I certainly should read Jakob Bernoulli's Ars Conjectandi again but let me share my concerns.

I'm just wondering when and how the Bernoulli distribution $Be(p)$ (and related distributions like the binomial $B(n,p)$) with real binomial proportion $p$ was introduced.

Indeed, classical probability theory defines probability as ratios

$p = \frac{{{\text{number of favorable cases}}}}{{{\text{number of possible cases}}}}$

and, subsequently, makes allowance only for rational probabilities like said binomial proportion $p$.

Bernoulli distributions with real binomial proportions can be defined within infinite frequentism, as the limit of relative frequencies. But it seems like (infinite) frequentism was introduced a long time after the Bernoulli distribution with real $p$.

Or we should start with a sequence of Bernoulli distributions with rational $p$'s and pass to the limit to build Bernoulli distributions with real $p$'s within classical probability theory, but I'm not aware of any construction like this.

So, assuming that infinite frequentism was introduced a long time after the Bernoulli distribution with real $p$, how did we go from Bernoulli distributions with rational $p$ to Bernoulli distributions with real $p$ within classical probability theory at the early time of Bernoulli? (today, we get them from the principle of maximum entropy, by constraining the first real moment $p$).

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    $\begingroup$ There is a dedicated SE community where you might find better scope. $\endgroup$ Jun 26 at 10:06
  • $\begingroup$ @User1865345-solidarityMods Thanks for the tip. My question is historical and mathematical at the same time. If I'm allowed to cross-post, I'll do it. $\endgroup$
    – Student
    Jun 26 at 10:23
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    $\begingroup$ Questions based on history pertaining to statistics is not off-topic here. It's that HSM is better suited for such question. Also, cross posting is discouraged by the community. $\endgroup$ Jun 26 at 10:24
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    $\begingroup$ @User1865345-solidarityMods I wouldn't have answered over there because I didn't know that place existed. Thanks for mentioning it! (Though not sure whether people there or here are more qualified for this; I think I'm not the only statistician interested in the history of our discipline.) $\endgroup$ Jun 26 at 16:19
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    $\begingroup$ (+1 to your post). As I said @ChristianHennig, history questions pertinent to statistics and probability are on-topic here and of course there too. It's just that I would have asked it there for I have already received some quality responses on probability there. $\endgroup$ Jun 26 at 16:27

1 Answer 1


Here are a few thoughts.

As far as I remember Bernoulli's work, the argument that Bernoulli gives doesn't rely on $p$ being rational (although he mentions, in line with what you write, that according to the concept of probability that he uses, it is).

The thing is that most people wouldn't think of interpreting the Bernoulli/Binomial distribution with non-rational $p$ in the generality required by frequentism as a new definition. Mathematically everything that Bernoulli did (in this case) still applies, so one can refer to Bernoulli's definition without bothering about the fact that Bernoulli meant this to apply to rational $p$ only.

Generally, in modern mathematics, there is a distinction between formal definitions and their interpretation when it comes to reality. Kolmogorov, when setting up his axiomatic foundation of probability, was well aware that at his time more than one interpretation of probability was around, and the mathematical axioms and definitions were meant to cover them all (or at least they were not meant to be exclusive to one particular interpretation).

Another interesting observation is that only around 1830, quite some time after Bernoulli and Bayes, remarks start to show up in the literature (Poisson and Cournot if I remember correctly; this could be checked in Stephen Stigler's work) that probability is understood differently by different people, namely epistemically (referring to uncertainty/lack of knowledge) and aleatory (referring to behaviour in real repeatable random processes). Earlier "definitions" or (as I'd rather say) interpretations of probability seem ambiguous to today's readers, and there is no indication that people at the time were aware of the tension between interpretations. (You can find papers with titles such as "Was Bayes a frequentist?" discussing the issue.) It seems that rather than having new concepts of probability introduced explicitly by definition, people became aware that probability was already used in different ways before anybody had bothered to distinguish them explicitly.

how did we go from Bernoulli distributions with rational p to Bernoulli distributions with real p

I'd think that this move was not made consciously in the early days. Bernoulli's requirement that $p$ should be rational wasn't mathematically necessary, so people wouldn't bother, and later, say, between 1850 and 1920, when frequentism was consciously set up starting from Venn, Bernoulli/Binomial were ready to be used with non-rational $p$ without any additional effort. Mathematically, one could argue, real $p$ was possible from the beginning, from what was done by Bernoulli himself, even though he didn't think of it.

  • $\begingroup$ Thanks a lot. I appreciate your input. According to mdpi.com/2227-7390/10/15/2680, Bernoulli already used to distinguish "two types of probability: a priori, objective, statistical or aleatory probability, calculated by deductive principles, which is the one that allows the study of phenomena with a finite number of cases; and a posteriori, subjective, empiric, or personal probability, which under inductive principles consists of the imperfect knowledge of events, contrastable with previous experience." $\endgroup$
    – Student
    Jun 27 at 12:15

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