Shapes for the variance (in purple) and the optimal proxy variance (in green) of the Beta distribution for varying parameters.

With my friend Olivier Marchal (mathematician, not filmmaker, nor the cop), we have just arXived a note on the sub-Gaussianity of the Beta and Dirichlet distributions.

The notion, introduced by Jean-Pierre Kahane, is as follows:

A random variable with finite mean is sub-Gaussian if there is a positive number such that:

Such a constant is called a proxy variance, and we say that is -sub-Gaussian. If is sub-Gaussian, one is usually interested in the optimal proxy variance:

Note that the variance always gives a lower bound on the optimal proxy variance: . In particular, when , is said to be *strictly* sub-Gaussian.

The sub-Gaussian property is closely related to the tails of the distribution. Intuitively, being sub-Gaussian amounts to having tails lighter than a Gaussian. This is actually a characterization of the property. Let . Then:

That equivalence clearly implies exponential upper bounds for the tails of the distribution since a Gaussian satisfies

That can also be seen directly: for a -sub-Gaussian variable ,

The polynomial function is minimized on at , for which we obtain

.

In that sense, the sub-Gaussian property of any compactly supported random variable comes for free since in that case the tails are obviously lighter than those of a Gaussian. A simple general proxy variance is given by Hoeffding’s lemma. Let be supported on with . Then for any ,

so is -sub-Gaussian.

Back to the Beta where , this shows the Beta is -sub-Gaussian. The question of finding the optimal proxy variance is a more challenging issue. In addition to characterizing the optimal proxy variance of the Beta distribution in the note, we provide the simple upper bound . It matches with Hoeffding’s bound for the extremal case , , where the Beta random variable concentrates on the two-point set (and when Hoeffding’s bound is tight).

In getting the bound , we prove a recent conjecture made by Sam Elder in the context of Bayesian adaptive data analysis. I’ll say more about getting the optimal proxy variance in a next post soon.

Cheers!

Julyan

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