基于HMM的拼音输入法

初始化： δ 1 ( i ) = π i b i ( o 1 ) , 1 ≤ i ≤ N \delta_1(i)=\pi_ib_i(o_1),1\le i\le N δ1​(i)=πi​bi​(o1​),1≤i≤N

Φ 1 ( i ) = 0 \Phi_1(i)=0 Φ1​(i)=0

迭代求解： δ t ( j ) = m a x 1 ≤ i ≤ N δ t − 1 ( i ) a i j b j ( o t ) \delta_t(j)=\mathrm{max}_{1\le i\le N}\delta_{t-1}(i)a_{ij}b_j(o_t) δt​(j)=max1≤i≤N​δt−1​(i)aij​bj​(ot​)

Φ t ( j ) = a r g m a x 1 ≤ i ≤ N δ t − 1 ( i ) a i j b j ( o t ) \Phi_t(j)=\mathrm{arg max}_{1\le i\le N}\delta_{t-1}(i)a_{ij}b_j(o_t) Φt​(j)=argmax1≤i≤N​δt−1​(i)aij​bj​(ot​)

终止： P ∗ = m a x 1 ≤ i ≤ N δ T ( i ) \mathrm{P^*}=\mathrm{max}_{1\le i\le N}\delta_T(i) P∗=max1≤i≤N​δT​(i)

q T ∗ = a r g m a x 1 ≤ i ≤ N δ T ( i ) q_T^*=\mathrm{argmax}_{1\le i\le N}\delta_T(i) qT∗​=argmax1≤i≤N​δT​(i)

最优路径（隐状态序列）回溯： q t ∗ = Φ t + 1 ( q t + 1 ∗ ) , t = T − 1 , T − 2 , . . . , 2 , 1 q_t^*=\Phi_{t+1}(q_{t+1}^*),t=T-1,T-2,...,2,1 qt∗​=Φt+1​(qt+1∗​),t=T−1,T−2,...,2,1