I was recently disturbed by my perception that, despite years of studying and debating probability problems, the LessWrong community as a whole has not markedly improved its ability to get the right answer on them.

I had expected that people would read posts and comments by other people, and take special note of comments by people who had a prior history of being right, and thereby improve their own accuracy.

But can that possibly work?  How can someone who isn't already highly-accurate, identify other people who are highly accurate?

Aumann's agreement theorem (allegedly) says that Bayesians with the same priors agree.  But it doesn't say that doing so helps.  Under what circumstances does revising your opinions, by updating in response to people you consider reliable, actually improve your accuracy?

To find out, I built a model of updating in response to the opinions of others.  It did, eventually, show that Bayesians improve their collective opinions by updating in response to the opinions of other Bayesians.  But this turns out not to depend on them satisfying the conditions of Aumann's theorem, or on doing Bayesian updating.  It depends only on a very simple condition, established at the start of the simulation.  Can you guess what it is?

I'll write another post describing and explaining the results if this post receives a karma score over 10.

In 2004, The United States government executed Cameron Todd Willingham via lethal injection for the crime of murdering his young children by setting fire to his house. 

In 2009, David Grann wrote an extended examination of the evidence in the Willingham case for The New Yorker, which has called into question Willingham's guilt. One of the prosecutors in the Willingham case, John Jackson, wrote a response summarizing the evidence from his current perspective. I am not summarizing the evidence here so as to not give the impression of selectively choosing the evidence.

A prior probability estimate for Willingham's guilt (certainly not a close to optimal prior probability) is the probability that a fire resulting in the fatalities of children was intentionally set. The US Fire Administration puts this probability at 13%. The prior probability could be made more accurate by breaking down that 13% of intentionally set fires into different demographic sets, or looking at correlations with other things such as life insurance data.

My question for Less Wrong: Just how innocent is Cameron Todd Willingham? Intuitively, it seems to me that the evidence for Willingham's innocence is of higher magnitude than the evidence for Amanda Knox's innocence. But the prior probability of Willingham being guilty given his children died in a fire in his home is higher than the probability that Amanda Knox committed murder given that a murder occurred in Knox's house.

Challenge question: What does an idealized form of Bayesian Justice look like? I suspect as a start that it would result in a smaller percentage of defendants being found guilty at trial. This article has some examples of the failures to apply Bayesian statistics in existing justice systems.

I present an algorithm I designed to predict which position a person would report for an issue on TakeOnIt, through Bayesian updates on the evidence of other people's positions on that issue. Additionally, I will point out some potential areas of improvement, in the hopes of inspiring others here to expand on this method.

For those not familiar with TakeOnIt, the basic idea is that there are issues, represented by yes/no questions, on which people can take the positions Agree (A), Mostly Agree (MA), Neutral (N), Mostly Disagree (MD), or Disagree (D). (There are two types of people tracked by TakeOnIt: users who register their own opinions, and Experts/Influencers whose opinions are derived from public quotations.)

The goal is to predict what issue a person S would take on a position, based on the positions registered by other people on that question. To do this, we will use Bayes' Theorem to update the probability that person S takes the position X on issue I, given that person T has taken position Y on issue I:

Really, we will be updating on several people T_j taking positions T_y on I:

Katja Grace has just presented an ingenious model, claiming that SIA combined with the great filter generates its own variant of the doomsday argument. Robin echoed this on Overcoming Bias. We met soon after Katja had come up with the model, and I signed up to it, saying that I could see no flaw in the argument.

Unfortunately, I erred. The argument does not work in the form presented.

First of all, there is the issue of time dependence. We are not just a human level civilization drifting through the void in blissful ignorance about our position in the universe. We know (approximately) the age of our galaxy, and the time elapsed since the big bang.

How is this relevant? It is relevant because all arguments about the great filter are time-dependent. Imagine we had just reached consciousness and human-level civilization, by some fluke, two thousand years after the creation of our galaxy, by an evolutionary process that took two thousand years. We see no aliens around us. In this situation, we have no reason to suspect any great filter; if we asked ourselves "are we likely to be the first civilization to reach this stage?" then the answer is probably yes. No evidence for a filter.

Imagine, instead, that we had reached consciousness a trillion years into the life of our galaxy, again via an evolutionary process that took two thousand years, and we see no aliens or traces of aliens. Then the evidence for a filter is overwhelming; something must have stopped all those previous likely civilizations from emerging into the galactic plane.

So neither of these civilizations can be included in our reference class (indeed, the second one can only exist if we ourselves are filtered!). So the correct reference class to use is not "the class of all potential civilizations in our galaxy that have reached our level of technological advancement and seen no aliens", but "the class of all potential civilizations in our galaxy that have reached our level of technological advancement at around the same time as us and seen no aliens". Indeed, SIA, once we update on the present, cannot tell us anything about the future.

But there's more.

Sometimes in an argument, an older opponent might claim that perhaps as I grow older, my opinions will change, or that I'll come around on the topic.  Implicit in this claim is the assumption that age or quantity of experience is a proxy for legitimate authority.  In and of itself, such "life experience" is necessary for an informed rational worldview, but it is not sufficient.

The claim that more "life experience" will completely reverse an opinion indicates that the person making such a claim believes that opinions from others are based primarily on accumulating anecdotes, perhaps derived from extensive availability bias.  It actually is a pretty decent assumption that other people aren't Bayesian, because for the most part, they aren't.  Many can confirm this, including Haidt, Kahneman, and Tversky.

When an opponent appeals to more "life experience," it's a last resort, and it's a conversation halter.  This tactic is used when an opponent is cornered.  The claim is nearly an outright acknowledgment of moving to exit the realm of rational debate.  Why stick to rational discourse when you can shift to trading anecdotes?  It levels the playing field, because anecdotes, while Bayesian evidence, are easily abused, especially for complex moral, social, and political claims.  As rhetoric, this is frustratingly effective, but it's logically rude.

Although it might be rude and rhetorically weak, it would be authoritatively appropriate for a Bayesian to be condescending to a non-Bayesian in an argument.  Conversely, it can be downright maddening for a non-Bayesian to be condescending to a Bayesian, because the non-Bayesian lacks the epistemological authority to warrant such condescension.  E.T. Jaynes wrote in Probability Theory about the arrogance of the uninformed, "The semiliterate on the next bar stool will tell you with absolute, arrogant assurance just how to solve the world's problems; while the scholar who has spent a lifetime studying their causes is not at all sure how to do this."

This is a variant built on Gary Drescher's xor problem for timeless decision theory.

You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.

Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.

He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:

"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."

What to do?

EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.