We obtain the optimal proxy variance for the sub-Gaussianity of Beta
distribution, thus proving upper bounds recently conjectured by Elder (2016).
We provide different proof techniques for the symmetrical (around its mean)
case and the non-symmetrical case. The technique in the latter case relies on
studying the ordinary differential equation satisfied by the Beta
moment-generating function known as the confluent hypergeometric function. As a
consequence, we derive the optimal proxy variance for the Dirichlet
distribution, which is apparently a novel result. We also provide a new proof
of the optimal proxy variance for the Bernoulli distribution, and discuss in
this context the proxy variance relation to log-Sobolev inequalities and
transport inequalities.

本文通过研究 Beta 分布的优化代理方差值，证明了 Elder (2016) 最近所推测的上界，并提供不同的证明技巧。同时，通过研究 Beta 生成函数所满足的一般微分方程，我们得到了狄利克雷分布的最优代理方差值，这显然是一个新的结果。我们还提供了伯努利分布最优代理方差值的新证明，并讨论了代理方差与 log-Sobolev 不等式和输运不等式之间的关系。