Aside from the other problems that have been pointed out, I will also take exception to calling an order of magnitude a rough estimate. An order of magnitude would be a rough estimate where you have actual numeric data to work with. In cases where you have to just make up the numbers, an order of magnitude is high precision—in some of these cases, extraordinarily high precision, far greater than you have any reason for claiming.
I’d say “an order of magnitude is a rough estimate” is a rough estimate. Remember, this is epistemic probability, so whether you
just think 76297 looks prime-ish and guess 9⁄10
mentally estimate the natural logarithm, quickly check whether 76297 is divisible by 2 or 3, and call it a 1⁄2 chance
can actually compute the Sieve of Eratosthenes with five nines of accuracy for it in ten seconds and call it a 1/10000 chance
You’re correct, as long as you’re not mis-reading your own degree of belief. To get into confidence about your degree of belief, I think we’d have to get into something like informal Dempster-Schafer theory—which, incidentally, I’d love to do.
Aside from the other problems that have been pointed out, I will also take exception to calling an order of magnitude a rough estimate. An order of magnitude would be a rough estimate where you have actual numeric data to work with. In cases where you have to just make up the numbers, an order of magnitude is high precision—in some of these cases, extraordinarily high precision, far greater than you have any reason for claiming.
I’d say “an order of magnitude is a rough estimate” is a rough estimate. Remember, this is epistemic probability, so whether you
just think 76297 looks prime-ish and guess 9⁄10
mentally estimate the natural logarithm, quickly check whether 76297 is divisible by 2 or 3, and call it a 1⁄2 chance
can actually compute the Sieve of Eratosthenes with five nines of accuracy for it in ten seconds and call it a 1/10000 chance
You’re correct, as long as you’re not mis-reading your own degree of belief. To get into confidence about your degree of belief, I think we’d have to get into something like informal Dempster-Schafer theory—which, incidentally, I’d love to do.