Baum-welch algorithm of Hidden Markov model

Baum-welch algorithm of hidden Markov model Preface

In the previous blog post, we learned about the probability calculation and prediction of hidden Markov models, but when we were prepared to understand learning problems, we found that the knowledge of EM algorithm was needed in learning problems, so we learned the EM algorithm and maximum likelihood estimation on the previous turnover, and got some own understanding to the solution of hidden variables. Now let's go back and learn the hidden Markov model learning problem. The relevant introduction of EM algorithm can be referred to the EM algorithm and its extension study notes. If you are not familiar with hidden Markov models, you can refer to the Hidden Markov learning Note (a). Learning Problems

The study of Hidden Markov model can be realized by supervised learning and unsupervised learning according to the training data including observation sequence and corresponding state sequence or only observation sequence. This section first introduces the supervised learning algorithm, and then introduces the unsupervised learning Algorithm--baum-welch algorithm (EM algorithm). Supervise learning issues

Assume that the training data contains an observation sequence of the same length as S and a corresponding state sequence {(O1,I1), (O2,I2),..., (Os,is)} \{(O_1,i_1), (o_2,i_2),..., (o_s,i_s) \}, Then the maximum likelihood estimation method can be used to estimate the parameters of the hidden Markov model, and the method is as follows.

1. Estimation of transfer probability AIJ A_{ij}
When the time t is at the state I moment t+1 shifts to J at a frequency of Aij A_{ij}, the estimated state transition probability is Aij A_{ij}
a^ij=aij∑nj=1aij,i=1,2,..., n,j=1,2,..., n \hat A_{ij} = \frac {A_{ij}}{\sum_{j=1}^{n}a_{ij}}, I =1,2,..., n,j =1,2,..., N
Directly according to the given O and I for the frequency statistics, in the seaweed model, we can count the number of weather transfer 100 days ago, such as in 100 days, statistics from sunny, sunny, sunny, cloudy number of times, sunny-> The number of rainy, respectively, as A1,a2,a3 a_1,a_2,a_3, then Asunny−>any state=[a1a1+a2+a3