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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 $\mathcal{X}$. It is defined wrt a distance d on $\mathcal{X}$ by

$W(P,Q)=\inf_{\pi\in\Pi(P,Q)}\int_{\mathcal{X}\times\mathcal{X}}d(x,y)\pi(\text{d}x,\text{d}y),$

where $\Pi(P,Q)$ is the set of probability measures on $\mathcal{X}\times \mathcal{X}$ with fixed marginal distributions respectively $P$ and $Q.$ Then, a probability measure $P$ is said to satisfy a transport inequality with positive constant $\sigma$, if for any probability measure $Q$ dominated by $P$,

$W(P,Q) \leq\sigma\sqrt{2 D(Q||P)},$

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

For a discrete space $\mathcal{X}$ equipped with the Hamming metric, $d(x,y) = \mathbf{1}_{\{x\neq y\}}$, the induced Wasserstein distance reduces to the total variation distance, $W(P,Q) = \Vert P-Q\Vert_{\text{TV}}$. In that setting, Ordentlich and Weinberger (2005) [3] proved the distribution-sensitive transport inequality:

$\Vert P-Q\Vert_{\text{TV}} \leq \sqrt{\frac{1}{g(\mu_P)}D(Q||P)},$

where the function $g$ is defined by $g(\mu)=\frac{1}{1-2\mu}\ln\frac{1-\mu}{\mu}$ and the coefficient $\mu_P$ is called the balance coefficient of $P$, and is defined by $\mu_P=\underset{A\subset \mathcal{X}}\max\min\{P(A),1-P(A)\}$. 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 $\frac{1}{2g(\mu)}=\frac{1-2\mu}{2\ln((1-\mu)/\mu)}$ for the Bernoulli with mean $\mu$, 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$(\alpha,\beta)$ random variables with fixed mean $\mu=\frac{\alpha}{\alpha+\beta}$ and with a sum $\alpha+\beta$ going to zero. This converges to a Bernoulli random variable with mean $\mu$, and we have shown that the limiting optimal proxy variance of such a sequence of Beta with decreasing sum $\alpha+\beta$ 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.