I still don't understand the philosophy of Bayesian probability

Brad DeLong is having an extremely fascinating conversation with an E.E. Doc Smith deus ex machina character, an emulation of a Princeton professor, looser emulations of two famous dead probabilists, and a made-up Greek mediator himself about the philosophy of Bayesian probability (see also here). DeLong focuses on the question of whether probabilities should be "sharp" - i.e., whether we should always say "I believe the probability of the event is x%" (as Bayesians always do), or whether we should say something along the lines of "I believe the probability of the event is between x% and y%."

But I want to focus on a deeper question, which is: What is a probability in the first place? I mean, sure, it's a number between 0 and 1 that you assign to events in a probability space. But how should we use that mathematical concept to represent events in the real world? What observable things should we represent with those numbers, and how should we assign the numbers to the things?

The philosophy of Bayesian probability says that probabilities should be assigned to beliefs. But are beliefs observable? Only through actions. So one flavor (the dominant flavor?) of Bayesian probability theory says that you observe beliefs by watching people make bets. As DeLong writes:

Thomas Bayes: It is simple. [Nate Silver assigning a 60% probability to a GOP takeover of the Senate in 2014] means that Nate Silver stands ready to bet on [Republican] Senate control next January at odds of 2-3. 

Thrasymakhos: “Stands ready”? 

Thomas Bayes: Yes. He stands ready to make a (small) bet that the Majority Leader of the Senate will [not] be a Republican on January 5, 2015 if he gets at least 2-3 odds, and he stands ready to make a (small) bet that the Majority Leader of the Senate will not be a Republican on January 5, 2015 if he gets at least 3-2 odds.

DeLong is very careful to write "a (small) bet". If he wrote "a bet", we would have to introduce Nate Silver's risk aversion into our interpretation of the observed action, if the bet size were large. DeLong is assuming that a small bet will get rid of Silver's risk aversion.

However, there's a problem: DeLong's assumption, though characteristic of the decision theory used in most economic models, does not fit the evidence. People do seem to be risk-averse over small gambles. One (probably wrong) explanation for this is prospect theory. Loss aversion (one half of prospect theory) makes people care about losing, no matter how small the loss is. To back out beliefs from bets, you need a model of preferences. And that model might be right for one person at one time, but wrong for other people and/or other times!

But isn't that just a practical, technological problem? Why do we need real-world observation in order to define a philosophical notion? Well, we don't. We already defined a probability as a real number between 0 and 1 (which gets assigned to the latter slot in the tuples that are the elements of a probability measure). That's fine. But the Bayesian philosophical definition of probability, if it is to be more than "a number between 0 and 1," seems like it has to include a scientific component. The Bayesian notion of "probability as belief" explicitly posits a believer, and ascribes the probability to that real, observable entity (note: This is also why I think the "Weak Axiom of Revealed Preference" is not an axiom). If we can't observe the probability, then it doesn't exist - or, rather, it goes back to just being "a number between 0 and 1".

So can't we just posit a hypothetical purely rational person, and define beliefs as his bet odds? Well, it seems to me that this will probably lead to circular reasoning. "Rational" will probably be defined, in part, as "taking actions based on Bayesian beliefs." But the Bayesian beliefs, themselves, will be defined based on the actions taken by the person! This means that imagining this purely rational person gets us nowhere. Maybe there's a way around this, but I haven't thought of it.

Does all this mean that the definition of Bayesian probability is logically incoherent? No. It means that defining Bayesian probability without reference to preferences (or other decision-theoretical rules that stand in for preferences) is scientifically useless. In physics, a particle that interacts with no other particles - and is hence unobservable, even indirectly - might as well not exist. So by the same token, I claim that Bayesian probabilities might as well not exist independently of other elements of the decision theory in which they are included. You can't chop decision theory up into two parts; it's all or nothing.

I assume philosophers and decision-theory people thought of this long ago. In fact, I'm probably wrong; there's probably some key concept I'm missing here.

But does it matter? Well, yes. If I'm right, it means the argument over whether stock prices swing around because of badly-formed beliefs or because of hard-to-understand risk preferences is pretty useless; there's no fundamental divide between "behavioral" and "rational" theories of asset pricing.

It's also going to bear on the more complicated question Brad is thinking about. If you're talking to people who make decisions differently than you do, it might not be a good idea to report a number whose meaning is conditional on your own decision-making process (which your audience does not know). So that could be a reason not to report sharp probabilities to the public, even if you would make your own decisions in the standard Bayesian-with-canonical-risk-aversion way. But what you should do instead, I'm not sure.