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There exist several interpretations of the concept of Probability:

https://en.wikipedia.org/wiki/Probability_interpretations

Being the assumption of Repeatability an important difference between them.

I was wondering if the interpretation of the concept of probability is still an open problem (or a problem of interest) in Philosophy of Science, and if there are any new/more recent definitions beyond the Frequentist and Bayesian definitions?

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    You can see SEP’s entry dedicated to the Interpretation of Probability for overview and references to current literature – Mauro ALLEGRANZA Sep 9 at 9:53
  • @MauroALLEGRANZA Thanks! That's an interesting reference! – Thomas Sep 9 at 10:36
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    It is not that defining probability is a problem. There are plenty of definitions, too many, and no consensus as to which one is "right" or "most fundamental". Or even if there is a "right" one at all rather than different ones for different purposes. That is something of a perennial problem. In addition to frequentism and Bayesianism (itself split into subjective and objective with shades in between) there are also evidentialism, propensities and best-systems, to name the main ones. – Conifold Sep 10 at 5:04
  • One approach is to give an operational definition to 'probability' and say that any quantity can be considered a probability iff it obeys the probability calculus. Frequencies do; credences do, at least to a good approximation; many other quantities do as well. Probability then needs to be interpreted as a specific quantity in order to make use of it, but any interpretation is satisfactory provided it obeys the calculus. – Bumble Oct 9 at 20:00
  • "Is defining the concept of Probability still an open problem in the Philosophy of Science?" Probably. – Cort Ammon Oct 11 at 2:08
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Besides questions along the lines of Frequentism vs. Subjectivism (as addressed in the Stanford Encyclopedia on Philosophy entry on interpretations of probability, mentioned in the comments above), there are also still problems where what is actually the "right" probability is still in dispute, such as the Sleeping Beauty problem.

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Yes, it absolutely is an open question, as can be seen in applications in quantum mechanics.

Like Unitarity, and challenges to it - though I am inclined to think they are definitional (along Cartwright's 'How The Laws Of Physics Lie' lines) rather than epistemic.

And the expanded probability ensemble, as briefly discussed in a question here with links, though sadly not answered yet https://physics.stackexchange.com/questions/11049/does-the-extended-probability-ensemble-interpretation-of-quantum-mechanics-make

Electron orbitals seem to have negative probabilities. This is a bit deceptive, because they only do so relationally, when a paired electron is there with opposite spin, yet somehow preserving in the total state that one electron is less likely to be where it's orbital partner is than located at the nucleus (because this wave-like behaviour is captured in the imaginary part, lost in observations).

Like entropy, we feel with probabilities we grasp an absolute, an intrinsic quality of a system, but find in practice it is often, usually, relative, about a change between ststems rather than absolute terms from first principles - reality, and useful conceptualisation/abstraction, is often too complex for that, and toy systems lead us to over-optimism.

Investigating the deep meaning of probabilities is key to future physics. The transition between fermions and bosons responsible for superconductiin & superfluidity for instance. Blackholes are now thought to be a (bosonic) superfluid that has an absolute maximum of entropy for a given volume. Penrose's Conformal Cyclic Cosmology seems to equate a pure photon soup to the opposite, a white hole. Somehow the transition between these (ie, a timeline including both the big bang, a white hole, and evaporating blackholes) has to either preserve entropy increase, and conservation of information, or show how these can be violated. These heuristics are the deepest principles of our understanding of the world, and we know there is an inconsistency. A better understanding the true implications of probability is clearly key to resolving this.

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