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.