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Those with experience may deny it, having suffered too long ago. But it stares you in the face with the somnolent, expressionless eyes of every student being exposed the first time. Probability notation is an abomination.

This difficulty is usually expressed as "$p$ stands for every thing".

$p$ represents four different functions in one equation:

$p(θ | x) = p(x | θ) p(θ) / p(x)$

If this were actual mathematical notation (yes, actual, and I mean that to sting), the things that look like variables would be variables and the things that look like functions would be functions. But none of that is the case. And overloading $p$ for both probability and conditional probability (which has two parameters) is just so extra.

I mean it's actually a lot worse than this but isn't this bad enough?

My question is, and it is entirely tendentious, who is the person to blame for this? I want to know their name(s) and pen a strongly worded letter to the New York Times, or find where they live and tape a note to their windshield that says "How Dare You".

But really, was there a single person who started using $p$ for absolutely everything in some quasi-mathematical manner. Or was it a long evolutionary process, first Markov's axioms, then someone used it for conditional probability, and then someone else started using it for whatever? Or something else?

Mitch
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  • Not that regular old mathematical notation is always so perfect (eg the conventions that $x,y,z$ are variables and $a,b,c$ are constants (that are unknown, which is sort of a kind of variable, right?)). But at least math relies a lot less on context to disambiguate. – Mitch Mar 31 '22 at 20:15
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    This is an interesting history question, so I hope to see an answer, but about your point about notation, I ask: If we used different letters for everything, would it be any better? I feel like you (or someone else) would have been complaining that there's too many letters. I think this is one of those cases where all possibilities are bad. – Maximal Ideal Mar 31 '22 at 20:40
  • "Why are there so many similar-sounding words about things in the world...?" :) – paul garrett Mar 31 '22 at 22:06
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    @MaximalIdeal 1) I think the 'well you mathematicians use too many symbols' was mentioned in one of the links.2) For Bayes rule and really elementary probabillty, the notation works fine. But it is different from usual mathematical practice. So the intro here is just a peeve/troll/clickbait to get the history of the notation. There's a better formula that shows multiple meanings... I'm looking for it. – Mitch Mar 31 '22 at 22:18
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    I don't know the answer, but if someone starts searching the literature, it's probably safe to start after 1920, since before the notation $p(x)$ was barely used. See: https://hsm.stackexchange.com/a/8273/3462 – Michael Bächtold Apr 01 '22 at 07:52
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    Andrey Kolmogorov; see Probability axioms. – Mauro ALLEGRANZA Apr 01 '22 at 09:43
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    @MauroALLEGRANZA: the question is not who came up with the notation $p(x)$, but who startet abbreviating what should be $p(X=x)$ with $p(x)$. Here $X$ is a random variable and $x$ one of its values. – Michael Bächtold Apr 01 '22 at 13:19
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    Also, why is "random variable $X$" what the rest of mathematics calls "function $f$"? :) For quite a few years, I thought there was something surely more exotic and ineffable about "random variable" than ... being a function on a set with a measure ("a probability space"!?!) – paul garrett Apr 03 '22 at 04:09
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    @paulgarrett I suppose it has to do with the fact, that the "analytic encoding" of random variables in terms of maps from a set with measure came much later than the idea of random variables itself. Random variables existed at least since Bernoulli, while the encoding you mention is probably from around 1930. Before that random variables were taken as basic concepts. This talk by Alex Simpson shows how a synthetic treatment of random variables can be made rigorous: https://youtu.be/XtsBsLM9ofk – Michael Bächtold Apr 04 '22 at 10:47
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    @MichaelBächtold, ah, I had not been aware that the terminology had existed so long prior... Interesting link, also! Thanks! – paul garrett Apr 04 '22 at 17:16

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