As often, I came a’X this 2007 paper following an X-validated (if poorly worded) question that linked to it. With weird arguments:
- Computing an unbiased estimate of the integral induced by an unnormalised density f is a poor sales argument, as the renormalised object is not a probability density. Call to the unbiased approximation of Bayes factors the paper does not!
- The proposed solution involves a replacement of the unnormalised density f by a substitute and normalised density h that leads to the identity
![\frac{f(y)}{\int f(x)\text dx}=\frac{hf(y)}{1-\int [h(x)-h(y)f(x)/f(x)]\text dx}=\frac{hf(y)}{1-a(y)}](https://s0.wp.com/latex.php?latex=%5Cfrac%7Bf%28y%29%7D%7B%5Cint+f%28x%29%5Ctext+dx%7D%3D%5Cfrac%7Bhf%28y%29%7D%7B1-%5Cint+%5Bh%28x%29-h%28y%29f%28x%29%2Ff%28x%29%5D%5Ctext+dx%7D%3D%5Cfrac%7Bhf%28y%29%7D%7B1-a%28y%29%7D&bg=000000&fg=B0B0B0&s=0&c=20201002)
with the assumption that the function a is between -1 and 1, which requires some insight about the connection between f and h.
- The unbiased estimator is an infinite power series in a(y). Requiring an increasing number of independent generations as the power grows.
- The series is unbiasedly terminated by playing roulette (if not Russian roulette!). Stopping when the k-th power estimate is below a certain bound. To be selected with no preliminary knowledge of the range of a.
- The uncertainty attached to the power series estimate is not evaluate and could reach the concerning level of an infinite variance.
that may explain for its limited popularity.
