i am indeed. I know that john pollock employed nonstandard analysis to come up with a combinatorial convergence theorem; I am also vaguely aware of certain attempts of von neumann to prove convergence on the continuum (i think)

Take a good look at Edward Nelson's Radically Elementary Probability Theory. I linked a pdf. It seems to handle convergence pretty well. It's a quick and fun read. It doesn't handle (4), but it gives an NSA viewpoint on the first 3.

https://web.math.princeton.edu/~nelson/books/rept.pdf

thanks for this, before i read them however, do you know what it is generally meant by the exact law of large numbers as is often expressed in nonstandard probability theory. Does it just mean the analogue of the strong law of large numbers for nonstandard probability, or something altogether different and stronger (that has perhaps no analogue in classical probability theory)

Dear Chris I remember scanning through nelsons books a few months back. On one of teh later pages i came across something which seemed do indicate a stronger form of convergence (ie convergence not merely with probability 1, but either something like with frequency convergence with certainty to some close value of the probability, or a combinatorial version where probability one was given an interpretation as almost all sequences trialled whilst still indeterministic. Ill have to re-read it and post the page number again to make see if i was misinterpreting him

I'm not sure what problems have motivated your questions, but if you are interested in a nonstandard explanation of why some probabilistic models work well in practice - along with a new class of assumptions that are sufficient to guarantee correctness (whenever the assumptions are satisfied) - you may want to have a look at my new book: Rethinking Randomness.  The website is www.RethinkingRandomness.com.   Chapter 9 (Limiting Properties of Samples) includes discussions of the Law of Large Numbers and the Ergodic Theorem.  Best, Jeff Buzen

Dear Jeff, I will definitely look into this.

When you say 'guarantee correctness' do you mean absolute convergence or having frequencies that approximately match probabilities with certainty in the infinite long run, or frequencies which with certainty match the probabilties within some bound of error, or something else.

I myself have theories of my own and i just trying to clarify the results that are already produced. In particular there are allusions to laws which guarantee to stronger forms of convergence then almost sure in 'nonstandard probability theory, at least from my cursory examination. Edward Nelson Ill get the page number soon alludes to one, and there others. THere is also the phrase, 'exact law of large numbers' which is often used in nonstandard probability logics such as in the works of keisler and hoover (i think these pertain to uncountable long sequences- or continuum strong laws but i am not sure)

May 1, 2016

Dear William,

My background is in the development of stochastic models for the prediction of computer system performance.  Practitioners who use these models first validate them by observing the performance of an actual system over an interval of time, measuring the values of the model's parameters, and then substituting these measured values into the equations of the model to see if predicted values of throughput, response time, etc are correct. 

If the system conforms to all the assumptions of my models, it is possible to guarantee with certainty that these calculations will yield values that are identical to their observed counterparts.  This is what I mean by "guaranteed correctness."

This level of certainty is, of course, impossible to achieve if the model is based on traditional stochastic assumptions (except in limiting cases - and even then with qualifications such as "almost surely" or "except for a set of measure zero"). 

To resolve this issue, I have developed an alternative/nonstandard framework for specifying the assumptions of models that incorporate uncertainty.   This alternative framework makes it possible to derive solutions that have the same form as the steady state solutions derived from traditional stochastic models.  However, all symbolic variables in these equations are interpreted as directly measurable quantities rather than random variables or parameters of underlying probability distributions. 

You can find more details in Chapter 1 of my book Rethinking Randomness.    I've uploaded the most critical portions of this chapter to the website www.RethinkingRandomness.com.   Excerpts from all other chapters have been uploaded as well. 

You can also download a working paper (The Observability Principle) that provides a compact and self-contained discussion of my latest thinking on this topic.  Go to the Supplementary Materials section of the website to download.

Best,

Jeff

Dear Jeff thanks for this; ive had been developing a single case propensity model that attempt to literally reduce probability to a ranked conditional indicative conditional logic (precisely what i think von neumann); by this i do not mean what cox, et jaynes, carnap and the other logicists/objectivsts would have you believe where they try to reduce to deductive logic making it apriori, but at the same time fail to do so because they make use of symmetry (so they put probability in to get probability out). I mean contingent physically necessary implications relations that could vary in different worlds (so they are contingent only in that sense) but purely logical in that probabilities reduces to the correlations induced by the probability inequalities theirein and nothing more (so probability reduces to the logic not logic +symmetry, but logic simpliciter). This is to resolve what it means for something to have a probability on the single case; where i make use of a defeasible indicative conditional P(A)>P(B) if and ONLY IF B ->A, where --> is the physical indicative implication relation which in  many cases will be finkish/non-monotonic/defeasible/counter-counter-factual (take for example A to more probable than ~A, you will see that i get a liar like sentence, which i give  a semantics for).    I use to make sense of contextuality and entanglement as well, and entanglement and contextuality also helps in making sense of these implications and the probability, But I am still grappling with the frequency connection (without becoming a frequentist or denying randomness/immunity to gambling systems). I have numerous ideas including worlds dying off (in the finite and infinite - if they diverge too much), abstruse place selection rules that continually change and cycle akin to the rule following paradox-- which may be induced by absolute non-standard arithmetic convervence. There are also other ideas including conceiving time and space to be fundamentally point like where everything is randomally and non locally distributed ab initio and we are following the path (i have this for modal space, and actual space but not normal space) and then there other ideas involving the contraints which quantum entanglements place on things. Ill go for the first solution for the moment but ill definitely look at that chapter and article of yours (ive already ordered your book)

Dear William,

I don't completely follow everything you are saying, but I have a few thoughts that are relevant to an important point in your last post.  You write: "This is to resolve what it means for something to have a probability on the single case."

Begin by considering what it means for an observable quantity to be uncertain.   In a purely deterministic universe, all observable quantities are determined by sets of factors that are, in principle, knowable.   However, these controlling factors may be beyond the ability of the observer/experimenter to control or even perceive. Thus, to the observer, such quantities are regarded as being uncertain.   The outcome of a coin toss or a spin of the roulette wheel are classic examples of this type of uncertainty.

 Quantum theory postulates the existence of a different type of uncertainty that could be called "intrinsic uncertainty."   In this case, it is assumed that there is no way to determine the value of an observed quantity in advance -- regardless of how much information the observer/experimenter has.

 In either case, traditional mathematical models of uncertainty are based on an assumption that I refer to (in Rethinking Randomness) as "the sampling premise."  Simply put, the sampling premise assumes that each observed outcome can be regarded as a sample drawn at random from an underlying probability distribution. This provides an enormously powerful starting point for most mathematical analyses of uncertainty.  [Among other things, the traditional Law of Large Numbers deals with the question of how likely it is that a large number of observations/samples will have the same properties as the underlying distribution from which they are drawn.]

 The arguments I present in Rethinking Randomness are based on a fundamentally different starting point.  Uncertainty is not characterized in terms of samples drawn at random from underlying probability distributions.  Instead, uncertain values are simply regarded as immaterial details.  All that matters are the observable proportions of populations (or sub-populations) as a whole.  There is no need to make any assumptions about "the single case"  (even though it is very tempting to do so).   Assumptions are instead expressed in terms of what I am calling "loose constraints."  These constraints apply to aggregates rather than individuals. 

[As a side note, it is possible to state and prove an analog to the Law of Large Numbers within this context (see Chapter 8), but once again the result is expressed in terms of directly observable proportions rather than abstract probabilities.]

 I'm not sure how much of this thinking is applicable to the problems you are working on, but I do think that the link between immateriality and uncertainty is absolutely critical when thinking about what it means "to have a probability on the single case".

Best,

Jeff

in my model is also have a proportion based strong laws- its combinatorial in nature and thus could be said to be more reductive-- are you familiar with the  work of john pollock who argued that this was how the weak law of large numbers should always have been applied - he also uses non standard analysis.

Simply said i think he holds that instead of 'probability 1, the lrf=probability value', we should have that ;almost all infinite sequences (in the combinatorial sense) will be such that their lrf=probability value'. Combinatorial in the sense that if each sequence has equal measure, such as in binary sequences almost all binary sequence have probability=1/2.

However, there are two kinds of almost all, there is : for some such infinite sequence, almost all possible outcome sequences it could develop into have lrf=probability, which is probably what we dont want as its based on a classical static conception of probability--- akin to the original measure theoretical law of large numbers with no real connections to frequencies (schacter calls it kolmogorovs aporia) this is to be distinguished with, almost all infinite sequences of IID trials , trialled pre outcome (supposing we could trial an uncountable number of infinite trials all simultaneously) are such that necessarily, have lrf=probability value. The latter concerns almost all trials, or sequences, the latter only concerns almost all possible outcomes for a single trial, sequence