Forward-backward algorithm

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Forward-backward algorithm. LING 572 Fei Xia 02/23/06. Outline. Forward and backward probability Expected counts and update formulae Relation with EM. HMM. A HMM is a tuple : A set of states S={s 1 , s 2 , …, s N }. A set of output symbols Σ ={w 1 , …, w M }.

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Forward-backward algorithm

LING 572

Fei Xia

02/23/06

Outline
• Forward and backward probability
• Expected counts and update formulae
• Relation with EM
HMM
• A HMM is a tuple :
• A set of states S={s1, s2, …, sN}.
• A set of output symbols Σ={w1, …, wM}.
• Initial state probabilities
• State transition prob: A={aij}.
• Symbol emission prob: B={bijk}
• State sequence: X1…XT+1
• Output sequence: o1…oT

oT

o1

o2

XT+1

XT

X1

X2

Decoding
• Given the observation O1,T=o1…oT, find the state sequence X1,T+1=X1 … XT+1 that maximizes P(X1,T+1 | O1,T).

 Viterbi algorithm

Notation
• A sentence: O1,T=o1…oT,
• T is the sentence length
• The state sequence X1,T+1=X1 … XT+1
• t: time t, range from 1 to T+1.
• Xt: the state at time t.
• i, j: state si, sj.
• k: word wk in the vocabulary

Forward and backward probabilities

Forward probability

The probability of producing oi,t-1 while ending up in state si:

Calculating forward probability

Initialization:

Induction:

Backward probability
• The probability of producing the sequence Ot,T, given that at time t, we are at state si.
Calculating backward probability

Initialization:

Induction:

Estimating parameters
• The prob of traversing a certain arc at time t given O: (denoted by pt(i, j) in M&S)
Expected counts

Sum over the time index:

• Expected # of transitions from state i to j in O:
• Expected # of transitions from state i in O:
Emission probabilities

Arc-emission HMM:

The inner loop for forward-backward algorithm

Given an input sequence and

• Calculate forward probability:
• Base case
• Recursive case:
• Calculate backward probability:
• Base case:
• Recursive case:
• Calculate expected counts:
• Update the parameters:

Relation to EM

Relation to EM
• HMM is a PM (Product of Multi-nominal) Model
• Forward-back algorithm is a special case of the EM algorithm for PM Models.
• X (observed data): each data point is an O1T.
• Y (hidden data): state sequence X1T.
• Θ (parameters): aij, bijk, πi.
Iterations
• Each iteration provides values for all the parameters
• The new model always improve the likeliness of the training data:
• The algorithm does not guarantee to reach global maximum.
Summary
• A way of estimating parameters for HMM
• Define forward and backward probability, which can calculated efficiently (DP)
• Given an initial parameter setting, we re-estimate the parameters at each iteration.
• The forward-backward algorithm is a special case of EM algorithm for PM model