Interview on ”Bayesian Statistics the Fun Way”

> Good. Then you know that any inference strategy that falls prey to Dutch Books is not rational. Right?

Do you even have an idea what "rational" means? There are people who argue that having cyclic preferences is not only rational, but even sometimes the only rational representation of evaluations. I'm not one of these, but just wanted to mention that things are not as simple as you lay them out.

If by "rational" you mean "fine for decision making", then I need to disappoint you. Dutch Books are not a working criterion for that. It is perfectly possible to make rational decisions with cyclic preferences. Your preferences need to weakly eligible and weak eligibility needs to be top-transitive (Hansson).

Weak eligibility: There are one or more alternatives such that there is no preferred alternative to them.

Top transitivity of weak eligibility: If a is weakly eligible and a~b, then b is also weakly eligible.

These are conditions on preferences. You can have similar conditions on subjective plausibility, of course, once you combine preferences and subjective plausibilities.

By the way, Expected Utility falls prey to Dutch Books. There is a money pump against every risk-averse or risk-seeking agent. Check out Wakker's book, which is much better than Jayne: Prospect Theory for risk and ambiguity. Anyway, EU is often considered rational and widely used, but according to your criterion it would be irrational. (In finance, the kind of Dutch Books are called "arbitrage" and exploited immediately, so the market prunes them away, but in other areas EU is used extensively. Are you maybe a finance guy???)

> For reference, Jaynes Desiderata:

Of course you can just claim "here is my list of postulates, and that's what 'rational' means", but that's not really an argument. The other theories I am talking about are also axiomatized. Take for example Fishburn's seminal work. According to your theory, Fishburn spent most of his life and efforts in decision making on irrational theories. I'm not convinced and rather be willing to talk about different kinds of rationality, if I'd be pressed to make a decision on that.

> (1) Degrees of plausibility are represented by real numbers. (And a continuity assumption.)

There is a vast array of literature on qualitative decision making for which this assumption does not hold. Lexicographic decision making does also not fulfill that requirement and there is a whole French-Belgium school on that, including axiomatizations and practical methods (tools like ELECTRE). For lexicographic decision making usually hyperreal numbers are used.

Qualitative decision making comes with a host of problems and limitations due to Arrow's Theorem, but lexicographic models can be very reasonable and even required if some of the authors in the field are right about some examples of seemingly irrational preferences. In any case, just to say that these axiomatized theories are irrational because "here are my axioms" is unacceptable. I'm sure not even Jayne does that.

As for the continuity assumption: There is a whole field of measurement theory that would tell you when you need it and when you don't need it, and I really don't see any non-measurement-theoretic way of defending such technical assumptions as rationality postulates independently. Again, just assuming these kind of things a bit too simple. After all, I can take any postulate and call it "rational", that's not a meaningful discussion of rationality, though.

> (3b) The robot always takes into account all of the evidence it has relevant to a question. It does not arbitrarily ignore some of the information, basing its conclusions only on what remains. In other words, the robot is completely non ideological.

This is an interesting principle, because even in probabilistic settings it completely controversial how to deal with conflicting evidence and how and when to revise beliefs in the face of evidence that directly conflicts with your existing beliefs.

It's a very vexing and complicated problem with many different solutions. It is definitely an underdetermined problem. One of the best discussions of it has evolved from criticisms of the corresponding update rule in the Dempster-Shafer theory of evidence, so it's worth taking a look at if you're really interested in this topic. But you seem to be hell-bent on taking Jayne's book as some sort of bible, which is weird. It's not as if any of the other approaches I've mentioned in my previous post are unknown or have been proposed by outsiders - it's almost impossible to not stumble across possibility theory (Dubois & Prade) or Halpern's work if you're doing AI research, for example.

> They're not just true, they're obvious.

Maybe for people who do not know the literature very well, but certainly not to me. Sorry. :(

Well, I agree mostly with you, except that I'm not a probabilist. There is an extensive discussion about what and what not Dutch books show, see Alan Hájek's work on that, which is really worth reading.

> Being either risk seeking or risk averse looks like a flaw too, though perhaps less severe than cyclic preferences.

Yes, I don't want to deny that this view is appealing. However, even if you use probabilistic representations of degrees of belief, you need to deal with ignorance and conflicting evidence in one way or another. Convex sets of probabilities can be shown to be able to represent many of the alternative approaches I've mentioned. There is also this "subjective logic" by Jøsang that is surprisingly nice despite it's silly name. Check it out, maybe the only quirk I have with it is that he mostly seems to re-brand many prior ideas, but the framework is interesting.

> Not even acknowledging the piece of data might be a good approximation in some cases, but in general it seems quite foolish. You don't just ignore a piece of evidence, you explain why it doesn't change your mind.

I agree with you, but at the same time we know from qualitative belief revision theory that there are many, many ways of dealing with conflicting evidence. Okay, we can rule out some of them, e.g. discarding all previous beliefs to learn the new evidence, but among the many less obviously flawed methods a choice needs to be made. The probabilistic setting doesn't help too much in that area, it actually makes it harder to see what's going on. As I've said, the problem is underdetermined.

> Then this book comes up, and provide justifications for my intuitions that were even stronger than I anticipated.

I'm definitely going to read it! However, I might already be tainted by other books on the subject and philosophical discussions. I really do think a belief representation ought not be closed under negation, i.e., I have strong Dempster-Shafer intuitions, and that some way of distinguishing ignorance from doubt is needed.

> Call it confirmation bias, but when I read that book, I already subscribed to probability theory as the correct way to think. I had for a long time. The intuition of probabilities being degrees of beliefs, I had for as long as I can remember.

Kudos to you for having such strong intuitions. It makes life easier. Maybe I'd be willing to buy into them for probabilities, but that wouldn't help me because of similar problems on the evaluative side on which most of my work focuses. On the evaluative side we have thought experiments like Spectrum Cases (Temkin, Rachels): Suppose A gives you extremely high pleasure for a month, B gives you a little bit less pleasure than A (barely noticeable) for 3 months, C gives you a little bit less pleasure than B (barely noticeable) for 9 months, and so on. Some people (not all) have the intuition that B is better than A, C is better than B, and so forth, until at some point, say Z, they would judge that A is better than Z. These thought experiments come in all varieties, can also be made about well-being and other notions of goodness and can be made as realistic as one wishes. Most people who want to keep "better than" transitive introduce some notion of significance, which is lexicographic decision making in disguise (significant value attributes always outrank insignificant value attributes). But okay, you were talking about probabilities only and already acknowledged the evaluative component could be discontinuous. (To be more precise, in this case the Archimedean axiom fails.) It's just that even if graded belief is purely probabilistic, these kind of preferences will complicate making decisions on the basis of your belief.

I agree about the tractability, too. Since you are a probabilist about graded belief, that already makes your life much easier than mine, though. Couldn't you just say that any heuristics are permissible in certain circumstances as shortcuts that - under these circumstances - are conducive to adequate probability approximations?

I didn't want to insinuate that there is anything wrong with being a probabilist, it's in the end a matter of intuitions, I merely wanted to point out that there are some fairly well-known authors who are not probabilists about graded belief in the narrow sense, e.g. Bouyssou, Fishburn, Vincke, Pirlot in decision making, people like Halpern, Dubois, Prade, Spohn and their scholars in A.I., and of course almost everybody in mathematical psychology such as Luce and Tversky. But as I've said, most of their generalizations can be represented by more complicated probability representations such as sets of probabilities.

Anyway, it was nice chatting with you!