 2 Given HMM parameters & observation sequence find most likely assignment of state sequence Vk T Vk T Compute recursively - probability of most likely sequence of states ending at state S T = k 22
Viterbi Decoding Compute probability Vk t recursively over t... Bayes rule Markov assumption S 1 O 1 S t-1 O t-1 S t O t 23
Viterbi Algorithm Can compute V t k for all k, t using dynamic programming: Initialize: V 1 k = p(o 1 S 1 =k)p(s 1 = k) for all k Iterate: for t = 2,, T for all k Termination: Traceback: 24
Computational complexity What is the running time for Forward, Backward, Viterbi? O(K 2 T) linear in T instead of O(K T ) exponential in T! 25
Learning Problem Given HMM with unknown parameters and observation sequence find parameters that maximize likelihood of observed data hidden variables state sequence But likelihood doesn t factorize since observations not i.i.d. EM (Baum-Welch) Algorithm: E-step Fix parameters, find expected state assignments M-step Fix expected state assignments, update parameters 26
Baum-Welch (EM) Algorithm Start with random initialization of parameters E-step Fix parameters, find expected state assignments Forward-Backward algorithm 27
Baum-Welch (EM) Algorithm Start with random initialization of parameters E-step -1 = expected # times in state i = expected # transitions from state i M-step = expected # transitions from state i to j 28
Some connections HMM vs Linear Dynamical Systems (Kalman Filters) HMM: States are Discrete Observations Discrete or Continuous Linear Dynamical Systems: Observations and States are multivariate Gaussians whose means are linear functions of their parent states (see Bishop: Sec 13.3) 29
HMMs.. What you should know Useful for modeling sequential data with few parameters using discrete hidden states that satisfy Markov assumption Representation - initial prob, transition prob, emission prob, State space representation Algorithms for inference and learning in HMMs Computing marginal likelihood of the observed sequence: forward algorithm Predicting a single hidden state: forward-backward Predicting an entire sequence of hidden states: viterbi Learning HMM parameters: an EM algorithm known as Baum- Welch 30