# Bias in R's random integers?

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## Bias in R's random integers?

 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#L808Many 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#L265Thanks! Carl -- http://carlboettiger.info        [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 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#L265I 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 very difficult to detect a probability difference that small, but if you define the partition of values into the high probability values vs the low probability values, you can probably detect the difference in a feasible simulation. Duncan Murdoch ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch (<[hidden email]>) escribió: > > 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". Iñaki > very difficult to detect a probability difference that small, but if you > define the partition of values into the high probability values vs the > low probability values, you can probably detect the difference in a > feasible simulation. > > Duncan Murdoch > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 In reply to this post by Duncan Murdoch-2 On Wed, 19 Sep 2018 at 13:43, Duncan Murdoch <[hidden email]> wrote: > > 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. I'm a bit alarmed by the logic here. Unbiased sampling seems basic for a statistical language. As a consumer of R I'd like to think that e.g. my bootstrapped p values are correct. Surely if the old results depend on the biased algorithm, then they are false results? -- Sent from Gmail Mobile         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 On 2018-09-19 09:40 AM, David Hugh-Jones wrote: > On Wed, 19 Sep 2018 at 13:43, Duncan Murdoch <[hidden email]> > wrote: > >> >> 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. > > > I'm a bit alarmed by the logic here. Unbiased sampling seems basic for a > statistical language. As a consumer of R I'd like to think that e.g. my > bootstrapped p values are correct. > Surely if the old results depend on the biased algorithm, then they are > false results? >    Balancing backward compatibility and correctness is a tough problem here.  If this goes into base R, what's the best way to do it?  What was the protocol for migrating away from the "buggy Kinderman-Ramage" generator, back in the day?   (Version 1.7 was sometime between 2001 and 2004).   I couldn't find the exact commit in the GitHub mirror: this is related ... https://github.com/wch/r-source/commit/7ad3044639fd1fe093c655e573fd1a67aa7f55f6#diff-dbcad570d4fb9b7005550ff630543b37=== ‘normal.kind’ can be ‘"Kinderman-Ramage"’, ‘"Buggy      Kinderman-Ramage"’ (not for ‘set.seed’), ‘"Ahrens-Dieter"’,      ‘"Box-Muller"’, ‘"Inversion"’ (the default), or ‘"user-supplied"’.      (For inversion, see the reference in ‘qnorm’.)  The      Kinderman-Ramage generator used in versions prior to 1.7.0 (now      called ‘"Buggy"’) had several approximation errors and should only      be used for reproduction of old results. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 In reply to this post by Iñaki Ucar On 19/09/2018 9:09 AM, Iñaki Ucar wrote: > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch > (<[hidden email]>) escribió: >> >> 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 ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 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. 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 <[hidden email]> wrote: > On 19/09/2018 9:09 AM, Iñaki Ucar wrote: > > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch > > (<[hidden email]>) escribió: > >> > >> 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 > -- 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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 In reply to this post by David Hugh-Jones-3 On 19/09/2018 9:40 AM, David Hugh-Jones wrote: > > > On Wed, 19 Sep 2018 at 13:43, Duncan Murdoch <[hidden email] > > wrote: > > >     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. > > > I'm a bit alarmed by the logic here. Unbiased sampling seems basic for a > statistical language. As a consumer of R I'd like to think that e.g. my > bootstrapped p values are correct. > Surely if the old results depend on the biased algorithm, then they are > false results? All Monte Carlo results contain Monte Carlo error.  Using the biased function will have some additional error, but for almost all simulations, it will be negligible compared to the Monte Carlo error.  I suspect the only simulations where the bias was anywhere near the same order of magnitude as the Monte Carlo error would be ones designed with this specific code in mind. Duncan Murdoch ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 That depends on the number of replications, among other things. Moreover, because of the bias, the usual formulae for uncertainty in estimates based on random samples, etc., are incorrect: sample() does not give a simple random sample. On Wed, Sep 19, 2018 at 9:15 AM Duncan Murdoch <[hidden email]> wrote: > On 19/09/2018 9:40 AM, David Hugh-Jones wrote: > > > > > > On Wed, 19 Sep 2018 at 13:43, Duncan Murdoch <[hidden email] > > > wrote: > > > > > >     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. > > > > > > I'm a bit alarmed by the logic here. Unbiased sampling seems basic for a > > statistical language. As a consumer of R I'd like to think that e.g. my > > bootstrapped p values are correct. > > Surely if the old results depend on the biased algorithm, then they are > > false results? > > All Monte Carlo results contain Monte Carlo error.  Using the biased > function will have some additional error, but for almost all > simulations, it will be negligible compared to the Monte Carlo error.  I > suspect the only simulations where the bias was anywhere near the same > order of magnitude as the Monte Carlo error would be ones designed with > this specific code in mind. > > 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 | @philipbstark         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 In reply to this post by Philip B. Stark 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 <[hidden email] > > wrote: > >     On 19/09/2018 9:09 AM, Iñaki Ucar wrote: >      > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch >      > (<[hidden email] >) >     escribió: >      >> >      >> 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 > > > > -- > 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 > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 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 <[hidden email]> 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 <[hidden email] > > > wrote: > > > >     On 19/09/2018 9:09 AM, Iñaki Ucar wrote: > >      > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch > >      > (<[hidden email] >) > >     escribió: > >      >> > >      >> 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 > > | > > @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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 On 19/09/2018 12:23 PM, Philip B. Stark wrote: > No, the 2nd call only happens when m > 2**31. Here's the code: Yes, you're right. Sorry! So the ratio really does come close to 2.  However, the difference in probabilities between outcomes is still at most 2^-32 when m is less than that cutoff.  That's not feasible to detect; the only detectable difference would happen if some event was constructed to hold an abundance of outcomes with especially low (or especially high) probability. As I said in my original post, it's probably not hard to construct such a thing, but as I've said more recently, it probably wouldn't happen by chance.  Here's one attempt to do it: Call the values from unif_rand() "the unif_rand() outcomes".  Call the values from sample() the sample outcomes. It would be easiest to see the error if half of the sample() outcomes used two unif_rand() outcomes, and half used just one.  That would mean m should be (2/3) * 2^32, but that's too big and would trigger the other version. So how about half use 2 unif_rands(), and half use 3?  That means m = (2/5) * 2^32 = 1717986918.  A good guess is that sample() outcomes would alternate between the two possibilities, so our event could be even versus odd outcomes. Let's try it:  > m <- (2/5)*2^32  > m > 2^31 [1] FALSE  > x <- sample(m, 1000000, replace = TRUE)  > table(x %% 2)       0      1 399850 600150 Since m is an even number, the true proportions of evens and odds should be exactly 0.5.  That's some pretty strong evidence of the bug in the generator.  (Note that the ratio of the observed probabilities is about 1.5, so I may not be the first person to have done this.) I'm still not convinced that there has ever been a simulation run with detectable bias compared to Monte Carlo error unless it (like this one) was designed specifically to show the problem. Duncan Murdoch > > (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 <[hidden email] > > 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 >     <[hidden email] >      >     >> wrote: >      > >      >     On 19/09/2018 9:09 AM, Iñaki Ucar wrote: >      >      > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch >      >      > (<[hidden email] >         >>) >      >     escribió: >      >      >> >      >      >> 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 >      > >      > >      > >      > -- >      > 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 >      > > > > > -- > 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 > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 Hi Duncan-- Nice simulation! The absolute difference in probabilities is small, but the maximum relative difference grows from something negligible to almost 2 as m approaches 2**31. Because the L_1 distance between the uniform distribution on {1, ..., m} and what you actually get is large, there have to be test functions whose expectations are quite different under the two distributions. Whether those correspond to commonly used statistics or not, I have no idea. Regarding backwards compatibility: as a user, I'd rather the default sample() do the best possible thing, and take an extra step to use something like sample(..., legacy=TRUE) if I want to reproduce old results. Regards, Philip On Wed, Sep 19, 2018 at 9:50 AM Duncan Murdoch <[hidden email]> wrote: > On 19/09/2018 12:23 PM, Philip B. Stark wrote: > > No, the 2nd call only happens when m > 2**31. Here's the code: > > Yes, you're right. Sorry! > > So the ratio really does come close to 2.  However, the difference in > probabilities between outcomes is still at most 2^-32 when m is less > than that cutoff.  That's not feasible to detect; the only detectable > difference would happen if some event was constructed to hold an > abundance of outcomes with especially low (or especially high) probability. > > As I said in my original post, it's probably not hard to construct such > a thing, but as I've said more recently, it probably wouldn't happen by > chance.  Here's one attempt to do it: > > Call the values from unif_rand() "the unif_rand() outcomes".  Call the > values from sample() the sample outcomes. > > It would be easiest to see the error if half of the sample() outcomes > used two unif_rand() outcomes, and half used just one.  That would mean > m should be (2/3) * 2^32, but that's too big and would trigger the other > version. > > So how about half use 2 unif_rands(), and half use 3?  That means m = > (2/5) * 2^32 = 1717986918.  A good guess is that sample() outcomes would > alternate between the two possibilities, so our event could be even > versus odd outcomes. > > Let's try it: > >  > m <- (2/5)*2^32 >  > m > 2^31 > [1] FALSE >  > x <- sample(m, 1000000, replace = TRUE) >  > table(x %% 2) > >       0      1 > 399850 600150 > > Since m is an even number, the true proportions of evens and odds should > be exactly 0.5.  That's some pretty strong evidence of the bug in the > generator.  (Note that the ratio of the observed probabilities is about > 1.5, so I may not be the first person to have done this.) > > I'm still not convinced that there has ever been a simulation run with > detectable bias compared to Monte Carlo error unless it (like this one) > was designed specifically to show the problem. > > Duncan Murdoch > > > > > (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 <[hidden email] > > > 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 > >     <[hidden email] > >      > >     >> wrote: > >      > > >      >     On 19/09/2018 9:09 AM, Iñaki Ucar wrote: > >      >      > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch > >      >      > (<[hidden email] > >     >     >>) > >      >     escribió: > >      >      >> > >      >      >> 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> > >      > > >       < > 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 > >     > >      > | > >      > @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 > > > > -- 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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

 One more thing, apropos this: I'm still not convinced that there has ever been a simulation run with detectable bias compared to Monte Carlo error unless it (like this one) was designed specifically to show the problem. I often use random permutations to simulate p-values to calibrate permutation tests. If I'm trying to simulate the probability of a low-probability event, this could matter a lot. Best wishes, Philip On Wed, Sep 19, 2018 at 12:52 PM Philip B. Stark <[hidden email]> wrote: > Hi Duncan-- > > Nice simulation! > > The absolute difference in probabilities is small, but the maximum > relative difference grows from something negligible to almost 2 as m > approaches 2**31. > > Because the L_1 distance between the uniform distribution on {1, ..., m} > and what you actually get is large, there have to be test functions whose > expectations are quite different under the two distributions. Whether those > correspond to commonly used statistics or not, I have no idea. > > Regarding backwards compatibility: as a user, I'd rather the default > sample() do the best possible thing, and take an extra step to use > something like sample(..., legacy=TRUE) if I want to reproduce old results. > > Regards, > Philip > > On Wed, Sep 19, 2018 at 9:50 AM Duncan Murdoch <[hidden email]> > wrote: > >> On 19/09/2018 12:23 PM, Philip B. Stark wrote: >> > No, the 2nd call only happens when m > 2**31. Here's the code: >> >> Yes, you're right. Sorry! >> >> So the ratio really does come close to 2.  However, the difference in >> probabilities between outcomes is still at most 2^-32 when m is less >> than that cutoff.  That's not feasible to detect; the only detectable >> difference would happen if some event was constructed to hold an >> abundance of outcomes with especially low (or especially high) >> probability. >> >> As I said in my original post, it's probably not hard to construct such >> a thing, but as I've said more recently, it probably wouldn't happen by >> chance.  Here's one attempt to do it: >> >> Call the values from unif_rand() "the unif_rand() outcomes".  Call the >> values from sample() the sample outcomes. >> >> It would be easiest to see the error if half of the sample() outcomes >> used two unif_rand() outcomes, and half used just one.  That would mean >> m should be (2/3) * 2^32, but that's too big and would trigger the other >> version. >> >> So how about half use 2 unif_rands(), and half use 3?  That means m = >> (2/5) * 2^32 = 1717986918.  A good guess is that sample() outcomes would >> alternate between the two possibilities, so our event could be even >> versus odd outcomes. >> >> Let's try it: >> >>  > m <- (2/5)*2^32 >>  > m > 2^31 >> [1] FALSE >>  > x <- sample(m, 1000000, replace = TRUE) >>  > table(x %% 2) >> >>       0      1 >> 399850 600150 >> >> Since m is an even number, the true proportions of evens and odds should >> be exactly 0.5.  That's some pretty strong evidence of the bug in the >> generator.  (Note that the ratio of the observed probabilities is about >> 1.5, so I may not be the first person to have done this.) >> >> I'm still not convinced that there has ever been a simulation run with >> detectable bias compared to Monte Carlo error unless it (like this one) >> was designed specifically to show the problem. >> >> Duncan Murdoch >> >> > >> > (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 < >> [hidden email] >> > > 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 >> >     <[hidden email] >> >      > > >     >> wrote: >> >      > >> >      >     On 19/09/2018 9:09 AM, Iñaki Ucar wrote: >> >      >      > El mié., 19 sept. 2018 a las 14:43, Duncan Murdoch >> >      >      > (<[hidden email] >> >     > >     >>) >> >      >     escribió: >> >      >      >> >> >      >      >> 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> >> >      > >> >       < >> 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 >> >     >> >      > | >> >      > @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 >> > >> >> > > -- > 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 > > -- 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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
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## Re: Bias in R's random integers?

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## Re: Bias in R's random integers?

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## Re: Bias in R's random integers?

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## Re: Bias in R's random integers?

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