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The notion of an individual's "true probability" or "true risk" of a certain outcome $Y$ is contested [1, 2, 3]. Nevertheless, it is often discussed and useful for certain kinds of analyses. However, all the works I can find only describe this quantity in words and do not define it mathematically.

How could one define and mathematically denote the "true individual probability" of a (binary) outcome $Y$ for a given individual? Is there a standard notation?

Is there notation for something like "P(outcome y=1 for individual i, given all there is to know about this individual and the state of the universe)" ?

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    $\begingroup$ Trivially there could be notation that you invented, but what use would that notation be? Notation only has point for expressing or deriving something that couldn't be expressed or derived more simply otherwise. $\endgroup$
    – Nick Cox
    Commented May 23, 2023 at 11:08

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Maybe you had problems with finding it because it is ambiguous. First of all, what is probability is a philosophical question, so there is no single answer to what it means. But that's a different story. If your quantity of interest is $X$, then the notation for the probability is simply $P(X)$. If you want to condition it on something, it's $P(X|Y,Z,\dots)$.

given all there is to know about this individual and the state of the universe

But what would it be? If you knew everything about every atom in the universe, or even everything on the sub-atomic level, with a perfect understanding of the universe, then without any uncertainty you would know what would happen, based on your knowledge and the knowledge of the laws of physics. In such a case, the probability would be always equal to one; it would be certain.

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  • $\begingroup$ "...then without any uncertainty you would know what would happen" while that may be true, it is still a contested hypothesis, as far as I know (due to quantum effects, if I understand correctly - not my area of expertise). In any case, this is slightly beyond the point, isn't it? Even if a property is 0 or 1, I might still want to define and denote it. ;-) $\endgroup$
    – Eike P.
    Commented May 25, 2023 at 8:39
  • $\begingroup$ @EikeP. but you want to condition on everything, so this assumes perfect knowledge of the universe. That's why, as I said, it starts getting philosophical, since the question arises if, beyond the laws of physics, there is some "randomness" that couldn't ever be explained by them. Chaotic systems are deterministic, but complicated, for comparison. But the mere fact that there is such a discussion means that having "a notation" for something not clearly defined would not solve any problem. $\endgroup$
    – Tim
    Commented May 25, 2023 at 9:06
  • $\begingroup$ Again, totally not my field, but isn't part of the argument explicitly that even with perfect knowledge of the universe, one cannot predict these quantum effects, i.e., they are truly random? In any case, how would you then define the aim of individual risk modeling, if not by referring to some underlying, true risk/probability that you're trying to estimate? $\endgroup$
    – Eike P.
    Commented May 25, 2023 at 13:10
  • $\begingroup$ @EikeP. as said in the answer, you condition on specific things, not on "everything". $\endgroup$
    – Tim
    Commented May 25, 2023 at 13:30

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