Anthropic reasoning is what leads people to believe in miracles. Rare events have a high probability of occurring if the number of observations is large enough. But whoever that rare event happens to will feel like it couldn’t have just happened by chance, because the odds of it happening to them was so large.
If you wait until the event occurs, and then start treating it as a random event from a single trial, forming your hypothesis after seeing the data, you’ll make inferential errors.
Imagine that there are balls in an urn, labeled with numbers 1, 2,...,n. Suppose we don’t know n. A ball is selected. We look at it. We see that it’s number x.
non-anthropic reasoning: all numbers between 1 and n were equally likely. I was guaranteed to observe some number, and the probability that it was close to n was the same as the probability that it was far from n. So all I know is that n is greater than or equal to x.
anthropic reasoning: A number as small as x is much less likely if n is large. Therefore, hypotheses with n close to x are more likely than hypotheses where n is much larger than x.
What you have labeled anthropic reasoning is actually straight-up Bayesian reasoning. Wikipedia has an article on the problem, but only discusses the Bayesian approach briefly and with no depth. Jaynes also talks about it early in PT:LOS. In any event, to see the logic of the math, just write down the likelihood function and any reasonable prior.
Anthropic reasoning is what leads people to believe in miracles. Rare events have a high probability of occurring if the number of observations is large enough. But whoever that rare event happens to will feel like it couldn’t have just happened by chance, because the odds of it happening to them was so large.
If you wait until the event occurs, and then start treating it as a random event from a single trial, forming your hypothesis after seeing the data, you’ll make inferential errors.
Imagine that there are balls in an urn, labeled with numbers 1, 2,...,n. Suppose we don’t know n. A ball is selected. We look at it. We see that it’s number x.
non-anthropic reasoning: all numbers between 1 and n were equally likely. I was guaranteed to observe some number, and the probability that it was close to n was the same as the probability that it was far from n. So all I know is that n is greater than or equal to x.
anthropic reasoning: A number as small as x is much less likely if n is large. Therefore, hypotheses with n close to x are more likely than hypotheses where n is much larger than x.
What you have labeled anthropic reasoning is actually straight-up Bayesian reasoning. Wikipedia has an article on the problem, but only discusses the Bayesian approach briefly and with no depth. Jaynes also talks about it early in PT:LOS. In any event, to see the logic of the math, just write down the likelihood function and any reasonable prior.
I suggest reading Radford Neal.
Yes, I’ve read that paper, and disagree with much of it. Perhaps I’ll take the time to explain my reasoning sometime soon