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I've been a statistician for a long time and have recently moved towards more information theoretic research. Because of this, the question of epistemic uncertainty in classical probability has been on my mind, particularly how to properly account for the separation of epistemic and aleatory uncertainty in simple probabilistic models.

For example, if we consider the random variable associated with a coin toss $X$, according to frequentist interpretations there is some objective $P(X=1) = p_f$ associated with the long run frequency of the outcome. This makes sense if the coin toss (or any other experiment) could truly be repeated in a completely identical way, but most physical processes are not time-invariant in such a way (or simply can't be repeated, like a particular sports match). From a Bayesian perspective $P(X=1) = p_b$ is not an objective quantity at all, but a degree of belief that may change with more or less information about the experiment.

My question is this:

For events that can not be infinitely repeated in an identical way, is there a way to account for both the event having some truly objective probability $p$ and having some transitional degree of belief under imperfect information $i$? Heavily abusing notation, something like $\lim_{i \to \inf} P(X=1 | i) = p$

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    $\begingroup$ Search this site for exchangeability $\endgroup$
    – kjetil b halvorsen ♦
    Commented Dec 9 at 17:11
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    $\begingroup$ Note that a Bayesian can believe in an objective reality, e.g., that $P(X=1) = p$ is a "Real Thing." The Bayesian process updates our beliefs about $p$, not $p$ itself. $\endgroup$
    – jbowman
    Commented Dec 9 at 17:17
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    $\begingroup$ Also, maybe this helps ? stats.stackexchange.com/questions/332026/… $\endgroup$
    – kjetil b halvorsen ♦
    Commented Dec 9 at 19:55
  • $\begingroup$ @jbowman "The posterior distribution describes the complete updated information about the unknown parameter. Remember that the Bayesian philosophy is a belief that there is not a unique true parameter value. Instead, the parameter is described by a probability distribution." (Qiang Wu, Paul Vos, in Handbook of Statistics, 2018) I have also heard Bayesians say that the PD approximates knowledge while the actual parameter is fixed in reality, but I'm not sure how to use this to information to separate types of uncertainty (aside from maybe a distributional measure on the prior). $\endgroup$
    – NotAGroupTheorist
    Commented Dec 9 at 22:07
  • $\begingroup$ In Bayesianism you can model the "true" probability $p$ as a latent random variable that represents the only aleatoric uncertainty of the system. $\endgroup$
    – cinch
    Commented 2 days ago

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