November 2018 - SIPTA News - lists.sipta.org

[SIPTA] WPMSIIP 2019

by COOLEN

Nov. 21, 2018

[SIPTA] Fw: IFSA-NAFIPS 2019 Second Call for papers

by Kreinovich

Nov. 21, 2018

[SIPTA] BigDat 2019: early registration December 8

by IRDTA

Nov. 20, 2018

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

by Abrams

Nov. 17, 2018

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

by Michael Smithson

Nov. 16, 2018

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

by Abrams

Nov. 16, 2018

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

by Marshall

Wow--thanks very much, everyone, for informative replies and discussion. Mik, Jay, Gert, and Marco, yes--your understanding accurately captures the general idea that I was thinking about. Thank you for the outlines of algorithms for generating the kind of sequence I had in mind. I may explore implementing something like this, but the discussion in thread is helping to clarify my thinking, and I wonder if I really do wonder whether I really do want a sequence that behaves as if it seems to be leading to a distinct liminf and limsup. That is certainly how I had been thinking about it. I need to think about it further. This discussion has been helpful for me in multiple respects. I had glanced at Gert and Jasper's paper earlier this year, I think, but I wasn't then asking the question I'm asking now, and had not read it at the time. I still have the paper in a reading queue, but I had forgotten about precisely what it was doing. Marco is right that it's finite sequences that interest me, but Gert sketched an algortithm in one of the emails, and I clearly need to read that paper regardless. (Mik, I am not sure I follow all of the reasoning in your last email below, but that's my fault. I need to read Gert and Jasper's paper first and then come back to these comments. By the way, Fine and his collaborators have quite a few papers on objective imprecise probability. The one I mentioned has been particularly interesting to me. Marco has a very interesting recent paper, too, and I mentioned several other authors--I'd be happy to send references if you'd like.) Best, Marshall > On Nov 13, 2018, at 3:11 PM, Professor Bickis <bickis(a)snoopy.usask.ca> wrote: > > A sequence such as Gert describes will generate subsequences whose limiting relative frequencies will not be identical. Nonetheless, such a sequence as a whole would still satisfy the strong law of large numbers, so that the relative frequency will (a.s.) converge to the average success probability. I am not sure that such a sequence is what the original question was asking. > >>> 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. > > If one restricts the sets over which averages are calculated, then one would observe the phenomenon of the frequency “not settling down”, but if the sets were allowed to grow without restriction, then the frequency would settle down to an average. > > It really depends on how one defines limits. In classical probability, the limit is defined in terms of a sequence of relative frequencies. For any epsilon>0, > can you find an n such that for all larger n the observed relative frequency is within epsilon of the limit. > > One could instead consider a net of finite sets (ordered by inclusion). Given epsilon, can one find a finite set such that for every finite superset the observed relative frequency is within epsilon of the limit. This criterion would fail for sequences of Gert’s type. It’s not immediately obvious to me that a strong law of large numbers would hold even for an i.i.d. sequence with this criterion. (An exercise for grad students, perhaps? :-)) > > Mik Bickis > >> On Nov 13, 2018, at 12:03 PM, Gert De Cooman <Gert.DeCooman(a)UGent.be> wrote: >> >> 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 >> >> _______________________________________________ >> 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. 15, 2018

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

by Marco Cattaneo

Dear all, I can only recommend the paper by Gert and Jasper, which includes several deep results about infinite sequences of 0's and 1's. However, as far as I understand, Marshall is interested in finite sequences of 0's and 1's that are generally accepted as "random", like the ones generated by PRNGs. Unfortunately, I do not know of any non-artificial way of generating such sequences for imprecise probabilities. Since iid sequences of 0's and 1's have relative frequencies that converge due to the law of large numbers, sequences for which the relative frequencies do not converge can be artificially obtained by dropping one (or both) of the two i's in iid. For instance, a sequence with relative frequency of 1's tending to oscillate between 0.6 and 0.8 can be obtained by keeping independence but varying the probability of 1's according to the following sequence: once 0.6, two times 0.9, six times 0.5, 18 times 0.9, 54 times 0.5, 162 times 0.9, and so on (alternating between 0.5 and 0.9 with block lengths increasing by threefold each time). By adding some randomness to the sequence of probabilities, this could be made less artificial, but I do not know of any work in which this kind of ideas is explicitly developed. Best wishes, Marco -----Original Message----- From: SIPTA <sipta-bounces(a)idsia.ch> On Behalf Of Gert De Cooman Sent: Tuesday, November 13, 2018 7:04 PM To: Dr. Alessandro Antonucci <alessandro(a)idsia.ch> Cc: sipta(a)idsia.ch Subject: Re: [SIPTA] Fwd: Are there imprecise analogues of pseudo-random number generators? 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 _______________________________________________ SIPTA mailing list SIPTA(a)idsia.ch http://mailman2.ti-edu.ch/mailman/listinfo/sipta

Nov. 14, 2018

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

by Professor Bickis

A sequence such as Gert describes will generate subsequences whose limiting relative frequencies will not be identical. Nonetheless, such a sequence as a whole would still satisfy the strong law of large numbers, so that the relative frequency will (a.s.) converge to the average success probability. I am not sure that such a sequence is what the original question was asking. >> 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. If one restricts the sets over which averages are calculated, then one would observe the phenomenon of the frequency “not settling down”, but if the sets were allowed to grow without restriction, then the frequency would settle down to an average. It really depends on how one defines limits. In classical probability, the limit is defined in terms of a sequence of relative frequencies. For any epsilon>0, can you find an n such that for all larger n the observed relative frequency is within epsilon of the limit. One could instead consider a net of finite sets (ordered by inclusion). Given epsilon, can one find a finite set such that for every finite superset the observed relative frequency is within epsilon of the limit. This criterion would fail for sequences of Gert’s type. It’s not immediately obvious to me that a strong law of large numbers would hold even for an i.i.d. sequence with this criterion. (An exercise for grad students, perhaps? :-)) Mik Bickis > On Nov 13, 2018, at 12:03 PM, Gert De Cooman <Gert.DeCooman(a)UGent.be> wrote: > > 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 > > _______________________________________________ > 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

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