From @t@rk @ending from @t@t@berkeley@edu  Wed Sep 19 18:23:07 2018
From: @t@rk @ending from @t@t@berkeley@edu (Philip B. Stark)
Date: Wed, 19 Sep 2018 09:23:07 -0700
Subject: [Rd] Bias in R's random integers?
In-Reply-To: <73c4667d-38ad-f5ef-468c-4c368e0fdc12@gmail.com>
References: <CAN_1p9wVX=AR8aYtC2LJo0bgN9cRW3XcbYJqnVHdCzA-QA1ceg@mail.gmail.com>
 <ffb74021-8d95-a820-8013-f0e51dc1a9b3@gmail.com>
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 <ece52410-ddb2-2198-9291-201475d3f46e@gmail.com>
 <CAA7KnHHEejYVn8NtKrY1KuKcJUq8n4LoJSDx9BZv-LJyC=r9yQ@mail.gmail.com>
 <73c4667d-38ad-f5ef-468c-4c368e0fdc12@gmail.com>
Message-ID: <CAA7KnHFcY5YJ2kTAfPJJ0PHwcWx7CumE49A4NgJWC2NK5CV=-A@mail.gmail.com>

No, the 2nd call only happens when m > 2**31. Here's the code:

(RNG.c, lines 793ff)

double R_unif_index(double dn)
{
    double cut = INT_MAX;

    switch(RNG_kind) {
    case KNUTH_TAOCP:
    case USER_UNIF:
    case KNUTH_TAOCP2:
cut = 33554431.0; /* 2^25 - 1 */
  break;
    default:
  break;
   }

    double u = dn > cut ? ru() : unif_rand();
    return floor(dn * u);
}

On Wed, Sep 19, 2018 at 9:20 AM Duncan Murdoch <murdoch.duncan at gmail.com>
wrote:

> On 19/09/2018 12:09 PM, Philip B. Stark wrote:
> > The 53 bits only encode at most 2^{32} possible values, because the
> > source of the float is the output of a 32-bit PRNG (the obsolete version
> > of MT). 53 bits isn't the relevant number here.
>
> No, two calls to unif_rand() are used.  There are two 32 bit values, but
> some of the bits are thrown away.
>
> Duncan Murdoch
>
> >
> > The selection ratios can get close to 2. Computer scientists don't do it
> > the way R does, for a reason.
> >
> > Regards,
> > Philip
> >
> > On Wed, Sep 19, 2018 at 9:05 AM Duncan Murdoch <murdoch.duncan at gmail.com
> > <mailto:murdoch.duncan at gmail.com>> wrote:
> >
> >     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 <mailto: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
> >     <
> https://www.stat.berkeley.edu/%7Estark/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
> >
> >
> >
> > --
> > Philip B. Stark | Associate Dean, Mathematical and Physical Sciences |
> > Professor,  Department of Statistics |
> > University of California
> > Berkeley, CA 94720-3860 | 510-394-5077 | statistics.berkeley.edu/~stark
> > <http://statistics.berkeley.edu/%7Estark> |
> > @philipbstark
> >
>
>

-- 
Philip B. Stark | Associate Dean, Mathematical and Physical Sciences |
Professor,  Department of Statistics |
University of California
Berkeley, CA 94720-3860 | 510-394-5077 | statistics.berkeley.edu/~stark |
@philipbstark

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[Rd] Bias in R's random integers?
Philip B. Stark