Something Bayesian. Despite my best effort I just do not get Bayesian probability, it more or less just does not make sense to me. Can you convince me otherwise? What is your best example of something with a probability that can not be analyzed in terms of frequencies or other proportions? And your Bayesian account of it must make sense, I am 90 % certain that P != NP and that is why I would take bets based on those odds does not cut it.
Someone walks out of a magic store holding a coin.
They propose a bet. If they flip it 100 times and the proportion of heads is within [0.4, 0.6], you win $100. If it's not, you pay $100. Do you take that bet?
Explanation: absent the magic store scenario, a `rational' person would take the bet. Your prior belief is that most coins are roughly unbiased. Given that they walked out of a magic store, you now have additional information. Maybe the coin is a trick coin. In that case, your belief that the coin is unbiased should be weaker, even if you don't know which direction the coin is biased in.
This illustrates two things: one, additional information (magic store) can update your beliefs. Two, a strong prior and a weak prior, in this case about the coin's bias, can lead to materially different decisions.
The bet does not really matter, the central question is whether they have a fair coin or if they are trying to me in some way. Even without the magic store, I would be very suspicious of anyone approaching random people with an offer like that.
So I would certainly consider it likely, that they are trying to trick me. But the probability I would assign to this, would still be rooted in some frequency, somewhere under the hood I would try to estimate the possible situations leading to such an offer and in which fraction of them I will be tricked.
If I am doing a good job with that, then repeatedly being in this situation should result in me getting tricked with the probability I cooked up. If I am bad at figuring out the possible states and their probabilities, then I they will not match.
The key operation of Bayesian inference is integrating information. This can be from the same source (an additional coin flip, for example) or from different sources (coin flips, plus auxiliary knowledge about where the coin came from, or the person's motives).
Calculated frequency is a point-estimate of bias. A Bayesian estimate is a distribution of belief over possible values of the bias, integrating all available information.
What's the probability that the sinking of the USS Maine in 1898 was accidental?
One could look at the likelihood of spontaneous coal fires and their effects, gather evidence about the activities surrounding the event, essentially trying to narrow down the set of possible states that would lead to the incident and then see what proportion of those states cause the sinking by fire, military action, or something else. But then attaching a number to that seems a dauting task.
Attaching a number to it isn't so daunting, if you view that number as a reflection of your own subjective certainty as to the truth of the proposition. There's no other "correct" number to shoot for; in reality it was either an accident or it wasn't; Nature knows which proposition is true but we don't. Our mission is only to find the figure that minimizes the logarithm of our error, if the truth were at some point to be revealed to us. The more evidence you gather and the more work you put in will get you a better minimum, but there's no reason to be afraid of putting even a rough quantitative number on it before doing that legwork; you'd just be doing it from a state of higher uncertainty, and your stated probability would have to be farther from either confident extreme (0% or 100%) accordingly.
Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?
Let us start with coin flips. You repeatedly flip a coin and the number of heads will come out to be about half the number of trials.
Where does that come from? It is not some intrinsic property of the coin, it comes from varying initial conditions. If you had enough precision when controlling your hand movements, you could in principle force an outcome with high probability.
But assuming you can not or at least do not do that, there is a certain set of initial states, some will lead to heads, some to tails, and each toss will start from a randomly selected initial state. So given my ignorance of the exact initial state, the coin will land heads with a probability equal to the number of initial states leading to heads divided by the number of initial states compatible with my observations of the initial state. [1]
Repeatedly tossing a coin will sample the set of initial states and the result will match the proportion of the number of states. At least as long as I am not wrong about the set of initial states.
The same applies to something like an election. I have imperfect knowledge about the state of the world but there is a set of states compatible with my knowledge about the world and certain subsets of them will lead to certain candidates to win.
[1] Maybe adjusted by some probability distribution over the initial states if they are not equally likely to be picked.