Or the coin being cheat, or some cheating or “non-random” effect in the situation. Delusional recollection of events.
How did I “rule out” the alternatives? When I imagine me doing that, I imagine me reasoning poorly. I go by Jaynes’ policy of having a catch all “something I don’t understand” hypothesis for multiple hypothesis testing. In this case, it would be “some agent action I can’t detect or don’t understand the mechanism of”. How did I rule that out?
Suppose it’s 1,000,000 coin flips, all heads. The probability of that is pretty damn low, and much much lower than my estimates for the alternatives, including the “something else” hypothesis. You can make some of that up with a sampling argument about all the “coin flip alternatives” one sees in a day, but that only takes you so far.
I don’t see how I would ever be confident that 1,000,000 came up all heads with “fair” coin flipping.
The probability of any specific sequence of 1M coin flips is “pretty damn low” in the same sense. The relevant thing here is not that that probability is low when they’re all heads, but that the probability of some varieties of “something else” is very large, relative to that low probability. Or, more precisely, what sets us thinking of “something else” hypotheses is some (unknown) heuristic that tells us that it looks like the probability of “something else” should be much bigger than the probability of chance.
(I guess the heuristic looks for excessive predictability. As a special case it will tend to notice things like regular repetition and copies of other sequences you’re familiar with.)
Or the coin being cheat, or some cheating or “non-random” effect in the situation. Delusional recollection of events.
How did I “rule out” the alternatives? When I imagine me doing that, I imagine me reasoning poorly. I go by Jaynes’ policy of having a catch all “something I don’t understand” hypothesis for multiple hypothesis testing. In this case, it would be “some agent action I can’t detect or don’t understand the mechanism of”. How did I rule that out?
Suppose it’s 1,000,000 coin flips, all heads. The probability of that is pretty damn low, and much much lower than my estimates for the alternatives, including the “something else” hypothesis. You can make some of that up with a sampling argument about all the “coin flip alternatives” one sees in a day, but that only takes you so far.
I don’t see how I would ever be confident that 1,000,000 came up all heads with “fair” coin flipping.
It’s a fair coin. It just has two heads on it.
The probability of any specific sequence of 1M coin flips is “pretty damn low” in the same sense. The relevant thing here is not that that probability is low when they’re all heads, but that the probability of some varieties of “something else” is very large, relative to that low probability. Or, more precisely, what sets us thinking of “something else” hypotheses is some (unknown) heuristic that tells us that it looks like the probability of “something else” should be much bigger than the probability of chance.
(I guess the heuristic looks for excessive predictability. As a special case it will tend to notice things like regular repetition and copies of other sequences you’re familiar with.)