hidden Markov chain and algorithms

Let (X1, Y1), ..., (Xn, Yn) be a sequence from a hidden Markov chain with the joint distribution n P(X1:n, Yl) = v(X1 )e(Y,IX,) IIP(XIXt-1)e(Y.IX-) t= 2 where Xt takes values in S = {1, ..., s} and Y, takes values in M = {1, ..., M}. Assume we have observations (yt)=1. Then the likelihood function is t=2 1. Write the normalized forward algorithm to compute the likelihood L. That is, derive a recursion for the normalized forward probabilities of(i) := P(xt = ily1:) and explain how to compute the likelihood using this recursion. (Hint: you may find defining Ct = P(Y1t-1) P(Yit) helpful.) 2. Suppose that the observed value yt for some 3