When a Beta random variable wants to act like a Bernoulli: convergence of optimal proxy variance.

In this third and last post about the Sub-Gaussian property for the Beta distribution [1] (post 1 and post 2), I would like to show the interplay with the Bernoulli distribution as well as some connexions with optimal transport (OT is a hot topic in general, and also on this blog with Pierre’s posts on Wasserstein ABC).

Let us see how sub-Gaussian proxy variances can be derived from transport inequalities. To this end, we need first to introduce the Wasserstein distance (of order 1) between two probability measures P and  Q on a space . It is defined wrt a distance d on  by

where  is the set of probability measures on with fixed marginal distributions respectively and Then, a probability measure is said to satisfy a transport inequality with positive constant , if for any probability measure dominated by ,

where is the entropy, or Kullback–Leibler divergence, between and . The nice result proven by Bobkov and Götze (1999) [2] is that the constant  is a sub-Gaussian proxy variance for P.

For a discrete space equipped with the Hamming metric, , the induced Wasserstein distance reduces to the total variation distance, . In that setting, Ordentlich and Weinberger (2005) [3] proved the distribution-sensitive transport inequality:

where the function is defined by and the coefficient is called the balance coefficient of , and is defined by . In particular, the Bernoulli balance coefficient is easily shown to coincide with its mean. Hence, applying the result of Bobkov and Götze (1999) [2] to the above transport inequality yields a distribution-sensitive proxy variance of for the Bernoulli with mean , as plotted in blue above.

In the Beta distribution case, we have not been able to extend this transport inequality methodology since the support is not discrete. However, a nice limiting argument holds. Consider a sequence of Beta random variables with fixed mean and with a sum going to zero. This converges to a Bernoulli random variable with mean , and we have shown that the limiting optimal proxy variance of such a sequence of Beta with decreasing sum is the one of the Bernoulli.

References

[1] Marchal, O. and Arbel, J. (2017), On the sub-Gaussianity of the Beta and Dirichlet distributions. Electronic Communications in Probability, 22:1–14, 2017. Code on GitHub.
[2] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. Journal of Functional Analysis, 163(1):1–28.
[3] Ordentlich, E. and Weinberger, M. J. (2005). A distribution dependent refinement of Pinsker’s inequality. IEEE Transactions on Information Theory, 51(5):1836–1840.