Hidden Markov Model

Hidden Markov Models

A Hidden Markov Model($N$,$\sum$,$\Theta$) consists of the following elements:

1.$N$ is a positive integer specifying the number of states in the model. Without loss of generality, we will take the $N$th state to be a special state, the final or stop state.

2.$\sum$ is a set of output symbols, for example $\sum =a,b$

3.$\Theta$ is a vector of parameters. It contains three types of parameters:
$\pi j$ for $j=1\dots N$ is the probability of choosing state $j$ as an initial state. Note that ${\sum }_{j=1}^N\pi j=1$

$a_{j,k}$ for $j=1\dots (N-1),k=1\dots N$

is the probability of transitioning from state j to state k. Note that for all j,

${\sum }_{k=1}^N{a}_{j,k}=1$

$b_j\left(o\right)$ for $j=1\dots N-1$, and $o\in \sum$,

is the probability of emitting symbol $o$ from state j. Note that for all $j$,

${\sum }_{o\in \sum }{b}_j\left(o\right)=1$.

Thus it can be seen that $\Theta$ is a vector of $N+(N-1)N+(N-1)|\sum |$ parameters.

An HMM specifies a probability for each possible $(x,y)$ pair, where x is a sequence of symbols drawn from $\sum$, and y is a sequence of states drawn from the integers $1\dots N-1$. The sequences $x$ and $y$ are restricted to have the same length. As an example, say we have an HMM with $N=3,\sum a,b$, and with some choice of the parameters $\Theta$. Take $x=a,a,b,b$ and $y=1,2,2,1$. Then in this case,
$P(x,y|\Theta )={\pi }_1{a}_{1,2}{a}_{2,2}{a}_{2,1}{a}_{1,3}{b}_1\left(a\right)b_2\left(a\right)b_2\left(b\right)b_1\left(b\right)$

Thus we have a product of terms specifying the probability of emitting each symbol from its associated state.

In general, if we have the sequence $x$, and the sequence $y$,

$P(x,y|\Theta )$

$={\pi }_{y1}{a}_{y_n,N}Pro{d}_{j=2}^n{a}_{y_{j-1,y_j}}Pro{d}_{j=1}^n{b}_{y_j}\left(x_j\right)$

Thus we see that the probability is a simple function of the prameters $\Theta$.