On the sub-Gaussianity of the Beta and Dirichlet distributions

By Olivier Marchal, Julyan Arbel. 2017

91 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Artificial Intelligence, Statistical and Nonlinear Physics

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Citations in the corpus

Variational Bayesian Reinforcement Learning with Regret Bounds (2018, background): “If the reward is categorical and the prior is Dirichlet then the posterior is also Dirichlet which is sub-Gaussian [51].”
On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables (2019, methodology): “This technique was used by Marchal and Arbel (2017) (Section 2.”
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions (2021, background): “We next state a result from [52], which gives a tractable bound for the subgaussian constant of the Beta distribution, as well as the exact but less explicit value which we will not need.”