Viterbi training

Initialize emission and transition probabilities to random numbers.

while (true)Do Viterbi decoding using current parametersSave current parameters as previous parameters.Re-estimate emission and transition parameters from the state path

decoded by Viterbi. (add pseduocounts, see next page).if sum of absolute difference between current and previous parameters

is tiny (e.g., < 0.00001), break;

endprint current parameter and P(sequence, viterbi path)Repeat the above procedure several times (with different

random seed), and compare P(sequence, viterbi path). Report the parameters learned that give the largest P.

Re-estimate parameters with pseudocounts

• Count number of transitions, n_xy, where x, y = {a, b}

• t_xy = (n_xy+c) / sum_x(n_xy+c)– e.g. t_ab = (n_ab +1) / (n_ab + n_aa + 2)

• Count number of symbols in each state, N_aX and N_bX, where X = A, C, G, T

• e_aX = (N_aX + 1) / (sum_X N_aX + 4)

• e_bX = (N_bX + 1) / (sum_X N_bX + 4)

Pseudocount

Backward-Forward algorithm:Compute sum of probabilities in log space

• Two probabilities x and y, x < y• lx = log(x), ly = log(y), (lx < ly)• z = x + y = y (1 + x/y)

lz = log(z) = log(x+y)= log(y) + log(1 + x/y)= ly + log(1 + exp(log(x)-log(y))= ly + log(1 + exp(lx – ly))

Also see page 4 in this doc: http://cs.utsa.edu/~jruan/teaching/cs5263_fall_2007/proj1.pdf

and page 77 of the handouts.