By the following definitions is it possible to divide aleatory uncertainty into these categories?
Please give some references to prove or disprove this idea.
Probabilistic uncertainty: applied when future circumstances are uncertain, but it is possible to project possible scenarios and assign a probability to each.
Deep uncertainty is applied when decision-makers cannot predict or agree on possible future scenarios or probabilities.
I don't think the question is formulated in a useful way.
Epistemic uncertainty is not fundamentally "probabilistic" as that word is used in statistics.
Aleatory uncertainty is by definition probabilistic
The error is assuming that all uncertainty is one or the other As just one example we introduced the concept of Teleological uncertainty to deal with Human intentions. Performance based fire safety regulation under intentional uncertainty
Vincent Brannigan, Carol Smidts
20 March 2000 Fire and Materials
Many technical variables used as inputs to fire models actually represent the output of unpredictable human decisions. Human decisions related to fire safety constitute a distinct problem for fire modelling that is not captured by epistemic or aleatory uncertainty, since it is often irreducible even with additional knowledge. Intentional (teleological) uncertainty is proposed to describe this uncertainty.
Another way to look at this problem is along the following lines:
Aleatory uncertainty: the (seemingly) inherent variability of a system which appears to us as being random. Note here the words 'seemingly' and 'appear to us', as this fully depends on the scale on which you are looking. These aleatory uncertainties are most appropriately modelled using the axioms of probability theory due to their random nature.
Epistemic uncertainty: uncertainty that belongs the analyst or analysis, and is caused by a lack of knowledge (e.g., due to limited sample size, knowing only the probability distribution without the condition on which it is based, not knowing correlations, etc.), or time/money (which cause limitations in testing campaigns, accuracy of the sensors, etc.). Since this uncertainty is not random, it is usually modelled using set-based approaches such as intervals or fuzzy numbers. Now note that is perfectly acceptable to model epistemic uncertainty using subjective probabilities, but keep in mind that in this case these probabilities have a fully different meaning, and also should be treated separately.
What you refer to as 'deep uncertainty' is when both cases occur at the same time. For further reference, I take the liberty to refer to one of my own papers:
[1] M. G. R. Faes, M. Daub, S. Marelli, E. Patelli, and M. Beer, ‘Engineering analysis with probability boxes: A review on computational methods’, Structural Safety, vol. 93, p. 102092, Nov. 2021, doi: 10.1016/j.strusafe.2021.102092.
I would like to add to the previous response by Matthias that the same rules apply to probabilities describing aleatoric or epistemic uncertainty. Their interpretation is certainly different. They are to be treated differently only in terms of your research objectives, e.g. information acquisition or mere uncertainty quantification. However, their calculus is the same. Hence fully probability-based descriptions have an advantage over the description of epistemic uncertainty with fuzzy sets or the like because it allows to cast the problem in a single coherent framework, in particular when you have mixed situations of both uncertainties (which is rather the rule than the exception). That is great, because it allows to compute with probabilities without caring TOO much about their origin or nature.
I too allow myself to refer to my own work:
[1] Ranftl, Sascha, and Wolfgang von der Linden. 2021. "Bayesian Surrogate Analysis and Uncertainty Propagation" Physical Sciences Forum 3, no. 1: 6. MaxEnt 2021 Proceedings. https://doi.org/10.3390/psf2021003006
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