Hidden Markov Models David Meir Blei November 1, 1999

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Hidden Markov Models David Meir Blei November 1, 1999. Graphical Model Circles indicate states Arrows indicate probabilistic dependencies between states. What is an HMM?. Green circles are hidden states Dependent only on the previous state

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Hidden Markov Models

David Meir Blei

November 1, 1999

Graphical Model

Circles indicate states

Arrows indicate probabilistic dependencies between states

What is an HMM?
Green circles are hidden states

Dependent only on the previous state

“The past is independent of the future given the present.”

What is an HMM?
Purple nodes are observed states

Dependent only on their corresponding hidden state

What is an HMM?
{S, K, P, A, B}

S : {s1…sN } are the values for the hidden states

K : {k1…kM } are the values for the observations

HMM Formalism

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{S, K, P, A, B}

P = {pi} are the initial state probabilities

A = {aij} are the state transition probabilities

B = {bik} are the observation state probabilities

HMM Formalism

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S

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Compute the probability of a given observation sequence

Given an observation sequence, compute the most likely hidden state sequence

Given an observation sequence and set of possible models, which model most closely fits the data?

Inference in an HMM

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Decoding

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Given an observation sequence and a model, compute the probability of the observation sequence

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Decoding

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Decoding

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Decoding

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Decoding

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Decoding

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Forward Procedure
• Special structure gives us an efficient solution using dynamic programming.
• Intuition: Probability of the first t observations is the same for all possible t+1 length state sequences.
• Define:

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Forward Procedure

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Forward Procedure

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Forward Procedure

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Forward Procedure

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Forward Procedure

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Forward Procedure

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Forward Procedure

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Forward Procedure
Backward Procedure

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Probability of the rest of the states given the first state

Decoding Solution

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Forward Procedure

Backward Procedure

Combination

Viterbi Algorithm

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The state sequence which maximizes the probability of seeing the observations to time t-1, landing in state j, and seeing the observation at time t

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Viterbi Algorithm

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Recursive Computation

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Viterbi Algorithm

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Compute the most likely state sequence by working backwards

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Parameter Estimation

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• Given an observation sequence, find the model that is most likely to produce that sequence.
• No analytic method
• Given a model and observation sequence, update the model parameters to better fit the observations.

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Parameter Estimation

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Probability of traversing an arc

Probability of being in state i

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Parameter Estimation

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Now we can compute the new estimates of the model parameters.

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The Most Important Thing

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We can use the special structure of this model to do a lot of neat math and solve problems that are otherwise not solvable.