> But your prior probability distribution is either (1) based on a lot of other data, so that the final result is not based on the single data point, or (2) garbage.

To do Bayesian reasoning at all, you need an initial prior probability distribution based on zero events, and all your posterior results are calculated from it. So if it's really "garbage" you're kind of in trouble regardless of how much data you have. In fact, it's easy to construct priors that cause Bayesian methods to give results that are obviously garbage by any standard even after updated with arbitrarily large amounts of data. This is often used as an argument for preferring frequentist statistics.

This is a fairly central aspect of Bayesian statistics: what can you pick as an "uninformative prior" for an unbounded distribution? A uniform distribution over the entire positive number line unfortunately isn't normalizable, so we unavoidably have some kind of falloff as we go to sufficiently low frequencies, and an exponential distribution is the least unreasonable thing to pick.

But, if we're completely uninformed and don't know anything about Earth or the universe, we might start with a prior exponential distribution whose median is at "one city-destroying-sized meteor explosion on Earth per nanosecond," not realizing that the Earth would be molten if this happened, or at "one city-destroying-meteor explosion on Earth per googol years", not realizing that the universe is vastly younger than that and in fact meteors do hit occasionally. Then, given the observed data that apparently a city was thus destroyed 3600 years ago, when cities covered maybe 0.001% of the land, and apparently less than ten cities have been destroyed since, and definitely none in the last 200 years even though cities spread to cover 1% of the land, the first of these gives us the posterior "city-destroying meteors happen every few hundred to every few thousand years" (even though it's not "based on a lot of other data" and the prior isn't very similar to that posterior, it succumbs to the evidence of those quintillions of nanoseconds when no cities got destroyed), while the second one gives us a much less reasonable estimate which could reasonably be described as "near infinite error bars" or "garbage".

So, in summary:

• you can compute the frequency of an event from a single observed occurrence;

• your prior does not have to derive from a lot of other data for this, not even the fact that the Earth is not currently molten;

• the only Bayesian way to derive a prior probability from a lot of other data is to use a prior that does not derive from any data, so any epistemology that rejects such priors as "garbage" necessarily rejects purely Bayesian reasoning entirely;

• such an uninformative prior can be obviously wrong in certain ways, to the point of absurdity, and still give you reasonable inference results; and

• these are very elementary facts about Bayesian statistics.

In short, the things you are saying are (with respect to Bayesian statistics) as incorrect as claims like "you can't trisect an angle," "you can't subtract 4 from 3," or "3 - 4 = -1 but there's no square root of -1."

> > Your untutored intuition

> It wasn't either, thanks, and its arrogant as fuck for you to assume this.

You evidently didn't know the elementary aspects of Bayesian statistics I explained above, so arrogant as fuck or no, I turned out to be right about that; I'd describe it more as an inference than an assumption. I probably can't teach you anything while you're in ego defense mode, but maybe I can keep you from misleading anybody else. I'm sorry I hurt your feelings, and if there is some way I could have totally dismissed your incorrect opinion without hurting your feelings, please tell me what it is so I can do it in the future.

> That has no impact on the uncertainty of the estimate from a single data point, only the actual estimate.

Yes, you're right, or very close anyway (since a proper prior for this problem necessarily depends on at least some kind of scale parameter, the shape of the results will vary slightly depending on at least the ratio between that scale parameter and the actual frequency estimate).