Hidden Markov Model Definition
Hidden Markov Model is a probability model about time series, which describes the process of randomly generating an unobservable random sequence of states from a hidden Markov chain, and then generating an observation from each state to generate a random sequence of observations.
The sequence of states randomly generated by the hidden Markov chain is called the state sequence; each state generates an observation, and the resulting random sequence of observations is called the observation sequence.
Each position of the sequence can in turn be seen as a moment.
Below we introduce some notations to represent these definitions:
Let Q be the set of all possible states and V be the set of all possible observations.
where N is the number of possible states and M is the number of possible observations.
State q is invisible, observation v is visible.
Applied to part-of-speech tagging, v stands for words, which can be observed. q represents the part of speech we want to predict (a word may correspond to multiple parts of speech) is a hidden state.
Applied to participles, v stands for words, which can be observed. q represents our label (B, E these labels, representing the beginning of a word, or the middle, etc.)
Applied to named entity recognition, v represents words, which can be observed. q represents our label (labels represent place words, time words, etc.)
For the methods mentioned above, interested students can refer to the corresponding information.
I is a state sequence of length T, and O is the corresponding observation sequence.
We can think of it as a training set given a word (O) + part of speech (I). Or a training set of a word (O) + word segmentation label (I).... With the training data, then with the training algorithm, many problems can be solved, and the problem will come later~
We continue to define A as the state transition probability matrix:
in,
is the probability of transitioning to state qj at time t+1 under the condition that time t is in state qi.
B is the observation probability matrix:
in,
is the probability of generating the observation vk (the so-called "emission probability") under the condition that the time t is in the state qj.
Therefore, in other materials, the commonly seen generation probability and emission probability are actually a concept.
π is the initial state probability vector:
in,
The hidden Markov model is determined by the initial state probability vector π, the state transition probability matrix A and the observation probability matrix B. π and A determine the state sequence, and B determine the observation sequence. Therefore, the Hidden Markov Model can be represented by a ternary notation, i.e.
\lambda =(A,B,\Pi ) is called the three elements of the hidden Markov model.
If a specific state set Q and observation sequence V are added, the quintuple of HMM is formed, which is also all the components of the hidden Markov model.
The three basic problems of Hidden Markov Chains are introduced as follows:
1. Find the probability of the occurrence of the observation sequence from the known model
2. Find the parameters of the model with the
known observation sequence 3. Find the possible state sequence from the known observation sequence