# What is the median of the Dirichlet Distribution?

The Dirichlet is a multivariate generalisation of the Beta distribution, and Wikipedia states that the median of the Beta distribution is the inverse of the incomplete Beta function evaluated at $\frac{1}{2}$, leading from the fact that the incomplete Beta function is the cumulative distribution function of the Beta distribution.

Is there a multivariate analogue to this arrangement?

• What definition of a multivariate median do you have in mind?
– whuber
Commented Jun 18, 2015 at 14:25
• @whuber a good point. I had been thinking along the lines of an intersection of hyperplanes, each of which is a median for one variable with the others fixed. I guess it's not immediately clear that such a thing should exist, nor what circumstances it would. I'd expect it to be unique if it did exist though. In respect of the link you gave, I guess "vector of marginal medians" fits that. Unfortunately, the linked chapter is behind a paywall.. Commented Jun 19, 2015 at 4:39
• I could not see the linked chapter either--but the information in the abstract alone is enough to document the existence of many kinds of multivariate median. If you're interested only in the marginal medians, then your question has already answered itself: aren't the marginals Beta distributed?
– whuber
Commented Jun 19, 2015 at 12:52
• @whuber Yes, they are, but finding a point of intersection may not be so trivial I think.. I found another (older) paper on multivariate medians that adds some explanation. I was hoping for something quick and easy. I have a paper to finish for now and other pressing things - perhaps I'll come back to this later.. Thank you for your thoughts and research (: Commented Jun 20, 2015 at 2:38
• The medians of the marginals do not generally intersect: consider the Dirichlet with parameters (1,1,1). The inverse beta of (1,2) at 0.5 is 0.293, which is less than the 1/3 that symmetry would require for intersection. Doesn't mean you can't normalize, as stats.stackexchange.com/questions/354356/… suggests, with a connection to another meaning. Commented Apr 15, 2019 at 22:46