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I don't know much about statistical uses of Bayesianism but can say something opinionated about the underlying philosophy. From a philosophical point of view,
by 13415 5y ago
I don't know much about statistical uses of Bayesianism but can say something opinionated about the underlying philosophy.
From a philosophical point of view, Bayesianism is fairly weak and lacks argumentative support. 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. Besides, priors are also often not known. As far as I can see, subjective utilities can only be considered rational as long as they match objective probabilities, i.e., if the agent responds in epistemically truth-conducive ways (using successful learning methods) to evidence and does not have strongly misleading and skewed priors.
I also reject the use of simple probability representations in decision theory, first because they do not adequately represent uncertainty, second because they make too strong rationality assumptions in the multiattribute case, and third because there are good reasons why evaluations of outcomes and states of affairs ought to be based on lexicographic value comparisons, not just on a simple expected utility principle. Generally speaking, Bayesians in this area tend to choose too simple epistemic representations and too simple value representations. The worst kind of Bayesians in philosophy are those who present Bayesian updating as if it was the only right way to respond to evidence. This is wrong on many levels, most notably by misunderstanding how theory discovery can and should work.
In contrast, frequentism is way more cautious and does not make weird normative-psychological claims about how our beliefs ought to be structured. It represents an overall more skeptical approach, especially when hypothesis testing is combined with causal models. A propensity analysis of probability may also sometimes make sense, but this depends on analytical models and these are not always available.
There are good uses of Bayesian statistics that do not hinge on subjective probabilities and any of the above philosophical views about them, and for which the priors are well motivated. But the philosophical underpinnings are weak, and whenever I read an application of Bayesian statistics I first wonder whether the authors haven't just used this method to do some trickery that might be problematic at a closer look.
I'd be happy if everyone would just use classical hypothesis testing in a pre-registered study with a p value below 1%.
- 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 agoDempster–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 agoDoes is somehow imply that the Dutch book argument is better developed than Cox's argument?
- spekcular 5y agoYou 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 agoI 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 agoThat'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.