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CS344 : Introduction to Artificial Intelligence

CS344 : Introduction to Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 24- Expressions for alpha and beta probabilities. r. q. A Simple HMM. a: 0.2. a: 0.3. b: 0.2. b: 0.1. a: 0.2. b: 0.1. b: 0.5. a: 0.4. Forward or α - probabilities.

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CS344 : Introduction to Artificial Intelligence

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

  2. r q A Simple HMM a: 0.2 a: 0.3 b: 0.2 b: 0.1 a: 0.2 b: 0.1 b: 0.5 a: 0.4

  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 σP(w1,n, Sn+1=si) = Σi=1 σ αi(n+1) σis the total number of states

  6. Recursive expression for α αj(t+1) =P(w1,t, St+1=sj) =Σi=1 σP(w1,t, St=si,St+1=sj) =Σi=1 σP(w1,t-1, St=sj) P(wt,St+1=sj|w1,t-1, St=si) =Σi=1 σP(w1,t-1, St=si) P(wt,St+1=sj|St=si) = Σi=1 σαj(t) P(wt,St+1=sj|St=si)

  7. The forward probabilities of “bbba”

  8. Backward or β-probabilities Let βi(t) be the probability of seeingwt,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 σP(wt-1,n, St=si |St-1=si) =Σi=1 σP(wt-1,St=sj|St-1=si)P(wt,n,|wt-1,St=sj, St-1=si) =Σi=1 σP(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)

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