Benja, I have never studied Solomonoff induction formally. God help me, but I’ve only read about it on the Internet. It definitely was what I was thinking of as a candidate for evaluating theories given evidence. But since I don’t really know it in a rigorous way, it might not be suitable for what I wanted in that hand-wavy part of my argument.
However, I don’t think I made quite so bad a mistake as highly-ranking the “we will observe some experimental result” theory. At least I didn’t make that mistake in my own mind ;). What I actually wrote was certainly vague enough to invite that interpretation. But what I was thinking was more along these lines:
[looks up color spectrum on Wikipedia and juggles numbers to make things work out]
The visible wavelengths are 380 nm -- 750 nm. Within that range, blue is 450 nm -- 495 nm, and red is 620 nm -- 750 nm.
Let f(x) be the decimal expansion of (x − 380nm)/370nm. This moves the visible spectrum into the range [0,1].
I was imagining that T3 (“the ball is visible”) was predicting
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign).”
while T1 (“the ball is red”) predicts
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign), and the digit immediately to the right is a 7.”
and T2 (“the ball is blue”) predicts
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign), and the digit immediately to the right is a 2.”
So I was really thinking of all the theories T1, T2, and T3 as giving precise predictions. It’s just that T3 opted not to make a prediction about something that T2 and T3 did predict on.
However, I definitely take the point that Solomonoff induction might still not be suitable for my purposes. I was supposing that T3 would be a “better” theory by some criterion like Solomonoff induction. (I’m assuming, BTW, that T3 did predict everything that T1 and T2 predicted for the first 20 results. It’s only for the 21st result that T3 didn’t give an answer as detailed as those of T1 and T2. ) But from reading your comment, I guess maybe Solomonoff induction wouldn’t even compare T3 to T1 and T2, since T3 doesn’t purport to answer all of the same questions.
If so, I think that just means the Solomonoff isn’t quite general enough. There should be a way to compare two theories even if one of them answers questions that the other doesn’t address. In particular, in the case under consideration, T1 and T2 are given to be “equally good” (in some unspecified sense), but they both purport to answer the same question in a different way. To my mind, that should mean that each of them isn’t really justified in choosing its answer over the other. But T3, in a sense, acknowledges that there is no reason to favor one answer over the other. There should be some rigorous sense in which this makes T3 a better theory.
Tim Freeman, I hope to reply to your points soon, but I think I’m at my “recent comments” limit already, so I’ll try to get to it tomorrow.
Benja, I have never studied Solomonoff induction formally. God help me, but I’ve only read about it on the Internet. It definitely was what I was thinking of as a candidate for evaluating theories given evidence. But since I don’t really know it in a rigorous way, it might not be suitable for what I wanted in that hand-wavy part of my argument.
However, I don’t think I made quite so bad a mistake as highly-ranking the “we will observe some experimental result” theory. At least I didn’t make that mistake in my own mind ;). What I actually wrote was certainly vague enough to invite that interpretation. But what I was thinking was more along these lines:
[looks up color spectrum on Wikipedia and juggles numbers to make things work out]
The visible wavelengths are 380 nm -- 750 nm. Within that range, blue is 450 nm -- 495 nm, and red is 620 nm -- 750 nm.
Let f(x) be the decimal expansion of (x − 380nm)/370nm. This moves the visible spectrum into the range [0,1].
I was imagining that T3 (“the ball is visible”) was predicting
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign).”
while T1 (“the ball is red”) predicts
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign), and the digit immediately to the right is a 7.”
and T2 (“the ball is blue”) predicts
“The only digit to the left of the decimal point in f(color of ball in nm) is a 0 (without a negative sign), and the digit immediately to the right is a 2.”
So I was really thinking of all the theories T1, T2, and T3 as giving precise predictions. It’s just that T3 opted not to make a prediction about something that T2 and T3 did predict on.
However, I definitely take the point that Solomonoff induction might still not be suitable for my purposes. I was supposing that T3 would be a “better” theory by some criterion like Solomonoff induction. (I’m assuming, BTW, that T3 did predict everything that T1 and T2 predicted for the first 20 results. It’s only for the 21st result that T3 didn’t give an answer as detailed as those of T1 and T2. ) But from reading your comment, I guess maybe Solomonoff induction wouldn’t even compare T3 to T1 and T2, since T3 doesn’t purport to answer all of the same questions.
If so, I think that just means the Solomonoff isn’t quite general enough. There should be a way to compare two theories even if one of them answers questions that the other doesn’t address. In particular, in the case under consideration, T1 and T2 are given to be “equally good” (in some unspecified sense), but they both purport to answer the same question in a different way. To my mind, that should mean that each of them isn’t really justified in choosing its answer over the other. But T3, in a sense, acknowledges that there is no reason to favor one answer over the other. There should be some rigorous sense in which this makes T3 a better theory.
Tim Freeman, I hope to reply to your points soon, but I think I’m at my “recent comments” limit already, so I’ll try to get to it tomorrow.