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N , β = 1, . . , N, E. Note that if for some α, β (0) (s) we had pαβ = 0, then the above estimation procedure ensures that pαβ = 0 for all s. Therefore, the connectivity of the underlying Markov chain obtained 50 3 Markov Chains and Hidden Markov Models on every iteration step does not contradict the a priori connectivity. 11). 17) can only be used if the corresponding denominator is non-zero; otherwise the new values of the relevant transition probabilities are set arbitrarily. Next, we will estimate the emission probabilities.

L ∈X ×qπ1 (x1 )pπ1 π2 qπ2 (x2 ) . . pπi−2 πi−1 qπi−1 (xi−1 )pπi−1 Gk qk (xi )pkl ql (xi+1 )pGl πi+2 ×qπi+2 (xi+2 ) . . ,πi−1 ∈X ×qπ2 (x2 ) . . ,πL ∈X ×qπi+2 (xi+2 ) . . pπL−1 pL qπL (xL )pπL 0 = fk (i)pkl ql (xi+1 )bl (i + 1) , P (x) where fk (i) and bl (i) are the quantities found from the forward and backward algorithms respectively. 19) for i = 1, . . , L − 1 and k, l = 1, . . , N . 6 Parameter Estimation for HMMs 49 respectively and is also called the posterior probability of states Gk and Gl at observations i and i + 1 respectively given x.

Set k = 3 and b = 1. Will the algorithm find an optimal alignment? What will change if we set k = 4 and b = 2? 5. 4). The score s(Mi ) of the column Mi is defined as follows: if the column contains three identical symbols, set s(Mi ) = 3; if it contains two identical symbols, but no gaps, set s(Mi ) = 2; if it contains three distinct symbols, but no gaps, set s(Mi ) = 1; if it contains exactly one gap, set s(Mi ) = −1; if it contains two gaps, set s(Mi ) = −2. Applying the relevant dynamic programming algorithm find all optimal alignments of the three sequences x = CAGC y = CT G z = T AC.

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