powering a probability [a Bernoulli factory tale]
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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 ργ since
![(1-[1-\rho])^\gamma=1+\gamma(1-\rho)+\frac{\gamma(\gamma-1)(1-\rho)^2}{2}+\cdots](https://www.r-bloggers.com/wp-content/plugins/jetpack/modules/lazy-images/images/1x1.trans.gif)
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
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