Not so. “X is guilty” is a very specific hypothesis and 0.99999999 is Very Confident, so general increases in uncertainty should make you think it’s less likely that “X is guilty” is true. For example, if I’m told I misread the question, since I will not be 0.99999999 confident on nearly every question, since I now have non-trivial probability mass on other questions, I should become less confident.
The result is that it takes a specific misreading to make you more confident and that most misreadings will make you less confident, so you should become less confident.
In the log-odds space, both directions look the same. You can wander up as easily as down.
I don’t know what probability space you have in mind for the set of all possible phenomena leading to an error, that would give a basis for saying that most errors will lie in one direction.
When I calculated the odds for the Euromillions lottery, my first calculation omitted to divide by a factor to account for there being no ordering on the chosen numbers, giving a probability for winning that was too small by a factor of 240. The true value is about 140 million to 1.
I have noted before that ordinary people, too ignorant to know that clever people think it impossible, manage to collect huge jackpots. It is literally news when they do not.
It’s not a random walk among probabilities, it’s a random walk among questions, which have associated probabilities. This results in a non-random walk downwards in probability.
The underlying distribution might be described best as “nearly all questions cannot be decided with probabilities that are as certain as 0.999999”.
There is a difference in “error in calculation” versus “error in interpreting the question”. The former affects the result in such a way that makes it roughly as likely to go up as down. If you err in interpreting the question, you’re placing higher probability mass on other questions, which you are less than 0.999999 certain about on average. Roughly, I’m saying that you expect regression to the mean effects to apply in proportion to the uncertainty. E.g. If I tell you I scored an 90% on my test for which the average was a 70%, then you expect me to score a bit lower on a test of equal difficulty. However, if I tell you that I guessed on half the questions, then you should expect me to score a lot lower than you did if you assumed I guessed on 0 questions.
I don’t know why the last comment is relevant. I agree that 1 in a million odds happen 1 in a million times. I also agree that people win the lottery. My interpretation is that it means “sometimes people say impossible when they really mean extremely unlikely”, which I agree is true.
I don’t know why the last comment is relevant. I agree that 1 in a million odds happen 1 in a million times. I also agree that people win the lottery. My interpretation is that it means “sometimes people say impossible when they really mean extremely unlikely”, which I agree is true.
The point was not that people win the lottery. It’s that when they do, they are able to update against the over 100 million-to-one odds that this has happened. “No, no,” say the clever people who think the human mind is incapable of such a shift in log-odds, “far more likely that you’ve made a mistake, or the lottery doesn’t even exist, or you’ve had a hallucination.” The clever people are wrong.
Anecdata: people who win large lotteries often express verbal disbelief, and ask others to confirm that they are not hallucinating. In fact, some even express disbelief while sitting in the mansion they bought with their winnings!
Right, but they don’t update to that from a single data point (looking at the winning numbers and their ticket once), they seek out additional data until they have enough subjective evidence to update to the very, very, unlikely event (and they are able to do this because the event actually happened). Probably hundreds of people think they won any given lottery at first, but when they double-check, they discover that they did not.
Not so. “X is guilty” is a very specific hypothesis and 0.99999999 is Very Confident, so general increases in uncertainty should make you think it’s less likely that “X is guilty” is true. For example, if I’m told I misread the question, since I will not be 0.99999999 confident on nearly every question, since I now have non-trivial probability mass on other questions, I should become less confident.
The result is that it takes a specific misreading to make you more confident and that most misreadings will make you less confident, so you should become less confident.
In the log-odds space, both directions look the same. You can wander up as easily as down.
I don’t know what probability space you have in mind for the set of all possible phenomena leading to an error, that would give a basis for saying that most errors will lie in one direction.
When I calculated the odds for the Euromillions lottery, my first calculation omitted to divide by a factor to account for there being no ordering on the chosen numbers, giving a probability for winning that was too small by a factor of 240. The true value is about 140 million to 1.
I have noted before that ordinary people, too ignorant to know that clever people think it impossible, manage to collect huge jackpots. It is literally news when they do not.
It’s not a random walk among probabilities, it’s a random walk among questions, which have associated probabilities. This results in a non-random walk downwards in probability.
The underlying distribution might be described best as “nearly all questions cannot be decided with probabilities that are as certain as 0.999999”.
There is a difference in “error in calculation” versus “error in interpreting the question”. The former affects the result in such a way that makes it roughly as likely to go up as down. If you err in interpreting the question, you’re placing higher probability mass on other questions, which you are less than 0.999999 certain about on average. Roughly, I’m saying that you expect regression to the mean effects to apply in proportion to the uncertainty. E.g. If I tell you I scored an 90% on my test for which the average was a 70%, then you expect me to score a bit lower on a test of equal difficulty. However, if I tell you that I guessed on half the questions, then you should expect me to score a lot lower than you did if you assumed I guessed on 0 questions.
I don’t know why the last comment is relevant. I agree that 1 in a million odds happen 1 in a million times. I also agree that people win the lottery. My interpretation is that it means “sometimes people say impossible when they really mean extremely unlikely”, which I agree is true.
The point was not that people win the lottery. It’s that when they do, they are able to update against the over 100 million-to-one odds that this has happened. “No, no,” say the clever people who think the human mind is incapable of such a shift in log-odds, “far more likely that you’ve made a mistake, or the lottery doesn’t even exist, or you’ve had a hallucination.” The clever people are wrong.
Anecdata: people who win large lotteries often express verbal disbelief, and ask others to confirm that they are not hallucinating. In fact, some even express disbelief while sitting in the mansion they bought with their winnings!
And yet, despite saying “Inconceivable!” they did collect their winnings and buy the mansion.
Right, but they don’t update to that from a single data point (looking at the winning numbers and their ticket once), they seek out additional data until they have enough subjective evidence to update to the very, very, unlikely event (and they are able to do this because the event actually happened). Probably hundreds of people think they won any given lottery at first, but when they double-check, they discover that they did not.