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Re: [SIPTA] Fwd: Are there imprecise analogues of pseudo-random number generators?

by Gert De Cooman

My reply to Alessandro, building on my previous message : Precise randomness is more than convergence of the sample mean for the sequence – which is what you use in your argument – it implies also convergence to the same limit of the sample mean for any (computably selected) subsequence. (This is the notion of computable stochasticity, which is still weaker than randomness, and therefore implied by it). It is easy to see that if the extreme points are visited in a computable fashion, we can computably select subsequences whose sample mean will converge to any of the extreme points. Or arbitrarily close to any convex mixture of them. So the sequence will then not be precisely random, contrary to what you suggest. Computable randomness (in the precise case) of a sequence is the stronger property that unless you use the exact precise probability as your model for betting on the sequence of digits, you can be made to lose an unbounded amount of money by someone who bets against you using some computable strategy. It is this (martingale-theoretic) characterisation of (computable) randomness that we generalise to the imprecise case in our paper. Every good wish, Gert Prof. Gert de Cooman Universiteit Gent (ELIS–IDLab) — Durham University (Department of Mathematical Sciences) Technologiepark - Zwijnaarde 914, 9052 Zwijnaarde, Belgium t: +3292645653 — iMessage: +32496832628 e: gert.decooman(a)UGent.be w: users.ugent.be/~gdcooma > Op 13 nov. 2018, om 16:33 heeft Alessandro Antonucci <alessandro(a)idsia.ch> het volgende geschreven: > > Dear Marshall, > > For an imprecise subjectivist “the imprecision is in the eye of the > learner”. A non-stationary RNG can jump from one sampling distribution > to another but, again, this is done according to another > (second-order) distribution. The sample will be indistinguishable from > another one sampled from the marginal of that hierarchical model. The > imprecision level is basically a parameter of your learning algorithm > (e.g., s for the IDM, alpha for likelihood-based approachs), but not > something you might learn from your data. > > These are my two cents. > > Bests, > Alessandro > _________________________________ > > Alessandro Antonucci > IDSIA > Dalle Molle Institute > for Artificial Intelligence > Via Cantonale (Galleria 2) > CH-6928, Manno-Lugano, CH > > mail: alessandro(a)idsia.ch > skype: alessandro.antonucci > tel: +41 916108515 > web: www.idsia.ch/~alessandro > _________________________________ > > > ---------- Forwarded message --------- > From: Michael Smithson <Michael.Smithson(a)anu.edu.au> > Date: mar 13 nov 2018 alle ore 14:55 > Subject: Re: [SIPTA] Are there imprecise analogues of pseudo-random > number generators? > To: <alessandro.antonucci(a)gmail.com> > > > Hi Marshall, > > It isn't clear to me what you want the long-run state to be. Do you > want p(1) to settle down to having, say, a uniform distribution on > [.6,.8]? Or for the relative frequency of 1's to end up strictly > confined between .6 and .8? Or are you thinking of having p(1) fixed > at .7 and some other random variable controlling the interval width > around .7 that eventually settles on a width of .2? Or ... ? > > Kind regards, > > --Mike > > ________________________________ > From: SIPTA <sipta-bounces(a)idsia.ch> on behalf of Abrams, Marshall > <mabrams(a)uab.edu> > Sent: Tuesday, 13 November 2018 3:18:37 PM > To: sipta(a)idsia.ch > Subject: [SIPTA] Are there imprecise analogues of pseudo-random number > generators? > > I would like to ask a theoretical and practical question. (No > conference announcement here!) > > tl;dr: Are there imprecise analogues of pseudo-random number generators? > > If that question doesn't make sense, please feel free to read on, with > my thanks for doing so. (Sorry the following is so long; I was worried > that what I was asking would be unclear otherwise.) > > I am interested in imprecise chance, that is, objective imprecise > probability; this idea has been discussed in different ways by > Terrence Fine and his colleagues, Marco Cattaneo, Luke Glynn, Alan > Hajek, Suppes and Zanotti, Stephan Hartmann, Igor Gorban, and probably > others. > > In an agent-based model, (precise) chance can be modeled using a PRNG. > Can this be generalized to base an ABM on a model of imprecise chance? > I had thought this would be impossible, but now I am not so sure. A > good PRNG such as a Mersenne Twister is of course not truly chancy, > and its output even has low Kolmogorov complexity, but it's still a > good way to model chance in many contexts. Why couldn't imprecise > chance be modeled in an ABM using a deterministic algorithm as well? > > What would the output of a pseudo imprecise random number generator > (PIRNG) look like? > > Well, suppose one wrote a function that used a normal PRNG to generate > 0's and 1's with probability 0.7 for 1. Then over most large sets of > outputs, we would find the frequency of 1 to be close to 0.7, and as > we increased the size of such a set, the frequency would usually get > closer to 0.7 > > Similarly, using a PIRNG, one ought to be able to write a function > that generates 0's and 1's with (for example) an interval probability > of [0.6, 0.8] for 1. Then over most large lets of outputs, the > frequency of 1 should lie in or near [0.6, 0.8], and as we increased > the size of such a set, the frequency would remain near [0.6, 0.8] but > would usually wander without settling down to any precise value within > that interval. > > Are there known algorithms that might have this kind of behavior? Or > could this kind of behavior be simulated using PRNGs for sequences > that are not extremely long? i.e. maybe one can write a function > using a PRNG such that over the not-too-long-run, sequences would > appear to be imprecisely chancy, but they would eventually settle down > to precise values in the very very long run. That might be good > enough for some ABMs. > > Or am I just confused? (I think this is not unlikely.) > > Any suggestions about what sort of literature I should look at or what > kinds of keywords to search on would be welcome. > > [In some parts of Fierens, Rego, and Fine's "A frequentist > understanding of sets of measures" (2009), it sounds as if such a > class of algorithms might be described, at least in a loose, general > way, but if that is so, I have not understood the article well enough > to see how what's given there does so. Their model uses sequence > specified to be of intermediate Kolmogorov complexity, which itself > seems like a difficult thing to generate.] > > > Thank you! > > Marshall > > > Marshall Abrams, Associate Professor > Department of Philosophy, University of Alabama at Birmingham > Email: mabrams(a)uab.edu; Phone: (205) 996-7483; Fax: (205) 975-6610 > Mail: HB 414A, 900 13th Street South, Birmingham, AL 35294-1260; Office: HB 418 > Website: http://members.logical.net/~marshall > > _______________________________________________ > SIPTA mailing list > SIPTA(a)idsia.ch > http://mailman2.ti-edu.ch/mailman/listinfo/sipta > _______________________________________________ > SIPTA mailing list > SIPTA(a)idsia.ch > http://mailman2.ti-edu.ch/mailman/listinfo/sipta

Nov. 13, 2018

Re: [SIPTA] Fwd: Are there imprecise analogues of pseudo-random number generators?

by Gert De Cooman

Dear Marshall I refer to our paper about imprecise randomness (Computable randomness is inherently imprecise, De Cooman and De Bock) in the latest ISIPTA proceedings: https://biblio.ugent.be/publication/8531435/file/8531437.pdf In the paper we show that non-stationarity is indeed a source of imprecise randomness. We also prove a number of convergence results. All this for sequences of 0-1, which is the simplest case. We show, for instance, that alternating between a two un-fair coins with probabilities p and q for heads (in a deterministic fashion) will produce an imprecise random sequence of 0’s and 1’s with characterising probability interval [p,q]. The meaning of this is explained in the paper, but the underlying idea is that anyone using an interval betting model that does not include [p,q] - so for instance a precise r in this interval - can be made to lose an infinite amount of money in an infinite sequence of betting events on the digits in the sequence (using a computable strategy); and anywhere who uses a betting model that includes [p,q] is immune to such computable strategies. So this suggests an algorithm in the more general case: use an algorithm that visits the externe points of a credal set deterministically (eg, cyclically) , and for each visit to an extreme point, use the corresponding random generator to generate an outcome. Whether this also works in the case of more than two possible outcomes, still remains to be investigated, however. Best wishes, Gert > Op 13 nov. 2018, om 16:33 heeft Alessandro Antonucci <alessandro(a)idsia.ch> het volgende geschreven: > > Dear Marshall, > > For an imprecise subjectivist “the imprecision is in the eye of the > learner”. A non-stationary RNG can jump from one sampling distribution > to another but, again, this is done according to another > (second-order) distribution. The sample will be indistinguishable from > another one sampled from the marginal of that hierarchical model. The > imprecision level is basically a parameter of your learning algorithm > (e.g., s for the IDM, alpha for likelihood-based approachs), but not > something you might learn from your data. > > These are my two cents. > > Bests, > Alessandro > _________________________________ > > Alessandro Antonucci > IDSIA > Dalle Molle Institute > for Artificial Intelligence > Via Cantonale (Galleria 2) > CH-6928, Manno-Lugano, CH > > mail: alessandro(a)idsia.ch > skype: alessandro.antonucci > tel: +41 916108515 > web: www.idsia.ch/~alessandro > _________________________________ > > > ---------- Forwarded message --------- > From: Michael Smithson <Michael.Smithson(a)anu.edu.au> > Date: mar 13 nov 2018 alle ore 14:55 > Subject: Re: [SIPTA] Are there imprecise analogues of pseudo-random > number generators? > To: <alessandro.antonucci(a)gmail.com> > > > Hi Marshall, > > It isn't clear to me what you want the long-run state to be. Do you > want p(1) to settle down to having, say, a uniform distribution on > [.6,.8]? Or for the relative frequency of 1's to end up strictly > confined between .6 and .8? Or are you thinking of having p(1) fixed > at .7 and some other random variable controlling the interval width > around .7 that eventually settles on a width of .2? Or ... ? > > Kind regards, > > --Mike > > ________________________________ > From: SIPTA <sipta-bounces(a)idsia.ch> on behalf of Abrams, Marshall > <mabrams(a)uab.edu> > Sent: Tuesday, 13 November 2018 3:18:37 PM > To: sipta(a)idsia.ch > Subject: [SIPTA] Are there imprecise analogues of pseudo-random number > generators? > > I would like to ask a theoretical and practical question. (No > conference announcement here!) > > tl;dr: Are there imprecise analogues of pseudo-random number generators? > > If that question doesn't make sense, please feel free to read on, with > my thanks for doing so. (Sorry the following is so long; I was worried > that what I was asking would be unclear otherwise.) > > I am interested in imprecise chance, that is, objective imprecise > probability; this idea has been discussed in different ways by > Terrence Fine and his colleagues, Marco Cattaneo, Luke Glynn, Alan > Hajek, Suppes and Zanotti, Stephan Hartmann, Igor Gorban, and probably > others. > > In an agent-based model, (precise) chance can be modeled using a PRNG. > Can this be generalized to base an ABM on a model of imprecise chance? > I had thought this would be impossible, but now I am not so sure. A > good PRNG such as a Mersenne Twister is of course not truly chancy, > and its output even has low Kolmogorov complexity, but it's still a > good way to model chance in many contexts. Why couldn't imprecise > chance be modeled in an ABM using a deterministic algorithm as well? > > What would the output of a pseudo imprecise random number generator > (PIRNG) look like? > > Well, suppose one wrote a function that used a normal PRNG to generate > 0's and 1's with probability 0.7 for 1. Then over most large sets of > outputs, we would find the frequency of 1 to be close to 0.7, and as > we increased the size of such a set, the frequency would usually get > closer to 0.7 > > Similarly, using a PIRNG, one ought to be able to write a function > that generates 0's and 1's with (for example) an interval probability > of [0.6, 0.8] for 1. Then over most large lets of outputs, the > frequency of 1 should lie in or near [0.6, 0.8], and as we increased > the size of such a set, the frequency would remain near [0.6, 0.8] but > would usually wander without settling down to any precise value within > that interval. > > Are there known algorithms that might have this kind of behavior? Or > could this kind of behavior be simulated using PRNGs for sequences > that are not extremely long? i.e. maybe one can write a function > using a PRNG such that over the not-too-long-run, sequences would > appear to be imprecisely chancy, but they would eventually settle down > to precise values in the very very long run. That might be good > enough for some ABMs. > > Or am I just confused? (I think this is not unlikely.) > > Any suggestions about what sort of literature I should look at or what > kinds of keywords to search on would be welcome. > > [In some parts of Fierens, Rego, and Fine's "A frequentist > understanding of sets of measures" (2009), it sounds as if such a > class of algorithms might be described, at least in a loose, general > way, but if that is so, I have not understood the article well enough > to see how what's given there does so. Their model uses sequence > specified to be of intermediate Kolmogorov complexity, which itself > seems like a difficult thing to generate.] > > > Thank you! > > Marshall > > > Marshall Abrams, Associate Professor > Department of Philosophy, University of Alabama at Birmingham > Email: mabrams(a)uab.edu; Phone: (205) 996-7483; Fax: (205) 975-6610 > Mail: HB 414A, 900 13th Street South, Birmingham, AL 35294-1260; Office: HB 418 > Website: http://members.logical.net/~marshall > > _______________________________________________ > SIPTA mailing list > SIPTA(a)idsia.ch > http://mailman2.ti-edu.ch/mailman/listinfo/sipta > _______________________________________________ > SIPTA mailing list > SIPTA(a)idsia.ch > http://mailman2.ti-edu.ch/mailman/listinfo/sipta

Nov. 13, 2018

[SIPTA] Fwd: Are there imprecise analogues of pseudo-random number generators?

by Alessandro Antonucci

Dear Marshall, For an imprecise subjectivist “the imprecision is in the eye of the learner”. A non-stationary RNG can jump from one sampling distribution to another but, again, this is done according to another (second-order) distribution. The sample will be indistinguishable from another one sampled from the marginal of that hierarchical model. The imprecision level is basically a parameter of your learning algorithm (e.g., s for the IDM, alpha for likelihood-based approachs), but not something you might learn from your data. These are my two cents. Bests, Alessandro _________________________________ Alessandro Antonucci IDSIA Dalle Molle Institute for Artificial Intelligence Via Cantonale (Galleria 2) CH-6928, Manno-Lugano, CH mail: alessandro(a)idsia.ch skype: alessandro.antonucci tel: +41 916108515 web: www.idsia.ch/~alessandro _________________________________ ---------- Forwarded message --------- From: Michael Smithson <Michael.Smithson(a)anu.edu.au> Date: mar 13 nov 2018 alle ore 14:55 Subject: Re: [SIPTA] Are there imprecise analogues of pseudo-random number generators? To: <alessandro.antonucci(a)gmail.com> Hi Marshall, It isn't clear to me what you want the long-run state to be. Do you want p(1) to settle down to having, say, a uniform distribution on [.6,.8]? Or for the relative frequency of 1's to end up strictly confined between .6 and .8? Or are you thinking of having p(1) fixed at .7 and some other random variable controlling the interval width around .7 that eventually settles on a width of .2? Or ... ? Kind regards, --Mike ________________________________ From: SIPTA <sipta-bounces(a)idsia.ch> on behalf of Abrams, Marshall <mabrams(a)uab.edu> Sent: Tuesday, 13 November 2018 3:18:37 PM To: sipta(a)idsia.ch Subject: [SIPTA] Are there imprecise analogues of pseudo-random number generators? I would like to ask a theoretical and practical question. (No conference announcement here!) tl;dr: Are there imprecise analogues of pseudo-random number generators? If that question doesn't make sense, please feel free to read on, with my thanks for doing so. (Sorry the following is so long; I was worried that what I was asking would be unclear otherwise.) I am interested in imprecise chance, that is, objective imprecise probability; this idea has been discussed in different ways by Terrence Fine and his colleagues, Marco Cattaneo, Luke Glynn, Alan Hajek, Suppes and Zanotti, Stephan Hartmann, Igor Gorban, and probably others. In an agent-based model, (precise) chance can be modeled using a PRNG. Can this be generalized to base an ABM on a model of imprecise chance? I had thought this would be impossible, but now I am not so sure. A good PRNG such as a Mersenne Twister is of course not truly chancy, and its output even has low Kolmogorov complexity, but it's still a good way to model chance in many contexts. Why couldn't imprecise chance be modeled in an ABM using a deterministic algorithm as well? What would the output of a pseudo imprecise random number generator (PIRNG) look like? Well, suppose one wrote a function that used a normal PRNG to generate 0's and 1's with probability 0.7 for 1. Then over most large sets of outputs, we would find the frequency of 1 to be close to 0.7, and as we increased the size of such a set, the frequency would usually get closer to 0.7 Similarly, using a PIRNG, one ought to be able to write a function that generates 0's and 1's with (for example) an interval probability of [0.6, 0.8] for 1. Then over most large lets of outputs, the frequency of 1 should lie in or near [0.6, 0.8], and as we increased the size of such a set, the frequency would remain near [0.6, 0.8] but would usually wander without settling down to any precise value within that interval. Are there known algorithms that might have this kind of behavior? Or could this kind of behavior be simulated using PRNGs for sequences that are not extremely long? i.e. maybe one can write a function using a PRNG such that over the not-too-long-run, sequences would appear to be imprecisely chancy, but they would eventually settle down to precise values in the very very long run. That might be good enough for some ABMs. Or am I just confused? (I think this is not unlikely.) Any suggestions about what sort of literature I should look at or what kinds of keywords to search on would be welcome. [In some parts of Fierens, Rego, and Fine's "A frequentist understanding of sets of measures" (2009), it sounds as if such a class of algorithms might be described, at least in a loose, general way, but if that is so, I have not understood the article well enough to see how what's given there does so. Their model uses sequence specified to be of intermediate Kolmogorov complexity, which itself seems like a difficult thing to generate.] Thank you! Marshall Marshall Abrams, Associate Professor Department of Philosophy, University of Alabama at Birmingham Email: mabrams(a)uab.edu; Phone: (205) 996-7483; Fax: (205) 975-6610 Mail: HB 414A, 900 13th Street South, Birmingham, AL 35294-1260; Office: HB 418 Website: http://members.logical.net/~marshall _______________________________________________ SIPTA mailing list SIPTA(a)idsia.ch http://mailman2.ti-edu.ch/mailman/listinfo/sipta

Nov. 13, 2018

Re: [SIPTA] Are there imprecise analogues of pseudo-random number generators?

by Professor Bickis

Hello Marshall: I think I understand what you are looking for. The law of large numbers says that with probability one, a sequence of i.i.d. random bits will settle down to a relative frequency corresponding to the probability of a 1. i.e., the lim sup and the lim inf of the relative frequency would both converge to the same number. What you are looking for is a generator of a sequence in which the lim sup is (e.g.) 0.8 and the lim inf is 0.6. Certainly such sequences exist, and with some ingenuity one could construct a way of generating such sequences. However, I am skeptical about what kind of probabilistic interpretation could be given to them, considering what the law of large numbers says. I imagine you would have to build in some kind of memory to the process, violating the independence of the trials. The obvious trick would be to vary the underlying probability that generates the sequence, but I don’t that would work. The relative frequency would just settle down to the average of the probabilities. My understanding of imprecise probability is that it is purely epistemic, but I note that you reference a (relatively) recent paper by Fine et al. about objective imprecise probability. I’ll have to look that up. These are just my immediate thoughts. It is an interesting theoretical question. I am not quite clear about what the application might be. Just an afterthought. There is a concept in analysis called “almost periodic sequences”, which are related to the phenomenon of quasi-crystals, and things like Penrose tilings. I don’t know much about these, but there may be a connection. Mik Bickis Professor Emeritus Department of Mathematics and Statistics University of Saskatchewan > On Nov 13, 2018, at 06:34 AM, Michael Smithson <Michael.Smithson(a)anu.edu.au> wrote: > > Hi Marshall, > It isn't clear to me what you want the long-run state to be. Do you want p(1) to settle down to having, say, a uniform distribution on [.6,.8]? Or for the relative frequency of 1's to end up strictly confined between .6 and .8? Or are you thinking of having p(1) fixed at .7 and some other random variable controlling the interval width around .7 that eventually settles on a width of .2? Or ... ? > Kind regards, > --Mike > From: SIPTA <sipta-bounces(a)idsia.ch> on behalf of Abrams, Marshall <mabrams(a)uab.edu> > Sent: Tuesday, 13 November 2018 3:18:37 PM > To: sipta(a)idsia.ch > Subject: [SIPTA] Are there imprecise analogues of pseudo-random number generators? > > I would like to ask a theoretical and practical question. (No conference announcement here!) > > tl;dr: Are there imprecise analogues of pseudo-random number generators? > > If that question doesn't make sense, please feel free to read on, with my thanks for doing so. (Sorry the following is so long; I was worried that what I was asking would be unclear otherwise.) > > I am interested in imprecise chance, that is, objective imprecise probability; this idea has been discussed in different ways by Terrence Fine and his colleagues, Marco Cattaneo, Luke Glynn, Alan Hajek, Suppes and Zanotti, Stephan Hartmann, Igor Gorban, and probably others. > > In an agent-based model, (precise) chance can be modeled using a PRNG. Can this be generalized to base an ABM on a model of imprecise chance? I had thought this would be impossible, but now I am not so sure. A good PRNG such as a Mersenne Twister is of course not truly chancy, and its output even has low Kolmogorov complexity, but it's still a good way to model chance in many contexts. Why couldn't imprecise chance be modeled in an ABM using a deterministic algorithm as well? > > What would the output of a pseudo imprecise random number generator (PIRNG) look like? > > Well, suppose one wrote a function that used a normal PRNG to generate 0's and 1's with probability 0.7 for 1. Then over most large sets of outputs, we would find the frequency of 1 to be close to 0.7, and as we increased the size of such a set, the frequency would usually get closer to 0.7 > > Similarly, using a PIRNG, one ought to be able to write a function that generates 0's and 1's with (for example) an interval probability of [0.6, 0.8] for 1. Then over most large lets of outputs, the frequency of 1 should lie in or near [0.6, 0.8], and as we increased the size of such a set, the frequency would remain near [0.6, 0.8] but would usually wander without settling down to any precise value within that interval. > > Are there known algorithms that might have this kind of behavior? Or could this kind of behavior be simulated using PRNGs for sequences that are not extremely long? i.e. maybe one can write a function using a PRNG such that over the not-too-long-run, sequences would appear to be imprecisely chancy, but they would eventually settle down to precise values in the very very long run. That might be good enough for some ABMs. > > Or am I just confused? (I think this is not unlikely.) > > Any suggestions about what sort of literature I should look at or what kinds of keywords to search on would be welcome. > > [In some parts of Fierens, Rego, and Fine's "A frequentist understanding of sets of measures" (2009), it sounds as if such a class of algorithms might be described, at least in a loose, general way, but if that is so, I have not understood the article well enough to see how what's given there does so. Their model uses sequence specified to be of intermediate Kolmogorov complexity, which itself seems like a difficult thing to generate.] > > > Thank you! > > Marshall > > > Marshall Abrams, Associate Professor > Department of Philosophy, University of Alabama at Birmingham > Email: mabrams(a)uab.edu; Phone: (205) 996-7483; Fax: (205) 975-6610 > Mail: HB 414A, 900 13th Street South, Birmingham, AL 35294-1260; Office: HB 418 > Website: http://members.logical.net/~marshall > > _______________________________________________ > SIPTA mailing list > SIPTA(a)idsia.ch > http://mailman2.ti-edu.ch/mailman/listinfo/sipta

Nov. 13, 2018

Re: [SIPTA] Are there imprecise analogues of pseudo-random number generators?

by Michael Smithson

Nov. 13, 2018

[SIPTA] AlCoB 2019: 1st call for papers

by IRDTA

Nov. 13, 2018

[SIPTA] Are there imprecise analogues of pseudo-random number generators?

by Abrams

Nov. 13, 2018

[SIPTA] FW: EUSFLAT 2019 - Call for papers

by Kreinovich

Nov. 10, 2018

[SIPTA] FW: open position in computational science

by Kreinovich

Nov. 8, 2018

[SIPTA] 2018 Lotfi A. Zadeh Pioneer Award

by Kreinovich

Dear Friends, I have just learned that Ronald R. Yager was aearded the 2018 Lotfi A. Zadeh Pioneer Award from the IEEE Systems, Man, and Cybernetics Society. The award is given to honor a person with outstanding and pioneering contributions to academic and/or industrial research in systems science and engineering, human-machine systems, and/or cybernetics. Dr. Yager is a pioneer in many areas of fuzzy research. He also had published several pioneering seminal papers on interval-valued probabilities and other imprecise probability topics. Congratulations to Dr. Yager!

Nov. 7, 2018