| > No, I was not speaking from a Bayesian perspective, I was laying out the propensity-theoretic explanation of probability. Unless you can explain this "propensity" in terms of actual physical properties, propensity by itself is… unjustified. The only domain I know of so far where we could possibly argue propensities are a thing is quantum mechanics. And even then it seems to rest on an anthropic argument: which universe am I living in? > Some people think that you need to explain why a die can be fair, A die by itself is not fair, right? A die might be balanced, and the way it is thrown it might have enough unpredictable variability to cause everyone in the room to think "uniform distribution over [1..6]". Likewise, a cryptographic pseudo random generator is unpredictable (and thus "fair"), to anyone who doesn't know its internal state. Even though the process itself is deterministic, it's just not computationally feasible to guess its output just from the observation of past inputs. (Though for this one I'm relying on the fact we're not logically omniscient.) > I'm an expert on this topic. Good. Then you know that any inference strategy that falls prey to Dutch Books is not rational. Right? To be fair, probability theory is not computationally tractable. I did not verify, but I guess any feasible approximation is vulnerable to some more or less subtle Dutch Books. Now the way you talk about Dutch Books sound like all the other strategies you mention are vulnerable, not just in practice, but in theory as well. They are thus not perfectly rational. Do their authors at least have the grace to admit this is a flaw that should be corrected? But then I suspect that correcting the flaw inevitably leads to probability theory itself: if you accept Jaynes three "desiderata" as required for any kind of rational reasoning, as he shows, the result is necessarily equivalent to probability theory as we know it (where probabilities are subjective assessments of plausibility, otherwise known as "degrees of belief"). I can only conclude that you do not accept Jayne's desiderata as necessary for correct inference. And this is the point where I look at you like you're not quite sane. For reference, Jaynes Desiderata: (1) Degrees of plausibility are represented by real
numbers. (And a continuity assumption.)
(2) Qualitative correspondence with common sense.
(explained in more detailed in the book)
(3a) If a conclusion can be reasoned out in more than
one way, then every possible way must lead to the
same result.
(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.
(3c) The robot always represents equivalent states of
knowledge by equivalent plausibility assignments.
That is, if in two problems the robot’s state of
knowledge is the same (except perhaps for the
labeling of the propositions), then it must assign
the same plausibilities in both.
Good luck convincing me (and I suspect, the majority of people, including frequentist statisticians), that we should reject any of these desiderata.I don't care it's reverse engineering, those desiderata match the way I think. I accept the conclusion that probability theory is the correct (albeit intractable) way to think, because I ultimately agree with the postulates it rests on. Vehemently so. They're not just true, they're obvious. If you don't accept them, then I can only give up, and remember what Yudkowsky once wrote: "How do you argue a rock into becoming a mind?" |
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. :(