Stephen Tashi said:
4. Does defining probability in terms of the decisions of a rational agent imply that only the Bayesian vew of probability is correct? Doesn't the dependence of probability on an agent's utility function imply there is no objective probability in physics?
A lot of the discussions about probability in QM ultimately are about the nature of probability, whether quantum or not. QM gives some additional twists to it, but the concept is pretty tricky classically, as well. My opinion about it is that the alternative to Bayesianism, frequentist probability, doesn't actually make complete sense. You can say that probabilities are relative frequencies in the limit as the number of trials goes to infinity. But there is nothing in the laws of physics that make it impossible for an infinite sequence of tosses of a fair coin to approach 1/3 or 9/10 or anything else (or to have no limit at all). The best you can say is that the probability of having a sufficiently large sequence of coin flips where the relative frequency differs appreciably from the probability for a single flip goes to zero as the number of flips goes to infinity. So the equation of relative frequency with probability only holds with probability 1. But the latter notion of probability isn't frequentist. To make sense of it in a frequentist approach, we would need infinitely many trials, each trial of which is an infinite sequence of coin flips.
There's another disturbing fact about even classical probability. Imagine that the universe is infinite, and with a certain quasi-periodic completeness property: there are infinitely many planets that are exactly like the Earth (at least in macroscopic detail). In this setting, if I flip a coin, it's expected that there are copies of the Earth where the copy of my flips a coin and gets "heads", and there are copies where the result is "tails". If I flip a coin 100,000 times, there will be Earths where all 100,000 were heads, and Earths where 100,000 were tails, and all other combinations in between. What we can probably say is that the density of Earths where between 45,000 and 55,000 were heads will be much higher than the density of Earths where the results will be outside of that range. But there will be Earths outside that range. No matter how unlikely a sequence of events is, as long as it's nonzero, there will be some place where that exact sequence happens. So the identification of probability with relative frequency will work for some Earths, but not others. If you say the life span of intelligent life on Earth is maybe bounded by 20 billion years, there will be Earths where the entire history of intelligent life will show departures of relative frequencies from probabilities.
What you could say is that there is a "typical" Earth-like world, where frequencies approach probabilities, and just not worry about atypical worlds. But that amounts to assuming that we are not in an atypical world. What's the reason for assuming that? No matter what evidence we can have for our world being typical, there will be copies of our Earth with the same evidence that are atypical. So at some point, you just have to assume, without evidence, that our world is typical (otherwise, probabilities are useless). But that assumption is really subjective---it's not a firm conclusion based on evidence.
So in my opinion, subjective beliefs always come into play in working with probability, even if it is just the subjective belief that probabilities will be objectively equal to relative frequencies.
