Baum-Welch算法就是EM算法，所以首先给出EM算法的Q函数
[sum_zP(Z|Y, heta')log P(Y,Z| heta)
]
换成HMM里面的记号便于理解
[Q(lambda,lambda') = sum_zP(I|O,lambda')log P(I,O|lambda)
]
根据状态序列和观测序列的联合分布
[egin{align*}
P(O,I|lambda) &= sum_IP(O|I,lambda)P(I|lambda)\
&= pi_{i_1}b_{i_1}(o_1)a_{i_1i_2}b_{i_2}(o_2)dots a_{i_{T-1}i_T}b_{i_T}(o_T)\
end{align*}]
代入上式后得
[egin{align*}
Q(lambda, lambda') &= sum_IP(I|O,lambda')logpi_{i_1}\ &+ sum_IP(I|O,lambda')logsum_{t=1}^Tb_{i_t}(o_t) \ &+ sum_IP(I|O,lambda')logsum_{t=2}^Ta_{i_{t-1}i_T}
end{align*}]
这便是E步，下面看看M步.
看Q函数得第一步， 由于带有约束
[sum_i^Npi_i = 1
]
这个时候就需要请出拉格朗日乘子了
[egin{align*}
L &= sum_IP(I|O,lambda')logpi_1 + gamma(sum_{i=1}^Npi_i -1)\
&= sum_{i=1}^NP(O,i_1=i|lambda')logpi_i + gamma(sum_{i=1}^Npi_i -1)\
end{align*}]
令(dfrac{partial L}{partialpi_i} = 0)得到
[egin{align*}
P(O, i_1 = i|lambda') + gamma pi_i &= 0\
P(O, i_1 = i|lambda') &= -gamma pi_i\
sum_{i=1}^NP(O, i_1 = i|lambda') &= -gamma sum_{i=1}^Npi_i\
gamma &= -P(O|lambda')
end{align*}]
回代，得到
[pi_i = dfrac{P(O, i_1=i|lambda')}{P(O|lambda')}
]
其他得参数同样可以得到