Table 2 The Viterbi decoding, 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 δ₁(i) = π _i b _i (o₁), ψ₁(i) = 0 for 1 ≤ i ≤ N,

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

• Initially β _T (i) = 1 for 1 ≤ i ≤ N,

• $\begin{array}{l}{\delta }_t(j)=\underset{1\le i\le N}{\max }[{\delta }_{t-1}(i)a_{i,j}]b_j(o_t),\hfill \\ {\psi }_t(j)=\underset{1\le i\le N}{\arg \max }[{\delta }_{t-1}(i)a_{i,j}]\hfill \end{array}$ for t = 2,..., T and 1 ≤ j ≤ 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,

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

• Finally $q_T^{\ast }=\underset{1\le i\le N}{\arg \max }[{\delta }_T(i)],q_t^{\ast }={\psi }_{t+1}(q_{t+1}^{\ast })$ for t = T - 1,..., 1 with optimal path $Q^{\ast }=\{q_1^{\ast },\mathrm{...},q_T^{\ast }\}$.

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

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