I thought of a problem which was related, but not the same, and it seems much harder. I don’t know how to start solving it.
The sheet may(let’s say 50%) have been switched by Oswald’s son Norbert, except he doesn’t know his numbers that well, so he just wrote “2d6” on the top and then filled it with nothing but the number 4, because 6-2 is 4. Oswald doesn’t notice this, since this is a bar and well, he’s drunk since Norbert’s been a handful lately.
Presumably, if this happens, at some point you will realize that buying information is not helping you in the slightest and stop. You are probably not going to pay 10,000 pounds for information 1 pound at a time on the 10,001st 4, because you will have probably considered the possibility that while it has an equal chance of being either, this doesn’t seem likely to be a distribution of a 2d6 or a 1d12.
How would you calculate a maximum amount in total you should be willing to pay for information on a potentially corrupted bet like this?
The only answer I can think of here is to take a prior over all possible sequence generating computer programs, weighting them for length, and then to jack up the ones that simulate 2d6 or 1d12, and then to use the numbers to update on that.
I don’t know if I’ve just described Solomonoff Induction or similar, but it sounds complicated, and yet I notice that if I’d just seen 10000 consecutive 4s I’d be pretty hot for the ‘always gives 4’ theory, and I wonder how I’d be doing that with my limited supply of slow neurons.
I thought of a problem which was related, but not the same, and it seems much harder. I don’t know how to start solving it.
The sheet may(let’s say 50%) have been switched by Oswald’s son Norbert, except he doesn’t know his numbers that well, so he just wrote “2d6” on the top and then filled it with nothing but the number 4, because 6-2 is 4. Oswald doesn’t notice this, since this is a bar and well, he’s drunk since Norbert’s been a handful lately.
Presumably, if this happens, at some point you will realize that buying information is not helping you in the slightest and stop. You are probably not going to pay 10,000 pounds for information 1 pound at a time on the 10,001st 4, because you will have probably considered the possibility that while it has an equal chance of being either, this doesn’t seem likely to be a distribution of a 2d6 or a 1d12.
How would you calculate a maximum amount in total you should be willing to pay for information on a potentially corrupted bet like this?
The only answer I can think of here is to take a prior over all possible sequence generating computer programs, weighting them for length, and then to jack up the ones that simulate 2d6 or 1d12, and then to use the numbers to update on that.
I don’t know if I’ve just described Solomonoff Induction or similar, but it sounds complicated, and yet I notice that if I’d just seen 10000 consecutive 4s I’d be pretty hot for the ‘always gives 4’ theory, and I wonder how I’d be doing that with my limited supply of slow neurons.