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