Conditional Probability - Markov chain

[Apteronotus]

Hi,

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

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

In the above we have assumed that the observations are discrete variables having [itex]K[/itex] states.

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

Thanks,

 

[Stephen Tashi]

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

 

[Apteronotus]

Brilliant!

Thank you.