Sub-Gaussian property for the Beta distribution (part 2)

December 20, 2017
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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 X with finite mean \mu=\mathbb{E}[X] is sub-Gaussian if there is a positive number \sigma 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}.

We focus on X being a Beta(\alpha,\beta) random variable. Its moment generating function \mathbb{E}[\exp(\lambda X)] is known as the Kummer function, or confluent hypergeometric function _1F_1(\alpha,\alpha+\beta,\lambda). So is \sigma^2-sub-Gaussian as soon as the difference function

u_\sigma(\lambda)=\exp\left(\frac{\alpha}{\alpha+\beta}\lambda+\frac{\sigma^2}{2}\lambda^2\right)-_1F_1(\alpha,\alpha+\beta,\lambda)

remains positive on \mathbb{R}. This difference function u_\sigma(\cdot) is plotted on the right panel above for parameters (\alpha,\beta)=(1,1.3). In the plot, \sigma^2 is varying from green for the variance \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 \frac{1}{4(\alpha+\beta+1)}, 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 \sigma^2 for which u_\sigma(\cdot) admits a double zero, as illustrated with the red curve (black dot). The left panel shows the curves with \mu = \frac{\alpha}{\alpha+\beta} varying, interpolating from green for \text{Var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} to blue for \frac{1}{4(\alpha+\beta+1)}, 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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