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PatReco: Hidden Markov Models Alexandros Potamianos Dept of ECE, Tech. Univ. of Crete Fall 2004-2005

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Markov Models: Definition Markov chains are Bayesian networks that model sequences of events (states) Sequential events are dependent Two non-sequential events are conditionally independent given the intermediate events (MM-1)

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Markov chains q1 q4q3q2 q0q1q4q3q2 q0q1q4q3q2 q0q1q4q3q2 MM-0 MM-1 MM-2 MM-3 … … … …

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Markov Chains MM-0: P(q 1,q 2.. q N ) = n=1..N P(q n ) MM-1: P(q 1,q 2.. q N ) = n=1..N P(q n |q n-1 ) MM-2: P(q 1,q 2.. q N ) = n=1..N P(q n |q n-1,q n-2 ) MM-3: P(q 1,q 2.. q N ) = n=1..N P(q n |q n-1,q n-2,q n-3 )

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Hidden Markov Models Hidden Markov chains model sequences of events and corresponding sequences of observations Events form an Markov chain (MM-1) Observations are conditionally independent given the sequence of events Each observation is directly connected with a single event (and conditionally independent with the rest of the events in the network)

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Hidden Markov Models q0q1q4q3q2 … o0o1o4o3o2 … P(o 0,o 1..o N, q 0,q 1..q N ) = n=0..N P(q n |q n-1 )P(o n |q n ) HMM-1

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Parameter Estimation The parameters that have to be estimated are the a-priori probabilities P(q 0 ) transition probabilities P(q n |q n-1 ) observation probabilities P(o n |q n ) For example if there are 3 types of events and continuous 1-D observations that follow a Gaussian distribution there are 18 parameters to estimate: 3 a-priori probabilities 3x3 transition probabilities matrix 3 means and 3 variances (observation probabilities)

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Parameter Estimation If both the sequence of events and sequences of observations are fully observable then ML is used Usually the sequence of events q 0,q 1..q N are non-observable in which case EM is used The EM algorithm for HMMs is the Baum- Welsh or forward-backward algorithm

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Inference/Decoding The main inference problem for HMMs is known as the decoding problem: given a sequence of observations find the best sequence of states: q = argmax q P(q|O) = argmax q P(q,O) An efficient decoding algorithm is the Viterbi algorithm

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Viterbi algorithm max q P(q,O) = max q P(o 0,o 1..o N, q 0,q 1..q N ) = max q n=0..N P(q n |q n-1 )P(o n |q n ) = max q N {P(o N |q N ) max q N-1 {P(q N |q N-1 )P(o N-1 |q N-1 ) … max q2 {P(q 3 |q 2 )P(o 2 |q 2 ) max q1 {P(q 2 |q 1 )P(o 1 |q 1 ) max q0 {P(q 1 |q 0 ) P(o 0 |q 0 ) P(q 0 )}}}…}}

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Viterbi algorithm 1 2 3 4 K.... time At each node keep only the best (most probable) path from all the paths passing through that node

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Deep Thoughts HMM-0 (HMM with MM-0 event chain) is the Bayes classifier!!! MMs and HMMs are poor models but simple and efficient computationally How do you fix this? (dependent observations?)

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Some Applications Speech Recognition Optical Character Recognition Part-of-Speech Tagging …

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Conclusions HMMs and MMs are useful modeling tools for dependent sequence of events (states or classes) Efficient algorithms exist for training HMM parameters (Baum-Welsh) and decoding the most probable sequence of states given an observation sequence (Viterbi) HMMs have many applications

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