I was thinking about the work I did when I worked in biostatistics at MD Anderson. This work was practical rather than mathematically elegant, useful in its time but not of long-term interest. However, one result came out of this work that I would call elegant, and that was a symmetry I found.
Let X be a beta(a, b) random variable and let Y be a beta(c, d) random variable. Let g(a, b, c, d) be the probability that a sample from X is larger than a sample from Y.
g(a, b, c, d) = Prob(X > Y)
This function often appeared in the inner loop of a simulation and so we spent thousands of CPU-hours computing its values. I looked for ways to evaluate this function more quickly, and along the way I found a symmetry.
The function I call g was studied by W. R. Thompson in 1933 [1]. Thompson noted two symmetries:
g(a, b, c, d) = 1 − g(c, d, a, b)
and
g(a, b, c, d) = g(d, c, b, a)
I found an additional symmetry:
g(a, b, c, d) = g(d, b, c, a).
The only reference to this result in a journal article as far as I know is a paper I wrote with Saralees Nadarajah [2]. That paper cites an MD Anderson technical report which is no longer online, but I saved a copy here.
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[1] W. R. Thompson. On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples. Biometrika, Volume 25, Issue 4. pp. 285 – 294.
[2] John D. Cook and Saralees Nadarajah. Stochastic Inequality Probabilities for Adaptively Randomized Clinical Trials. Biometrical Journal. 48 (2006) pp 256–365.
The post Beta inequality symmetries first appeared on John D. Cook.