Table 1 Forward and backward procedures.

α _t (i) ≡ p(o₁,..., o _t |q _t = S _i , λ)

β _t (i) ≡ p(o_t+1,..., o _T |q _t = S _i , λ)

• Initially α₁(i) = π _i b _i (o₁) for 1 ≤ i ≤ N,

• Initially β_T(i) = 1 for 1 ≤ i ≤ N,

• ${\alpha }_t(j)=\left[{\displaystyle {\sum }_{i=1}^N{\alpha }_{t-1}(i)a_{i,j}}\right]b_j(o_t)$ for t = 2, 3,..., T and 1 ≤ j ≤ N,

• ${\beta }_t(i)={\displaystyle {\sum }_{j=1}^N{\alpha }_{i,j}{b}_j(o_{t+1}){\beta }_{t+1}(j)}$ for t = T - 1,...,1 and 1 ≤ i ≤ N,

• Finally $p(O|\lambda )={\displaystyle {\sum }_{i=1}^N{\alpha }_T(i)}$ is the sequence likehood.

• Finally $p(O|\lambda )={\displaystyle {\sum }_{i=1}^N{\pi }_i{b}_i(o_1){\beta }_1(i)}$.