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

2. 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

3. 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

4. Initial condition on αi(t)
1.0 if i=1
αi(t)=
0 otherwise

5. 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

6. 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)

7. The forward probabilities of “bbba”
Time Ticks 1 2
INPUT ε b
bb bbb bbba
1.0
0.2
0.0
0.1
P(w,t)
0.3

8. 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)

9. Probability of the observation using β
P(w1,n)=β1(1)

10. 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)