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I am wondering whether the distinction into epistemic and aleatoric uncertainty really makes sense. The way I have understood it (and Wikipedia seems to define it) the distinction is:

  • epistemic uncertainty is inperfection of the model, which may be alleviated by improving process representation.
  • aleatoric uncertainty is inperfection of the data to which we apply our model, so even a model with (hypothetical) zero epistemic uncertainty might still yield uncertain predictions due to aleatoric input uncertainty.

I wonder whether fundamentally aleatoric uncertainty isn't just another type of epistemic uncertainty. For example, if I measure the temperature with a mercury thermometer, I am not really measuring temperature: I am observing the temperature-based expansion of mercury in a glass tube, then compare its rise to a scale at its side, and convert this length into a temperature using a regression obtained by some previous experiment.

This process contains a lot of assumptions, so in a sense it constitutes a model of its own. It seems to me that any aleatoric uncertainty we obtain from repeated temperature measurements could seemingly be equally well explained with epistemic uncertainty of the thermometer model. The 'hard' uncertainty limit in the universe found in quantum mechanics might be an exception, but if this is a fundamental property of the universe, one might equally well consider it a hard limit on epistemic uncertainty.

Is there a nuance I have missed? Are there any papers or books on the topic you could recommend?

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  • You would have (absolute) aleatoric uncertainty when measuring position of an electron, due to Heisenberg's uncertainty principle. Even if we disregard "a lot of assumptions" there will be fundamental "imperfection" in the data. But generally, the separation is just relative to a model. The "lot of assumptions" involved in measurements (production of data), and models utilized there, are irrelevant as far as judging imperfections of the model (under consideration) that receives them as inputs, and the uncertainty it contributes itself.
    – Conifold
    Apr 23, 2020 at 10:44
  • Thank you for the comment! Heisenberg's uncertainty principle, if my high-school physics memory doesn't betray me, would be similar to the quantum mechanics example I mentioned above. The electron has a true location, we just can't get it because the process of measuring it destroys the information we are after: that's what I would call a 'hard limit' of uncertainty. If this hard limit always exists, then it is impossible to find a model with zero epistemic uncertainty, as its perfection must be tested against reality. As a consequence, I would argue that the distinction is still not required.
    – J.Galt
    Apr 23, 2020 at 11:13
  • This is not a place for arguing, just describing what is out there. That he process of measuring "destroys the information" was the initial view of Bohr, it is long discarded. There is no information to destroy, it is generated by the measurement itself, along with uncertainty. In any case, aleatoric uncertainty for a given model can be epistemic in a broader context. This is a special case of distinguishing internal from external, which is always relative.
    – Conifold
    Apr 23, 2020 at 11:47
  • Certainly, it wasn't my intention to argue - just to clarify the question as best as I can within a character limit. If I interpret your response correctly, then you suggest that the distinction between aleatoric and epistemic uncertainty is a subjective question of where to draw the line, to define the 'internal' against the 'external'. So with a broad enough view, everything is epistemic? Given the ubiquity of these concepts in science, I imagine this topic was discussed before: is there any mathematical or philosophical literature exploring this topic fundamentally you can recommend?
    – J.Galt
    Apr 23, 2020 at 12:11
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    "The electron has a true location, we just can't get it because the process of measuring it destroys the information we are after" That would be a "hidden variables interpretation", one which posits a quantum system has additional properties beyond those assigned to it by the quantum state. Bell's theorem shows that no local hidden variables theory can explain quantum behavior (one where influences cannot travel faster than light), and although Bohmian mechanics replicates quantum predictions, it is fundamentally untestable because the hidden variables are not measurable even in principle.
    – Hypnosifl
    Apr 23, 2020 at 19:39

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Probability is a sufficiently entangled topic to engage with. However, by adopting a methodological stance, we can have a clear view of the mentioned issue without getting bogged down in its details:

Basically, a concept, as one of its essential functionalities, conduces us to behave or act in certain ways toward, and to have certain attitudes to the matters it is involved in. Let us call them methods. So, for example, a concept X possesses M1 and M2 as its constituent methods. Conversely, disparities observed in such outward manifestations hint at a need for a conceptual differentiation. Suppose we follow methods M1, M2 and a perceptibly different M3, organised around the concept X. In that case, we have to deal with X itself in order to improve our understanding. Strategies may vary: We can introduce a new concept Y beside X, or we can qualify X to obtain XY and XZ. By the same token, concepts can be coalesced, as is the case for electromagnetism and space-time continuum.

Let us focus scientific phenomena. A divergence occurs when we attempt to quantify and alter their probabilities. While we can reduce uncertainty by improving our observational and experimental techniques and theoretical tools (chaos theory is a nice example), we face the fact that an irreducible uncertainty is retained within definite bounds. This can be compared to the situation to that of universal physical constants; we can increase precision of their values (viz., reduce epistemic uncertainty about them) just as we calculate more and more digits of the number pi, but they are out there. Hence, we are led to draw the conclusion that the uncertainty is inherent in the phenomena, and alluding to the metaphor of dice throwing (one of the favourite examples in textbooks), the probability due to it is called aleatoric (also referred to as ontic), and the other one is called epistemic. It should be remarked that the irreducibility of uncertainty does not necessarily entail that it is inaccessible to us.

As far as philosophy is concerned, I am of the opinion that, at the present stage, we are in need of further categories of probability, rather than coalescence of them in order to have a better grasp of it. Finding them illuminating and thought-provoking for future work, I would recommend two books on the historical development of the idea of probability with a philosophical outlook:

  • The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference by Ian Hacking,

  • Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective by Jan von Plato.

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