CS344 : Introduction to Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 24- Expressions for alpha and beta probabilities

A Simple HMM

q r

a: 0.3

b: 0.1

a: 0.2

b: 0.1

b: 0.2

b: 0.5

a: 0.2

a: 0.4

Forward or α-probabilities

Let αi(t) be the probability of producing w1,t-1, while ending up in state si

αi(t)= P(w1,t-1,St=si), t>1

Initial condition on αi(t)

αi(t)=

1.0 if i=1

0 otherwise

Probability of the observation using αi(t)

P(w1,n)

=Σ1 σ P(w1,n, Sn+1=si)

= Σi=1 σ αi(n+1)

σ is the total number of states

Recursive expression for ααj(t+1)

=P(w1,t, St+1=sj)

=Σi=1 σ P(w1,t, St=si, St+1=sj)

=Σi=1 σ P(w1,t-1, St=sj)

P(wt, St+1=sj|w1,t-1, St=si)

=Σi=1 σ P(w1,t-1, St=si)

P(wt, St+1=sj|St=si)

= Σi=1 σ αj(t) P(wt, St+1=sj|St=si)

Time Ticks 1 2 3 4 5

INPUT ε b

bb bbb bbba

1.0 0.2 0.05 0.017 0.0148

0.0 0.1 0.07 0.04 0.0131

P(w,t) 1.0 0.3 0.12 0.057 0.0279

)(tq

)(tr

The forward probabilities of “bbba”

Backward or β-probabilities

Let βi(t) be the probability of seeing wt,n, given that the state of the HMM at t is si

βi(t)= P(wt,n,St=si)

Probability of the observation using β

P(w1,n)=β1(1)

Recursive expression for ββj(t-1)

=P(wt-1,n |St-1=sj)

=Σj=1 σ P(wt-1,n, St=si |St-1=si)

=Σi=1 σ P(wt-1, St=sj|St-1=si) P(wt,n,|wt-1,St=sj,

St-1=si)

=Σi=1 σ P(wt-1, St=sj|St-1=si) P(wt,n, |St=sj)

(consequence of Markov Assumption)= Σj=1

σ P(wt-1, St=sj|St-1=si) βj(t)