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".
Having sat on a jury (for a rather dull case of a failed burglary), I concur with this.
Jury confidentiality is taken seriously in the UK, so I can't comment on our deliberations, but the consensus was that it was him wot dunnit. He looked resigned rather than indignant when the verdict was read out, so with that and the evidence I'm as sure as I need to be that we got it right. I couldn't put a number on it, but 0.000001 is way smaller than a reasonable doubt.
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.
You just need to be confident enough that, given one million similar cases, you would make only one mistake.
This still seems crazy confident to me though. I do think there are hypothetical people who could do it, but I don't currently have strong reason to believe there actually exist even trained rationalists that could do it, even if they were extremely careful every single time.
Given a million evaluations of the video chain-of-custody or DNA evidence, you expect there are people who would not make a mistake (or, be actively deceived by an adversary, or have forgotten to eat lunch and not noticed they're tired?) even twice?
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.]
If I buy a ticket in the Euromillions lottery, I am over 0.99999999 sure I will lose. (There are more than 100 million possible draws.)
Yes, see response to Dagon. But, 0.99999999 seems overconfident to me. You have to account not only for "I might be insane" (what are the base rates on that?), but simpler things like "I misread the question or had a brain fart."
Like, there's an old LW chat log where someone claims they can be 99.999% confident about whether a low-digit number is prime. Then someone challenges them to answer "prime or not?" for ~100 numbers. And then like 25 questions in they get one wrong. 0.99999999 is a Really God Damn Confident.
Even if you include esoteric options, like being a Boltzmann brain, you can have negatives with way more probability than 999999/1000000. It's EASY to be more certain than that on "will I fail to win the next powerball drawing". And more certain still on "did I fail to win the previous powerball drawing".
Some recursive positives approach 1 - "I exist". Tautologies remain actually 1: P -> P.
But for random human-granularity events where you have only very indirect evidence, you're right. 99% would be surprising, 95% would take a fair bit of effort.
what's an example of a complicated bet that you shouldn't take even if you're real damn certain?
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).
If X has confessed, how can he be on trial?
1. X confesses to police, but later claims that the confession was cooerced, and asks for a trial.
2. X confesses to some part of the crime "I was holding the knife that penetrated the deceased" but not all of it "but I was sleepwalking at the time, so it's not Murder" or "but I was in a jealous rage at the time, so it's not pre-meditated"
3. X confesses, but the prosecutor believes that other people were involved (regardless of the status of X) and is holding a joint trial for all the accused.
by the way, it's an underrated aspect of Bayeseanism that encountering the question _is_ evidence. The prior for even having a trial on a topic that could approach certainty is extremely low. If the evidence existed that would get you even to 99%, they'd bargain out of having a trial.
I am a prosecutor.
Yes, trials occur when there are viable outcomes on all sides. But also:
1. One of the people involved in a criminal plea negotiations is irrational. (Prosecutor, defense attorney, defendant.) Defendants sometimes go to trial on crushing cases. Attorneys may want their clients to take a deal, and it doesn't happen.
2. There's no benefit to pleading for either side. If you've got a second degree murder with a fixed-by-statute sentence and the defendant is stone cold good for it, the prosecution may not offer a deal for the defense to take, and then the defense gets a trial because there's no harm to the client
In most jurisdictions, prosecutors win a very high percentage of trials because there are a lot of very good cases that go to trial.
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.)