Title says it all.
I have seen both "the hyperparameter of the Dirichlet distribution" and "the parameter of the Dirichlet distribution"
What are the differences?
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A hyperparameter is a parameter for the (prior) distribution of some parameter.
So for a simple example, let's say we state that the variance parameter $\tau^2$ in some problem has a uniform prior on $(0,\theta)$.
(I personally would be unlikely to do such a thing, but it happens; I might in some very particular circumstance)
Then $\tau^2$ is a parameter (in the distribution of the data) and $\theta$ is a hyperparameter.
If we then in turn specify a (prior) distribution for $\theta$ (e.g. that it's Gamma with mean 100 and shape parameter 2), that's a hyperprior - a prior distribution on a parameter of a prior distribution.