Markov Chain

1st-order Markov Chain: The time $i$ value depends only on the $i-1$ value. Transition probability:

$$a_{st}=P(x_i=t|x_{i-1}=s)$$

Probability of an instance $x$:

$$P(x)=P(x_1)\prod _{i=2}^L{a}_{x_{i-1}{x}_1}$$

To use this, you should make transition probability for all the classes.

$$S(x)=\log \left(\frac{P(x|+)}{P(x|-)}\right)=\sum _{i=1}^L\frac{\log (a_{x_{i-1},x_i}^{+})}{\log (a_{x_{i-1},x_i}^{-})}=\sum _{i=1}^L{\beta }_{x_{i-1},x_i}$$

Then,

$$\text{If }S(x)=\{\begin{array}{ll}& \ge \lambda ,\text{then x∈ω+}\\ & <\lambda ,\text{then x∈ω−}\end{array}$$

In here, $\lambda$ is called the threshold value. Its value highly depends on the circumstances of the problem. It can be based on the cost of misclassification, prior probabilities, or fixed at 0.