Understanding Hidden Markov Models (reproduced)

Set S1,s2,s3 ... Represents a signal from a source of information. O1, O2, O3 ... Is the signal received by the receiver. Communication in the decoding is based on the received signal O1, O2, O3 ... Restore the sent signal s1,s2,s3 ....
So how do you guess what the speaker wants to say, based on the information received? We can solve these problems by using the Hidden Markov model called "Hidden Markov models". Taking speech recognition as an example, when we observe the speech signal O1,o2,o3, we have to speculate on the s1,s2,s3 of the sentences sent from the set of signals. Obviously, we should find the most probable one in all possible sentences. It is described in mathematical language, that is, in the known O1,o2,o3,... case, the conditional probability is obtained
P (S1,s2,s3,... | O1,o2,o3 ....) The sentence that reaches the maximum value S1,s2,s3,...
Of course, the probability above is not easy to find directly, so we can calculate it indirectly. By using the Bayesian formula and omitting a constant term, the above equation can be converted to the equivalent
P (O1,o2,o3,... | S1,s2,s3 ....) * P (S1,S2,S3,...)
which
P (O1,o2,o3,... | S1,s2,s3 ....) To express a word s1,s2,s3 ... Be read as O1,o2,o3,... The possibility, while
P (s1,s2,s3,...) denotes string s1,s2,s3,... The possibility of being able to become a reasonable sentence, so the meaning of this formula is to send the signal for S1,S2,S3 ... The probability of this sequence multiplied by s1,s2,s3 ... itself can be a possibility of a sentence, to draw the probability.
(The reader may ask here whether you are now complicating the problem because the formula is longer.) Don't worry, we'll simplify the problem now. We do two assumptions here:
First, S1,s2,s3,... is a Markov chain, that is, si only determined by si-1 (see series one);
Second, the first time of the receiving signal Oi only by the sending signal Si (also known as the independent output hypothesis, P (O1,o2,o3,... | S1,s2,s3 ....) = P (o1|s1) * p (O2|S2) *p (O3|S3) ....
Then we can easily use the algorithm Viterbi find the maximum value of the above formula, and then find the sentence to be recognized S1,S2,S3,...。
The model that satisfies the above two assumptions is called the Hidden Markov model. We use the word "implied" because of the state S1,S2,S3,... is not directly observable.
The application of Hidden Markov model is far more than in speech recognition. In the formula above, if we put S1,S2,S3,... As Chinese, put O1,o2,o3,... As the corresponding English, then we can use this model to solve machine translation problems, if we put O1,o2,o3,... This model can be used to solve the recognition of printed and handwritten characters as the image characteristics of scanned text.
P (O1,o2,o3,... | S1,s2,s3 ....) Depending on the name of the application, it is called the "acoustic model" in speech recognition, and in machine translation It is the "acoustic model" and in the middle of the spelling is the "error correction model" (Correction models). and P (S1,s2,s3,...) is the language model we mentioned in the series one.

Before using the hidden Markov model to solve the problem of language processing, we should train the model first. The usual training methods were proposed by Boehm (Baum) in the 60 's and named after him. The successful application of hidden Markov model in the early processing of language problems is speech recognition. In the 70 's, Fred Jelinek (Jarinik) of IBM and Carnegie Mellon University, Jim and Janet Baker, a brother of Kai-Fu Lee, independently proposed using the hidden Markov model to identify speech, the error rate of speech recognition compared to artificial intelligence and pattern matching The method is reduced by three times times (from 30% to 10%). In the 80 's, Dr. Kai-Fu Lee insisted on adopting the framework of hidden Markov model, and successfully developed the world's first large vocabulary continuous speech recognition system (SPHINX).

Understanding Hidden Markov Models (reproduced)