The Hidden Markov Models
We introduce a simple HMM through an example.
We consider the process of choosing a die of four faces (a tetrahedron) and recording the number of dots by
throwing the die.
Suppose we have two dice A and B, each of them has four faces (1,2,3 and 4).
Moreover, Die A is fair and Die B is bias.
The probability distributions of dots obtained by throwing dice A and B are given in the table below.
| Die |
1 |
2 |
3 |
4 |
| A |
1/4 |
1/4 |
1/4 |
1/4 |
| B |
1/6 |
1/6 |
1/3 |
1/3 |
| Table 1 |
|---|
Each time a die is chosen, with probability a, Die A is chosen and with probability ( 1 - a ),
Die B is chosen. This process is hidden because we don't know which die is chosen. The
value of a is to be determined and it characterized how fair the game is (the closer the
value of a to 1, the more "fair" the game is). The chosen die is then thrown and the
number of dots (this is observable) obtained is recorded. The following is a possible
realization of the process:
A,1,A,2,B,1,B,2,A,1,B,2.
We note that the whole process of the HMM can be modeled by a classical Markov chain
model with the transition probability matrix P being given by
| P1 = |
|
|
A |
B |
|
1 |
2 |
3 |
4 |
|
| A |
æ
ç
ç
ç
ç
ç
è |
0 |
0 |
 |
1/4 |
1/4 |
1/4 |
1/4 |
ö
÷
÷
÷
÷
÷
ø |
| B |
0 |
0 |
1/6 |
1/6 |
1/3 |
1/3 |
| |
| 1 |
a |
1 - a |
0 |
0 |
0 |
0 |
| 2 |
a |
1 - a |
0 |
0 |
0 |
0 |
| 3 |
a |
1 - a |
0 |
0 |
0 |
0 |
| 4 |
a |
1 - a |
0 |
0 |
0 |
0 |
|
It is clear that each state of the Markov chain has a period of two.
Here if we are given the value of a then one can simulate the whole process easily
because it is just a simple Markov chain process. However, very often the value of a
is unknown and we may only be given a sequence of observable states.
An important question is the following:
how to estimate a from a given sequence of observable states?
|
