(Roughly) the way Cosma Shalizi puts it:
Because "integrating over the posterior distribution is the whole point of Bayesian decision theory", a Bayesian cannot be uncertain about the probability of an observable. Bayesians are uncertain about the values of parameters. Bayesians are uncertain about the truth of hypotheses. But they cannot be uncertain about the probabilities of observables--and thus they cannot be uncertain about whether to take bets or not.
Can a Bayesian come close to being uncertain about the probability of an observable?
A Bayesian can say: My posterior over the parameters is such that even a small amount of new information would massively change my assessment of the probability of this observable.
A Bayesian can therefore say: Since discovering that there is another mind out there with a different information set than I have that thinks the probability is X is new information that would greatly move my assessment of the probability toward X.
A Bayesian can say: New information is arriving all the time. Our current assessment of the probability is based on little information. If we possibly can, we should neither accept nor reject this bet but should instead wait for the new information to arrive.
Do those statements that Bayesian can (and often should) make together cover the essentials of what we mean when we say that we are uncertain about the probability of an observable? Do they capture what we are gesturing at when we talk about "Knightian Uncertainty"?
The possible set of answers that could have been given to the question as of November 1, 2012 of whether or not you should make a (small-stakes) bet on Barack Obama at odds of 4-1 were:
- Yes: the probability that Obama will win is greater than 80%.
- It's a fair bet: the probability that Obama will win is 80%.
- No: the probability that Obama will win is less than 80%.
- The probability is uncertain--it's not a bad bet if you are betting against Nature, but if you are betting against another mind you will probably lose.
- The probability is uncertain--your best strategy is to wait for today's information and then see if you want to make the bet tomorrow.
Is there anything else than (4) and (5) that a rational somebody who wants to start their answer with not "yes", "no", or "it's fair" but with "the probability is uncertain" could want to end their answer with?
I would have been profoundly depressed if back when I was 19 somebody had told me that: "You think you are depressed now because you do not understand 'Knightian Uncertainty'. But I tell you that when you are 54 you will still not understand 'Knightian Uncertainty'. In fact, when you are 54, you will not only not understand it, you will be so confused that you will be uncertain about whether 'Knightian Uncertainty' is or is not a sensible concept."
- Cosmos Elysée (2009): On the Certainty of the Bayesian Fortune-Teller
- Brad Delong: Elementary Philosophy of Probability and the War on Nate Silver: The (Not Very) Honest Broker for the Week of August 2, 2014
- Sky Masterson: An Ear Full of Cider
- Adam Elga (2010): Subjective Probabilities Should Be Sharp