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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,

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# Conditional Probability - Markov chain

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