Does it make sense to speak of probabilities only when you have numerous enough trials?
No, probability theory also has non-frequency applications.
Can we speak of probabilities for singular, non-repeating events?
Yes. This is the core of a Bayesian approach to decision making. The usual interpretation is that the probabilities reflect your state of knowledge about events rather than frequencies of actual event outcomes. Try starting with the LW wiki article on Baesian probability and the blog posts linked therefrom.
Assigning a non-repeating event the probability P means that, for a well-calibrated agent, if you assign 100 different things this probability then 100 * P of them will actually occur. I believe this is a standard interpretation of Bayesian probability, and it puts things in terms of frequencies of actual event outcomes.
ETA: Alternatively, one may think of Bayesian probability as the answer to the question “if I believed this statement, then in which fraction P of all plausible worlds in which I ended up with this information would I be correct”?
I have to disagree with this interpretation. The whole point is that the frequency interpretation of probability can be a specific case of the Bayesian (probability = belief) interpretation, but not vice versa.
If I say I belief in the existence of aliens with 0.2 belief i think its non-intuitive and unrealistic that what im really saying is, “i think aliens exist in 20% of all plausible worlds”. Apart from the difficulty in clearly defining ‘plausible’ the point of Bayesianism is that this simply represents my state of knowledge/belief.
I find Bayesian probability to be meaningless unless you connect it to a pseudo-frequentist interpretation somehow. Sure, you can say “Bayesian probability measures my subjective belief in something”, but in that case, what does having a 20% subjective belief in something actually mean, and how’s it different from having an 80% subjective belief? You need some scheme of translating it from a meaningless number to an expectation, and all such translations (e.g. in terms of betting behavior) always end up being pseudo-frequentist somehow.
The traditional way of defining the degree of a belief held by some agent is by finding what the agent thinks is a fair wager on the proposition. Is that pseudo-frequentist in a way I’m not seeing?
No, probability theory also has non-frequency applications.
Yes. This is the core of a Bayesian approach to decision making. The usual interpretation is that the probabilities reflect your state of knowledge about events rather than frequencies of actual event outcomes. Try starting with the LW wiki article on Baesian probability and the blog posts linked therefrom.
Assigning a non-repeating event the probability P means that, for a well-calibrated agent, if you assign 100 different things this probability then 100 * P of them will actually occur. I believe this is a standard interpretation of Bayesian probability, and it puts things in terms of frequencies of actual event outcomes.
ETA: Alternatively, one may think of Bayesian probability as the answer to the question “if I believed this statement, then in which fraction P of all plausible worlds in which I ended up with this information would I be correct”?
I have to disagree with this interpretation. The whole point is that the frequency interpretation of probability can be a specific case of the Bayesian (probability = belief) interpretation, but not vice versa.
If I say I belief in the existence of aliens with 0.2 belief i think its non-intuitive and unrealistic that what im really saying is, “i think aliens exist in 20% of all plausible worlds”. Apart from the difficulty in clearly defining ‘plausible’ the point of Bayesianism is that this simply represents my state of knowledge/belief.
I find Bayesian probability to be meaningless unless you connect it to a pseudo-frequentist interpretation somehow. Sure, you can say “Bayesian probability measures my subjective belief in something”, but in that case, what does having a 20% subjective belief in something actually mean, and how’s it different from having an 80% subjective belief? You need some scheme of translating it from a meaningless number to an expectation, and all such translations (e.g. in terms of betting behavior) always end up being pseudo-frequentist somehow.
The traditional way of defining the degree of a belief held by some agent is by finding what the agent thinks is a fair wager on the proposition. Is that pseudo-frequentist in a way I’m not seeing?
Obviously, this needs more discussion but the kind of thought I was trying to motivate was the following:
How is that saying a non-repeating singular event has a very small probability of occurring different from saying it will not happen?
This was motivated by the lottery paradox. Questions like, when you buy a lottery ticket, you don’t believe you will win, so why are you buying it?
Examples like these sort of pull my intuitions towards thinking no, it doesn’t make sense to speak of probabilities for certain events.