Thanks Mike.  The paper looks potentially interesting for me.  I'll have to start reading it and get into the details to figure that out.  I'll think about the beta distribution idea, too.

(I just looked up "almost periodic sequence", which Mik suggested earlier.  That seems potentially relevant, too.)

There's no question that my thinking about what I wanted had vaguenesses and confusions, and this discussion has been very helpful, but my thinking continues to evolve.  

Concerning your earlier 11/13 questions about what I wanted, I wouldn't require that the frequencies remain strictly within an interval.  Marco Cattaneo's 2017 PMLR paper, on a different, related topic, inspired me to think that that would be an overly restrictive requirement.  Nor do I want to require that frequencies settle down to a particular distribution.

Of course the particular strategy that I described in my last email does imply that frequencies settle down in the limit, in fact to single, precise values, although I view that model not as realizing objective imprecise probabilities--it can't--but as modeling them.  At least, that's the idea I am exploring.

The new point that I have come to see, I think, is that limiting behavior is not directly relevant to the difference between precisely probabilistic behavior as usually conceived for real-world processes on reasonable time scales, and the sorts of behavior that Fine et al. and Gorban have characterized as illustrating or as best modeled by objective imprecise probability.  

"As usually conceived": It's a common practice to use limiting behavior as an indicator of what happens in the observable moderately long run, even though we know that in general one could have any limit at all, or no limit, consistent with any behavior in an observable (i.e. finite) sequence.  This use of limiting behavior to predict observable behavior works in practice given the kinds of models that are usually used in science.  Maybe the Fine et al. and Gorban examples, and other ones that I am thinking about, should not be characterized in terms of limiting behavior of a process.  That's what I am thinking now.

On Nov 16, 2018, at 5:01 PM, Michael Smithson <Michael.Smithson@anu.edu.au> wrote:

Hi Marshall,
Your latest email suggests that perhaps I wasn't mistaken when I asked for clarification about what kind of sequence you had in mind...  Anyhow, if you want to further "fuzz" the probabilities for the two biased coins, you could have their respective probabilities follow distributions (e.g., betas).  If you want their distributions to behave like "conjugate" lower-upper probabilities you could use the distributions that Parker Blakey and I wrote about in our IJAR paper that appeared this year (which I presented at the 2017 ISIPTA meeting). 
Kind regards,
--Mike
From: SIPTA <sipta-bounces@idsia.ch> on behalf of Abrams, Marshall <mabrams@uab.edu>
Sent: Saturday, 17 November 2018 6:03:31 AM
To: sipta@idsia.ch
Subject: Re: [SIPTA] Are there imprecise analogues of pseudo-random number generators?
 
This discussion has clarified for me that what I need is not a sequence with distinct liminf and limsup.  In fact I understand the examples of natural physical processes that have been proposed as involving imprecise chance as ones in which there is no tendency to settle down to stable frequencies within the short or medium term (e.g.  quartz oscillator flicker noise in Grize and Fine 1987, or many examples in Igor Gorban's two recent books in English).  For all we know, if these processes could be defined precisely enough that their infinite extensions could be predicted, the relative frequencies might indeed tend toward single limits.

The fact that such processes exist in nature and that some of them might involve largely deterministic processes (e.g. air and water temperature fluctuations or sea wave heights in Gorban's books) suggests that it might be possible in principle to generate such patterns algorithmically.


I still need to read some of the papers that have been suggested, but in the mean time I am now thinking about a very crude method for generating this kind of pattern.  Here is a simple illustration:

Step 0: Start with an empty list of (no) coin toss outcomes.

Step 1: Flip a fair coin to choose another coin which is biased with either p=0.4 or p=0.6 for heads.  Also toss a 1000-sided die of uniform density to choose an integer n between 1 and 1000, inclusive.

Step 2: Then toss (or spin) the chosen biased coin n times and add the outcomes to the end of the list.

Go back to step 1, iterating steps 1 and 2 until there is a moderately long list of coin toss outcomes, say 50,000, if that is enough data to drive one's ABM or other simulation.


With this particular example, the limit of the relative frequency of heads in the limit will be 0.5 with probability 1, because the single-coin subsequences of lengths n=1 through 1000 have equal probability, and the two biased coins have equal probability of being chosen for such a subsequence.  But in shorter runs, frequencies will not appear to tend to 0.5, but will fluctuate in the neighborhood of [0.4, 0.6].

This can all be translated into code using PRNGs (in which case it's a deterministic model of a probabilistic model used to model something that's not probabilistic!).  I've started doing this.

Variations are possible.  There can be more than two coins, chosen with unequal probabilities.  The subsequence length choice could be done differently, e.g. using a Poisson distribution with no maximum length.  Regardless, the subsequence length choice process might have to be tuned to the typical numbers of trials in your simulation, since if there are many trials, frequencies might settle toward the ultimate limit, which might not be what's wanted for the simulation.  (As noted, this is a very crude method.)


Does this make (some) sense?

Thanks very much, again.


Marshall

Marshall Abrams, Associate Professor 
Department of Philosophy, University of Alabama at Birmingham
Email: mabrams@uab.edu; Phone: (205) 996-7483;  Fax: (205) 975-6610
Mail: HB 414A, 900 13th Street South, Birmingham, AL 35294-1260;  Office: HB 418