Hidden Markov Models

by Nikolai Shokhirev

Up Algorithms

Introduction

Hidden Markov models are widely used in science, engineering and many other areas (speech recognition, optical character recognition, machine translation, bioinformatics, computer vision, finance and economics, and in social science). 

Definition: The Hidden Markov Model (HMM) is a variant of a finite state machine having a set of hidden states, Q, an output alphabet (observations), O, transition probabilities, A, output (emission) probabilities, B, and initial state probabilities, Π. The current state is not observable. Instead, each state produces an output with a certain probability (B). Usually the states, Q, and outputs, O, are understood, so an HMM is said to be a triple, ( A, B, Π ).

Formal Definition:

Hidden states Q = { qi }, i = 1, . . . , N .

Transition probabilities A = {aij = P(qj at t +1 | qi at t)}, where P(a | b) is the conditional probability of a given b, t = 1, . . . , T is time, and qi in Q. Informally, A is the probability that the next state is qj given that the current state is qi.

Observations (symbols) O = { ok }, k = 1, . . . , M .

Emission probabilities B = { bik = bi(ok) = P(ok | qi) }, where ok in O. Informally, B is the probability that the output is ok given that the current state is qi.

Initial state probabilities Π = {pi = P(qi at t = 1)}.

HMM

The model is characterized by the complete set of parameters: Λ = {A, B, Π }.

Canonical problems

There are 3 canonical problems to solve with HMMs:

  1. Given the model parameters, compute the probability of a particular output sequence. This problem is solved by the Forward and Backward algorithms (described below).
  2. Given the model parameters, find the most likely sequence of (hidden) states which could have generated a given output sequence. Solved by the Viterbi algorithm and Posterior decoding.
  3. Given an output sequence, find the most likely set of state transition and output probabilities. Solved by the Baum-Welch algorithm

Forward Algorithm

Let αt(i) be the probability of the partial observation sequence Ot = {o(1), o(2), ... , o(t)} to be produced by all possible state sequences that end at the i-th state. 

αt(i) = P(o(1), o(2), ... , o(t) | q(t) = qi ).

Then the unconditional probability of the partial observation sequence is the sum of αt(i) over all N states. 

Observed and hidden sequences

The Forward Algorithm is a recursive algorithm for calculating αt(i) for the observation sequence of increasing length t . First, the probabilities for the single-symbol sequence are calculated as a product of initial i-th state probability and emission probability of the given symbol o(1) in the i-th state. Then the recursive formula is applied. Assume we have calculated αt(i) for some t. To calculate αt+1(j), we multiply every αt(i) by the corresponding transition probability from the i-th state to the j-th state, sum the products over all states, and then multiply the result by the emission probability of the symbol o(t+1). Iterating the process, we can eventually calculate αT(i), and then summing them over all states, we can obtain the required probability.

Formal Definition

Initialization:

  α1(i) = pi bi(o(1)) , i =1, ... , N

Recursion:

here  i =1, ... , N , t =1, ... , T - 1

Termination:

Backward Algorithm

In a similar manner, we can introduce a symmetrical backward variable βt(i) as the conditional probability of the partial observation sequence from o(t+1) to the end to be produced by all state sequences that start at i-th state (3.13). 

βt(i) = P(o(t+1), o(t+2), ... , o(T) | q(t) = qi ).

The Backward Algorithm calculates recursively backward variables going backward along the observation sequence. The Forward Algorithm is typically used for calculating the probability of an observation sequence to be emitted by an HMM, but, as we shall see later, both procedures are heavily used for finding the optimal state sequence and estimating the HMM parameters.

Formal Definition

Initialization:

  βT (i) = 1 , i =1, ... , N

According to the above definition, βT(i) does not exist. This is a formal extension of the below recursion to t = T.

Recursion:

here  i =1, ... , N , t = T - 1,  T - 2 , . . . , 1

Termination:

Obviously both Forward and Backward algorithms must give the same results for total probabilities P(O) = P(o(1), o(2), ... , o(T) ).

 

Posterior decoding

There are several possible criteria for finding the most likely sequence of hidden states. One is to choose states that are individually most likely at the time when a symbol is emitted. This approach is called posterior decoding. 

Let λ t(i) be the probability of the model to emit the symbol o(t) being in the i-th state for the given observation sequence O

λ t(i) = P( q(t) = qi | O ).

It is easy to derive that

λ t(i) = αt(i) βt(i) / P( O ) , i =1, ... , N , t =1, ... , T

Then at each time we can select the state q(t) that maximizes λ t(i). 

q(t) = arg max {λ t(i)}

Posterior decoding works fine in the case when HMM is ergodic, i.e. there is transition from any state to any other state. If applied to an HMM of another architecture, this approach could give a sequence that may not be a legitimate path because some transitions are not permitted. 

Viterbi algorithm

The Viterbi algorithm chooses the best state sequence that maximizes the likelihood of the state sequence for the given observation sequence.

Let δ t(i) be the maximal probability of state sequences of the length t that end in state i and produce the t first observations for the given model.

δ t(i) = max{P(q(1), q(2), ..., q(t-1) ; o(1), o(2), ... , o(t) | q(t) = qi ).}

 The Viterbi algorithm is a dynamic programming algorithm that uses the same schema as the Forward algorithm except for two differences:

  1. It uses maximization in place of summation at the recursion and termination steps.
  2. It keeps track of the arguments that maximize δ t(i) for each t and i, storing them in the N by T matrix ψ. This matrix is used to retrieve the optimal state sequence at the backtracking step.

Initialization:

δ1(i) = pi bi(o(1)) 
ψ1(i) = 0 , i =1, ... , N

According to the above definition, βT(i) does not exist. This is a formal extension of the below recursion to .

Recursion:

δt ( j) = max i [δt - 1(i) aij] b j (o(t)) 
ψt( j) = arg max i [δt - 1(i) aij]  

Termination:

p* = max i [δT( i )] 
q*T = arg max i [δT( i )]  

Path (state sequence) backtracking:

 q*t = ψt+1( q*t+1) , t = T - 1,  T - 2 , . . . , 1

 

Baum-Welch algorithm

Let us define ξ t(i, j), the joint probability of being in state q i at time t and state q j at time t +1 , given the model and the observed sequence:

 ξ t(i, j) = P(q(t) = q i, q(t+1) = q j | O, Λ)

The figure below illustrates the calculation of  ξ t(i, j).

Joint probability paths.

Therefore we get

The probability of output sequence can be expressed as

The probability of being in state q i at time t:

Estimates

Initial probabilities:

Transition probabilities:

Emission probabilities:

In the above equation Σ * denotes the sum over t so that o(t) = ok.

Implementation

In progress ...

Source code

Forward abd Backward Algorithms , Viterbi Algorithm , Posterior decoding, and Baum-Welch Algorithm is available here (Delphi code - uHMM.pas) or here Forward  Backward and Viterbi Algorithm , Posterior decoding (C++ code - HMM.h and HMM.cpp). The code is not optimized (for educational purpose only).

Kamen Kelchev ( kkelchev@bulsyst.com ) refactored uHMM.pas and made a testing program for the Baum-Welch Algorithm. The code is available in the download section below.

References

  1. Hidden Markov model, http://www.nist.gov/dads/HTML/hiddenMarkovModel.html 
  2. Hidden Markov model, http://en.wikipedia.org/wiki/Hidden_Markov_model
  3. Lawrence R. Rabiner, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77 (2), p. 257286, February 1989.
  4. V. Petrushin. Hidden Markov Models: Fundamentals and Applications. Part 2: Discrete and Continuous Hidden Markov Models, Online Symposium for Electronics Engineers. 2000 
  5. Narada Warakagoda, Baum-Welch Algorithm. http://jedlik.phy.bme.hu/~gerjanos/HMM/node11.html

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ŠNikolai Shokhirev, 2001-2010