Conditional Probability - Markov chain

[Apteronotus]

Hi,

I was reading about Markov chains and came across the following statement:

"The conditional distribution p(x_n|x_{n-1}) will be specified by a set of K-1 parameters for each of the K states of x_{n-1} giving a total of K(K-1) parameters."

In the above we have assumed that the observations are discrete variables having K states.

I understand that x_{n-1} can have K states, but why K-1 parameters for each state? And what are those parameters?

Thanks,

 

[Stephen Tashi]

There are K different probabilities in the set of values p(x_n|x_{n-1}) and you could call each of these numbers a parameter. Since these probabilities must sum to 1, you only have to specify K-1 of them and this will determine the value of "the last one".

 

[Apteronotus]

Brilliant!

Thank you.