Bias in R's random integers?
On 19/09/2018 9:09 AM, I?aki Ucar wrote:
El mi?., 19 sept. 2018 a las 14:43, Duncan Murdoch (<murdoch.duncan at gmail.com>) escribi?:
On 18/09/2018 5:46 PM, Carl Boettiger wrote:
Dear list, It looks to me that R samples random integers using an intuitive but biased algorithm by going from a random number on [0,1) from the PRNG to a random integer, e.g. https://github.com/wch/r-source/blob/tags/R-3-5-1/src/main/RNG.c#L808 Many other languages use various rejection sampling approaches which provide an unbiased method for sampling, such as in Go, python, and others described here: https://arxiv.org/abs/1805.10941 (I believe the biased algorithm currently used in R is also described there). I'm not an expert in this area, but does it make sense for the R to adopt one of the unbiased random sample algorithms outlined there and used in other languages? Would a patch providing such an algorithm be welcome? What concerns would need to be addressed first? I believe this issue was also raised by Killie & Philip in http://r.789695.n4.nabble.com/Bug-in-sample-td4729483.html, and more recently in https://www.stat.berkeley.edu/~stark/Preprints/r-random-issues.pdf, pointing to the python implementation for comparison: https://github.com/statlab/cryptorandom/blob/master/cryptorandom/cryptorandom.py#L265
I think the analyses are correct, but I doubt if a change to the default is likely to be accepted as it would make it more difficult to reproduce older results. On the other hand, a contribution of a new function like sample() but not suffering from the bias would be good. The normal way to make such a contribution is in a user contributed package. By the way, R code illustrating the bias is probably not very hard to put together. I believe the bias manifests itself in sample() producing values with two different probabilities (instead of all equal probabilities). Those may differ by as much as one part in 2^32. It's
According to Kellie and Philip, in the attachment of the thread referenced by Carl, "The maximum ratio of selection probabilities can get as large as 1.5 if n is just below 2^31".
Sorry, I didn't write very well. I meant to say that the difference in
probabilities would be 2^-32, not that the ratio of probabilities would
be 1 + 2^-32.
By the way, I don't see the statement giving the ratio as 1.5, but maybe
I was looking in the wrong place. In Theorem 1 of the paper I was
looking in the ratio was "1 + m 2^{-w + 1}". In that formula m is your
n. If it is near 2^31, R uses w = 57 random bits, so the ratio would be
very, very small (one part in 2^25).
The worst case for R would happen when m is just below 2^25, where w
is at least 31 for the default generators. In that case the ratio could
be about 1.03.
Duncan Murdoch