One time, someone asked me what my name was. I said, “Mark Xu.” Afterward, they probably believed my name was “Mark Xu.” I’m guessing they would have happily accepted a bet at 20:1 odds that my driver’s license would say “Mark Xu” on it.
The prior odds that someone’s name is “Mark Xu” are generously 1:1,000,000. Posterior odds of 20:1 implies that the odds ratio of me saying “Mark Xu” is 20,000,000:1, or roughly 24 bits of evidence. That’s a lot of evidence.
Seeing a Wikipedia page say “X is the capital of Y” is tremendous evidence that X is the capital of Y. Someone telling you “I can juggle” is massive evidence that they can juggle. Putting an expression into Mathematica and getting Z is enormous evidence that the expression evaluates to Z. Vast odds ratios lurk behind many encounters.
One implication of the Efficient Market Hypothesis (EMH) is that is it difficult to make money on the stock market. Generously, maybe only the top 1% of traders will be profitable. How difficult is it to get into the top 1% of traders? To be 50% sure you're in the top 1%, you only need 200:1 evidence. This seemingly large odds ratio might be easy to get.
On average, people are overconfident, but 12% aren't. It only takes 50:1 evidence to conclude you are much less overconfident than average. An hour or so of calibration training and the resulting calibration plots might be enough.
Running through Bayes’ Rule explicitly might produce a bias towards middling values. Extraordinary claims require extraordinary evidence, but extraordinary evidence might be more common than you think.
One particularly important thing I got out of this post was crystallizing a complaint I sometimes have about people using anthropic reasoning. If someone says there's trillion-to-1 evidence for (blah) based on anthropics, it's actually not so crazy to say "well I don't believe (blah) anyway, based on the evidence I get from observing the world", it seems to me.
Or less charitably to myself, maybe this post is helping me rationalize my unjustified and unthinking gut distrust of anthropic reasoning :-P
Anyway, great post.
ADDED 3½ YEARS LATER: Whoops, that’s not a good argument. Generically in Bayesian rationality, if you encounter trillion-to-1 evidence for X over Y, and then later encounter trillion-to-1 evidence for Y over X, then the wrong thing to do is apply Bayes’s rule and revert to the prior, whereas the right thing to do is to recognize that this situation should basically never happen, and therefore it’s overwhelmingly likely that at least one of those two purported pieces of evidence is actually wrong or misinterpreted. (H/t something Eliezer wrote, probably.)
Portions of this are taken directly from Three Things I've Learned About Bayes' Rule.
One time, someone asked me what my name was. I said, “Mark Xu.” Afterward, they probably believed my name was “Mark Xu.” I’m guessing they would have happily accepted a bet at 20:1 odds that my driver’s license would say “Mark Xu” on it.
The prior odds that someone’s name is “Mark Xu” are generously 1:1,000,000. Posterior odds of 20:1 implies that the odds ratio of me saying “Mark Xu” is 20,000,000:1, or roughly 24 bits of evidence. That’s a lot of evidence.
Seeing a Wikipedia page say “X is the capital of Y” is tremendous evidence that X is the capital of Y. Someone telling you “I can juggle” is massive evidence that they can juggle. Putting an expression into Mathematica and getting Z is enormous evidence that the expression evaluates to Z. Vast odds ratios lurk behind many encounters.
One implication of the Efficient Market Hypothesis (EMH) is that is it difficult to make money on the stock market. Generously, maybe only the top 1% of traders will be profitable. How difficult is it to get into the top 1% of traders? To be 50% sure you're in the top 1%, you only need 200:1 evidence. This seemingly large odds ratio might be easy to get.
On average, people are overconfident, but 12% aren't. It only takes 50:1 evidence to conclude you are much less overconfident than average. An hour or so of calibration training and the resulting calibration plots might be enough.
Running through Bayes’ Rule explicitly might produce a bias towards middling values. Extraordinary claims require extraordinary evidence, but extraordinary evidence might be more common than you think.