I was curious to re-read the chat log, and had to do some digging on archive.org to find it. The guy made 17 bets about numbers being prime, and lost the bet on the 17th bet. 

Transcript here

Sequence article that referenced it here.

Interesting followup by Chris Halliquist here:

If it's not clear why this doesn't follow consider the anecdote Eliezer references in the quote above, which runs as follows: A gets B to agree that if 7 is not prime, B will give A $100. B then makes the same agreement for 11, 13, 17, 19, and 23. Then A asks about 27. B refuses. What about 29? Sure. 31? Yes. 33? No. 37? Yes. 39? No. 41? Yes. 43? Yes. 47? Yes. 49? No. 51? Yes. And suddenly B is $100 poorer.

Now, B claimed to be 100% sure about 7 being prime, which I don't agree with. But that's not what lost him his $100. What lost him his $100 is that, as the game went on, he got careless. If he'd taken the time to ask himself, "am I really as sure about 51 as I am about 7?" he'd probably have realized the answer was "no." He probably didn't check  he primality of 51 as carefully as I checked the primality of 53 at the beginning of this post. (From the provided chat transcript, sleep deprivation may have also had something to do with it.)

If you tried to make 10,000 statements with 99.99% certainty, sooner or later you would get careless. Heck, before I started writing this post, I tried typing up a list of statements I was sure of, and it wasn't long before I'd typed 1 + 0 = 10 (I'd meant to type 1 + 9 = 10. Oops.) But the fact that, as the exercise went on, you'd start including statements that weren't really as certain as the first statement doesn't mean you couldn't be justified in being 99.99% certain of that first statement.

I do think this is an important counterpoint, but still, while I agree that if a person actually thought carefully about each prime number, they'd have made it much farther than a 1-out-of-17 failure rate, I'd still bet against them successfully making 10,000 careful statements without ever screwing up in some dumb way.

[ Question ]

How to convince Y that X has committed a murder with >0.999999 probability?

by Colin Tang 1 min read19th May 202034 comments


Suppose X has murdered someone with a knife, and is being tried in a courthouse. Two witnesses step forward and vividly describe the murder. The fingerprints on the knife match X's fingerprints. In fact, even X himself confesses to the crime. How likely is it that X is guilty?

It's easy to construct hypotheses in which X is innocent, but which still fit the evidence. E.g. X has an enemy, Z, who bribes the two witness to give false testimony. Z commits the murder, then plants X's fingerprints on the knife (handwave; assume Z is the type of person who will research and discover methods of transplanting fingerprints). X confesses to the murder which he did not commit because of the plea deal.

Is there any way to prove to Y (a single human) that X has committed the murder, with probability > 0.999999? (Even if Y witnesses the murder, there's a >0.000001 chance that Y was hallucinating, or that the supposed victim is actually an animatronic, etc.)

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People don't generally form beliefs with that level of precision. "beyond a reasonable doubt" is the usual instruction, for exactly this reason. And the underlying belief is "appears likely enough that it's preferable to hold the person publicly responsible".

Six nines of reliability sounds like a lot, and it's more than is usually achieved in criminal cases, but it's hardly insurmountable. You just need to be confident enough that, given one million similar cases, you would make only one mistake. A combination of recorded video and DNA evidence, with reasonably good validation of the video chain of custody and of the DNA evidence-processing lab's procedures, would probably clear this bar.

My short answer is "you probably can't." >0.999999 is just a lot of certainty. 

There might exist particularly-well-calibrated humans who can have a justified >.0.999999 probability in a given murder trial, but my guess is that most Well Calibrated People still probably sort of cap-out in justified confidence at some point, based on what the human mind can reasonably process. After that, I think it makes less sense to think in terms of exact probabilities and more sense to think in terms of "real damn certain, enough that it's basically certain for practical purposes, but you wouldn't make complicated bets based on it."

(I'm curious what Well Calibrate Rationalists think is the upper bound of how certain they can be about anything)

[Edit: yes, there are specific domains where you can fully understand a mathematical question, where you can be confident something won't happen apart from "I might be insane or very misguided about reality" reasons.]

This problem is known in the philosophy of science as the underdetermination problem. Multiple hypotheses can fit the data. If we don't assign a priori probabilties to hypotheses, we will never reach a conclusion. For example, the hypothesis that (a) Stephen Hawking lived till 2018 against (b) There was a massive conspiracy by his relatives and friends to take his existence after his death in 1985. (That was an actual conspiracy theory). No quantity of evidence can refute the second theory. We can always increase the number of conspirators. The only reason we choose (1) over (2) is the implausibility of (2).

Even with very strong evidence, such as a video of the crime taking place, there will nevertheless be an associated baseline uncertainty, so it would be difficult indeed to convince someone that a murder took place with an estimated >0.999999 probability.

If X has confessed, how can he be on trial?