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> The underlying idea of probabilism - that degrees of belief have to be represented by probability measures - is in my opinion wrong for many reasons. Basicall

by kgwgk 5y ago

> The underlying idea of probabilism - that degrees of belief have to be represented by probability measures - is in my opinion wrong for many reasons. Basically the only well-developed arguments for this view are Dutch book arguments, which make a number of questionable assumptions. Why don't you consider Cox's theorem - and related arguments - well-developed? https://en.wikipedia.org/wiki/Cox%27s_theorem https://en.wikipedia.org/wiki/Cox%27s_theorem

spekcular 5y ago
Dempster–Schafer theory is the obvious counterexample to "degrees of belief have to be represented by probability measures." https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory
- kgwgk 5y ago
  Does is somehow imply that the Dutch book argument is better developed than Cox's argument?
  - spekcular 5y ago
    You asked, "Why don't you consider Cox's theorem - and related arguments - well-developed?" I consider Cox's argument not well-developed because D–S theory shows the postulates miss useful and important alternatives. So it fails as an argument for a particular interpretation of probability.
    - kgwgk 5y ago
      I quoted 13415 saying that the only well-developed arguments were […] and asked him why didn’t he consider […] well-developed - compared to the former. I apologize if the scope of the question was not clear.
13415 5y ago
That's an excellent question. The answer is that I don't really count such kind of theorems as positive arguments. They are more like indicators that carve out the space of possible representations of rational belief and basically amount to reverse-engineering when they are used as justifications. Savage does something similar in his seminal book, he stipulates some postulates for subjective plausibility that happen to amount to full probability (in a multicriteria decision-making setting). He motivates these postulates, including fairly technical ones, by finding intuitively compelling examples. But you can also find intuitively compelling counter-examples. To mention some alternative epistemic representations that could or have also been axiomatized: Dempster-Shafer theory, possibility theory by Dubois/Prade, Halpern's generalizations (plausibility theory), Haas-Spohn ranking theory, qualitative representations by authors like Bouyssou, Pirlot, Vincke, convex sets of probability measures, Jøsang's "subjective logic", etc. Some of them are based on probability measures, others are not. (You can find various formal connections between them, of course.) The problem is that presenting a set of axioms/postulates and claiming they are "rational" and others aren't is really just a stipulation. Moreover, in my opinion a good representation of epistemic states should at least account for uncertainty (as opposed to risk), because uncertainty is omnipresent. That can be done with probability measures, too, of course, but then the representation becomes more complicated. There is plenty of leeway for alternative accounts and a more nuanced discussion.
- kgwgk 5y ago
  Thanks. I found interesting that you like the Dutch book arguments more than the axiomatic ones. > Moreover, in my opinion a good representation of epistemic states should at least account for uncertainty (as opposed to risk), because uncertainty is omnipresent. Maybe I'm misunderstading that remark because the whole point of Bayesian epistemology is to address uncertainty - including (but definitely not limited to) risk. See for example Lindley's book: Understanding Uncertainty. Now, we could argue that this theory doesn't help when the uncertainty is so deep that it cannot be modelled or measured in any meaningful way. But it's useful in many settings which are not about risk. One couple of examples from the first chapter of the aforementioned book: "the defendant is guilty", "the proportion of HIV [or Covid!] cases in the population currently exceeds 10%".

by kgwgk 5y ago

spekcular 5y ago

Dempster–Schafer theory is the obvious counterexample to "degrees of belief have to be represented by probability measures." https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory

kgwgk 5y ago
Does is somehow imply that the Dutch book argument is better developed than Cox's argument?
- spekcular 5y ago
  You asked, "Why don't you consider Cox's theorem - and related arguments - well-developed?" I consider Cox's argument not well-developed because D–S theory shows the postulates miss useful and important alternatives. So it fails as an argument for a particular interpretation of probability.
  - kgwgk 5y ago
    I quoted 13415 saying that the only well-developed arguments were […] and asked him why didn’t he consider […] well-developed - compared to the former. I apologize if the scope of the question was not clear.

13415 5y ago

That's an excellent question. The answer is that I don't really count such kind of theorems as positive arguments. They are more like indicators that carve out the space of possible representations of rational belief and basically amount to reverse-engineering when they are used as justifications. Savage does something similar in his seminal book, he stipulates some postulates for subjective plausibility that happen to amount to full probability (in a multicriteria decision-making setting). He motivates these postulates, including fairly technical ones, by finding intuitively compelling examples. But you can also find intuitively compelling counter-examples. To mention some alternative epistemic representations that could or have also been axiomatized: Dempster-Shafer theory, possibility theory by Dubois/Prade, Halpern's generalizations (plausibility theory), Haas-Spohn ranking theory, qualitative representations by authors like Bouyssou, Pirlot, Vincke, convex sets of probability measures, Jøsang's "subjective logic", etc. Some of them are based on probability measures, others are not. (You can find various formal connections between them, of course.) The problem is that presenting a set of axioms/postulates and claiming they are "rational" and others aren't is really just a stipulation. Moreover, in my opinion a good representation of epistemic states should at least account for uncertainty (as opposed to risk), because uncertainty is omnipresent. That can be done with probability measures, too, of course, but then the representation becomes more complicated. There is plenty of leeway for alternative accounts and a more nuanced discussion.

kgwgk 5y ago
Thanks. I found interesting that you like the Dutch book arguments more than the axiomatic ones. > Moreover, in my opinion a good representation of epistemic states should at least account for uncertainty (as opposed to risk), because uncertainty is omnipresent. Maybe I'm misunderstading that remark because the whole point of Bayesian epistemology is to address uncertainty - including (but definitely not limited to) risk. See for example Lindley's book: Understanding Uncertainty. Now, we could argue that this theory doesn't help when the uncertainty is so deep that it cannot be modelled or measured in any meaningful way. But it's useful in many settings which are not about risk. One couple of examples from the first chapter of the aforementioned book: "the defendant is guilty", "the proportion of HIV [or Covid!] cases in the population currently exceeds 10%".