HMMs Revisited 3 How do we adjust the model parameters n=(A, B, to maximize P(o2)? Assume a topology of states X Want to learn p(xilx ak), and p(zilx) from a sequence O(z1, a1,Z2, a zr. a · Approximate idea Assume we have O(z1, a, X, Z2, a2, x2, . Z,, aT, X,) Use counting to estimate p(xix, ak), and p(zixi) ·Re- estimate x1,X2,x · Repeat How to do this efficiently and correctly Remember hmm Basic Problem 1 a1(1)=xp(二1|x) ∑a()p(x,|x,a)p(=1x) J p(O|)=∑a( i=1 Backwards terms Bn()=1 月(1=∑p(x|x2a1)p(1x)B1(HMMs Revisited 3) How do we adjust the model parameters λ=(A,B,π) to maximize P(O|λ)? ● Assume a topology of states X ● Want to learn p(xi |xj , ak), and p(zi |xj ) from a sequence O(z1,a1, z2,a2,…, zT,aT) ● Approximate idea: ● Assume we have O(z1,a1,x1,z2,a2,x2,…, zT,aT,xT) ● Use counting to estimate p(xi |xj , ak), and p(zi |xj ) ● Re-estimate x1,x2,…,xT ● Repeat ● How to do this efficiently and correctly? Remember HMM Basic Problem 1 ( ) ( | ) 1 i 1 i α i = π p z x ( ) ( ) ( | , ) ( | ) 1 | | 1 1 t i X j t t j i t i j p x x a p z x + = + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ α = ∑α ∑= = | | 1 ( | ) ( ) X i t p O λ α i β T (i) =1 ( ) ( | , ) ( | ) ( ) 1 1 | | 1 i p x x a p z x j t j t X j t j i t + + = β = ∑ β Backwards terms: