Sub-Gaussian property for the Beta distribution (part 2)
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Left: What makes the Beta optimal proxy variance (red) so special? Right: The difference function has a double zero (black dot).
As a follow-up on my previous post on the sub-Gaussian property for the Beta distribution [1], I’ll give here a visual illustration of the proof.
A random variable 
with finite mean ![\mu=\mathbb{E}[X]](https://s0.wp.com/latex.php?latex=%5Cmu%3D%5Cmathbb%7BE%7D%5BX%5D&bg=fff&%23038;fg=222&%23038;s=0 "\mu=\mathbb{E}[X]")
is sub-Gaussian if there is a positive number 
such that:
![\mathbb{E}[\exp(\lambda (X-\mu))]\le\exp\left(\frac{\lambda^2\sigma^2}{2}\right)\,\,\text{for all } \lambda\in\mathbb{R}.](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5B%5Cexp%28%5Clambda+%28X-%5Cmu%29%29%5D%5Cle%5Cexp%5Cleft%28%5Cfrac%7B%5Clambda%5E2%5Csigma%5E2%7D%7B2%7D%5Cright%29%5C%2C%5C%2C%5Ctext%7Bfor+all+%7D+%5Clambda%5Cin%5Cmathbb%7BR%7D.&bg=fff&%23038;fg=222&%23038;s=0 "\mathbb{E}[\exp(\lambda (X-\mu))]\le\exp\left(\frac{\lambda^2\sigma^2}{2}\right)\,\,\text{for all } \lambda\in\mathbb{R}.")
We focus on X being a Beta
random variable. Its moment generating function ![\mathbb{E}[\exp(\lambda X)]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5B%5Cexp%28%5Clambda+X%29%5D&bg=fff&%23038;fg=222&%23038;s=0 "\mathbb{E}[\exp(\lambda X)]")
is known as the Kummer function, or confluent hypergeometric function 
. So X is 
-sub-Gaussian as soon as the difference function
remains positive on 
. This difference function 
is plotted on the right panel above for parameters 
. In the plot, is varying from green for the variance ![\text{Var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}](https://s0.wp.com/latex.php?latex=%5Ctext%7BVar%7D%5BX%5D%3D%5Cfrac%7B%5Calpha%5Cbeta%7D%7B%28%5Calpha%2B%5Cbeta%29%5E2%28%5Calpha%2B%5Cbeta%2B1%29%7D&bg=fff&%23038;fg=222&%23038;s=0 "\text{Var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}")
(which is a lower bound to the optimal proxy variance) to blue for the value 
, a simple upper bound given by Elder (2016), [2]. The idea of the proof is simple: the optimal proxy-variance corresponds to the value of for which admits a double zero, as illustrated with the red curve (black dot). The left panel shows the curves with 
varying, interpolating from green for to blue for , with only one curve qualifying as the optimal proxy variance in red.
References
[1] Marchal and Arbel (2017), On the sub-Gaussianity of the Beta and Dirichlet distributions. Electronic Communications in Probability, 22:1–14, 2017. Code on GitHub.
[2] Elder (2016), Bayesian Adaptive Data Analysis Guarantees from Subgaussianity, https://arxiv.org/abs/1611.00065
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