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Probability only deals with a very specific type of uncertainty: stochasticity. This is by far not the only type of uncertainty. Compare e.g. uncertainty as va

Probability only deals with a very specific type of uncertainty: stochasticity. This is by far not the only type of uncertainty. Compare e.g. uncertainty as vagueness. Probability is entirely ill-equipped to deal with this kind of uncertainty. E.g. consider needing to make a distinction between class A and B (real life example: heart muscle vs papillary muscle). Now consider the existence of a 'hybrid' class between the two (e.g. a cellular structure that is neither heart, nor papillary muscle, but a mixture of the two). Any probabilistically stated problem that attempts to define uncertainty of the hybrid tissue, will fail even before starting, because it will attempt to provide a stochastic intepretation, when in fact sampling from that pixel should always deterministically lead to the exact same result, albeit uncertain in terms of vagueness. This type of uncertainty is better addressed using fuzzy measures other than probability (which is only a subset of more general fuzzy theory). See Klir & Folger for nice discussion on this (I forget the title, but it is called something like Uncertainty in Fuzzy Theory Applications).

nextaccountic 6y ago
That's fascinating! > This type of uncertainty is better addressed using fuzzy measures other than probability (which is only a subset of more general fuzzy theory). Do you mean that both probability theory and possibility theory are subsumed by fuzzy logic? What about works that integrate fuzzy logic and probability? eg. https://arxiv.org/abs/1610.09156 https://arxiv.org/abs/1610.09156 employs both fuzzy logic and probability; can those be understood in terms of fuzzy logic alone? > See Klir & Folger for nice discussion on this (I forget the title, but it is called something like Uncertainty in Fuzzy Theory Applications). Do you mean the 1988 book "Fuzzy Sets, Uncertainty, and Information" by George J. Klir and Tina A. Folger? If you have any other references on this I would be glad to read! edit: regarding the my first question, I found this https://arxiv.org/abs/1302.4953 https://arxiv.org/abs/1302.4953 Fuzzy Logic and Probability Petr Hajek, Lluis Godo, Francesc Esteva In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical two-valued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a {em fuzzy} logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truth-values of some suitable fuzzy propositions associated to the crisp ones. Moreover, suggestions and examples of how to extend the formalism to cope with conditional probabilities and with other uncertainty formalisms are also provided.
- tpoacher 6y ago
  Sorry for the delay, just seeing this. Yes, effectively the entirety of probability theory can be reduced to Cox's axioms of probability, which themselves are specialised instances of fuzzy set/logic theory. Therefore in theory any probabilistic formulation can also be expressed in fuzzy terms with suitable constraints. E.g. I think what is implied by your final quote is the distinction between 'crisp' events and non-crisp ones. Probability only deals with crisp events. In other words, if you have a probability distribution over events, those events are required to be mutually exclusive and exhaustive for Cox's axioms of probability to apply. A fuzzy formulation can express the same problem but without that constraint, e.g. events may be 'fuzzy' (in that you might consider an 'event' with an associated 'probability / fuzzy membership in some set' of its own, but also the definition of the event itself is 'leaking out' into neighbouring events. As an example, consider a set of integers {1 .. 10}, and you want to define a probability distribution for each of the numbers on that set. A fuzzy formulation could allow you to generalise this space further using 'fuzzy integers', expressing the notion e.g. 'approximately ten'. You can then then talk about the probability of 'approximately ten' in the set (where the definition of 'approximately' is not necessarily stochastic). I'm not an expert in any of this, alas, but it does seem to me that exposure to fuzzy theory in the public seems to follow a pareto distribution, where 99% of people have only heard of ridiculously elementary uses of fuzzy theory (and therefore think it's some sort of poor-man's probability theory), which probably only accounts for something like 1% of the field, and the remaining 99% of fuzzy theory is only dealt with by 1% of people who are more preoccupied with abstract math formalizations and have no interest in practical applications (which is a shame).

nextaccountic 6y ago

That's fascinating! > This type of uncertainty is better addressed using fuzzy measures other than probability (which is only a subset of more general fuzzy theory). Do you mean that both probability theory and possibility theory are subsumed by fuzzy logic? What about works that integrate fuzzy logic and probability? eg. https://arxiv.org/abs/1610.09156 https://arxiv.org/abs/1610.09156 employs both fuzzy logic and probability; can those be understood in terms of fuzzy logic alone? > See Klir & Folger for nice discussion on this (I forget the title, but it is called something like Uncertainty in Fuzzy Theory Applications). Do you mean the 1988 book "Fuzzy Sets, Uncertainty, and Information" by George J. Klir and Tina A. Folger? If you have any other references on this I would be glad to read! edit: regarding the my first question, I found this https://arxiv.org/abs/1302.4953 https://arxiv.org/abs/1302.4953 Fuzzy Logic and Probability Petr Hajek, Lluis Godo, Francesc Esteva In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical two-valued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a {em fuzzy} logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truth-values of some suitable fuzzy propositions associated to the crisp ones. Moreover, suggestions and examples of how to extend the formalism to cope with conditional probabilities and with other uncertainty formalisms are also provided.

tpoacher 6y ago
Sorry for the delay, just seeing this. Yes, effectively the entirety of probability theory can be reduced to Cox's axioms of probability, which themselves are specialised instances of fuzzy set/logic theory. Therefore in theory any probabilistic formulation can also be expressed in fuzzy terms with suitable constraints. E.g. I think what is implied by your final quote is the distinction between 'crisp' events and non-crisp ones. Probability only deals with crisp events. In other words, if you have a probability distribution over events, those events are required to be mutually exclusive and exhaustive for Cox's axioms of probability to apply. A fuzzy formulation can express the same problem but without that constraint, e.g. events may be 'fuzzy' (in that you might consider an 'event' with an associated 'probability / fuzzy membership in some set' of its own, but also the definition of the event itself is 'leaking out' into neighbouring events. As an example, consider a set of integers {1 .. 10}, and you want to define a probability distribution for each of the numbers on that set. A fuzzy formulation could allow you to generalise this space further using 'fuzzy integers', expressing the notion e.g. 'approximately ten'. You can then then talk about the probability of 'approximately ten' in the set (where the definition of 'approximately' is not necessarily stochastic). I'm not an expert in any of this, alas, but it does seem to me that exposure to fuzzy theory in the public seems to follow a pareto distribution, where 99% of people have only heard of ridiculously elementary uses of fuzzy theory (and therefore think it's some sort of poor-man's probability theory), which probably only accounts for something like 1% of the field, and the remaining 99% of fuzzy theory is only dealt with by 1% of people who are more preoccupied with abstract math formalizations and have no interest in practical applications (which is a shame).