Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A) = 1 \iff A = \Omega$. One consequence of this is that now the definition $$P(A|B) := \frac{P(A \cap B)}{P(B)}$$ is valid for any $A, B \in 2^{\Omega}$, so we can avoid all of the rigamarole with conditional expectations that arises in the standard formulation, just like how in non-standard analysis we have that derivatives are actually quotients. We also can measure every element of $2^\Omega$ so we don't even need to bother with $\sigma$-algebras.

What I like so much about this is that it removes many of the initial hurdles to dealing with measure-theoretic framework, such how if $X \sim \mathcal N(0,1)$ then $P(X = 0)=P(X = \textrm{"blue"}) = 0$, i.e. we can't distinguish impossible outcomes from "almost never" outcomes.

My question: are there ever practical uses of non-Kolmogrovian probability, and if not, what's so special about Kolmogrov's framework that it's the only one that we can or do use to compute actual quantities of interest? By practical uses I mean real-life probability computations, like coin tosses and die rolls or like machine learning predictions such as estimating the conditional probability of defaulting on a loan.

My guess would be that it comes down to how all of these models agree on the finite cases (like the probability of getting 3 heads in a row for a coin toss) and it's only on the infinite and not-so-practical cases that we see disagreement (like the probability of tossing a coin and getting heads forever, and cases where $\sigma$-additivity can lead to counterintuitive results), but I'm not sure about this.

Update: @pash has pointed out that with hyperreal-valued extensions we have the transfer principle. Does this completely settle the issue? The Kolmogorov formulation has different axioms though so can't I expect that there are true statements in one formulation that are false in the other? And regardless, this still doesn't seem to answer my question of what is so special about Kolmogorov's formulation that it is the root of all of these non-Archimedian extensions.

  • $\begingroup$ What do you mean by "practical uses"? It's not true that "Kolmogorov's framework [is] the only one that we can use to compute actual quantities of interest": you can do everything in a nonstandard formulation of probability that you can do in the conventional formulation, up to working with uncountable sample spaces in which the uncountability happens to matter for your problem. (In nonstandard formulations, spaces that conventionally would be countably infinite are modeled as hyperfinite/nonstandard finite). Historically NSA has seen its widest use in probability theory and its applications. $\endgroup$ – pash Jul 24 '17 at 23:46
  • $\begingroup$ @pash Thanks a lot for the comment. For the paper that i linked to, their axioms aren't the same as Kolmogorov's so can we still appeal to the transfer principle like this? I'm getting way out of my depth with this, so i may be completely wrong, but if we were to take the exact same Kolmogorov axioms but with a hyperreal-valued measure then i would definitely expect agreement, but how do i know this formulation agrees? As for "practical uses" I know that's vague but I'm picturing "real-life" combinatorics-style computations like coin tosses and die rolls $\endgroup$ – alfalfa Jul 25 '17 at 14:30
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    $\begingroup$ You are probably interested in Nelson's book Radically Elementary Probability Theory. Nelson uses the Internal Set Theory formulation of NSA, though he does not make this 100% explicit. What you find in going through it is that dealing with internal vs. external sets is not so different from the kinds of hassles you encounter in standard probability theory. There is somehow "conservation of difficulty" there. You gain some real elegance, though, for example $dB=\pm \sqrt{dt}$ is a meaningful statement per se rather than shorthand for something else. $\endgroup$ – Ian Jul 25 '17 at 15:52
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    $\begingroup$ have you seen this? $\endgroup$ – Masacroso Jul 25 '17 at 21:33
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    $\begingroup$ @Masacroso no i haven't, but that looks like it exactly answers my question! $\endgroup$ – alfalfa Jul 27 '17 at 18:42

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