CS344 : Introduction to Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 24- Expressions for alpha and beta probabilities

A Simple HMM r q a: 0.2 a: 0.3 b: 0.2 b: 0.1 a: 0.2 b: 0.1 b: 0.5

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) 1.0 if i=1 αi(t)= 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 α αj(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)

The forward probabilities of “bbba” 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) 0.3 0.12 0.057 0.0279

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 β βj(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)