powering a probability [a Bernoulli factory tale]

Starting from an X validated question on finding an unbiased estimator of an integral raised to a non-integer power, I came across a somewhat interesting Bernoulli factory solution! Thanks to Peter Occil’s encyclopedic record of cases, pointing out to Mendo’s (2019) solution for functions of ρ that can be expressed as power series. Like ρ^γ since

$(1-[1-\rho])^\gamma=1+\gamma(1-\rho)+\frac{\gamma(\gamma-1)(1-\rho)^2}{2}+\cdots$

which rather magically turns into the reported algorithm

Set k=1
Repeat the following process, until it returns a value x:
 1. Generate a Bernoulli B(ϱ) variate z; if z=1, return x=1
 2. Else, with probability γ/k, return x=0
 3. Else, set k=k+1 and return to 1.

since

$\rho^\gamma=\rho+(1-\rho)(1-\gamma)\big\{\rho+\frac{(1-\rho)(2-\gamma)}{2}\big[\rho+\cdots$

This entry was posted on April 21, 2023 at 12:23 am and is filed under Books, R, Statistics with tags Bernoulli factory, cross validated, Monte Carlo integration. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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powering a probability [a Bernoulli factory tale] – Data Science Austria Says:
April 21, 2023 at 6:35 pm

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