I say this with trepidation, since Peter and Eliezer have both already read this, but...
As an epistemic rationalist, I would say that 1⁄2 is a better approximation than 0, because the Kullback-Leibler Divergence is (about) 1 bit for the former, and infinity for the latter.
(If the probability distribution peaked at 1⁄2, it would be not-completely-unreasonable to use a flat distribution, and express a probability as a fixed-point number between 0 and 1. In that case, it would take 60 bits to express 10^-18. With floating point, you’d get a good approximation with 7 bits.)
But you’re not really making a fair comparison. You’re comparing “probability distribution centered on 1/2” with “0, no probability distribution”. If the “centered on 0” choice doesn’t get to have a distribution, neither should the “centered on 1/2″ choice. Then both give you a divergence of infinity.
The KL-divergence comparison assumes use of a probability distribution. The probability distribution that peaks at zero is going to be able to represent 1E-18 with many fewer bits than the one that peaks at 1⁄2. So zero wins in both cases, and there is no demonstrated conflict between epistemic and instrumental rationality.
I was talking about a discrete probability distribution over two possible states: {meteorite, no meteorite}. You seem to be talking about something else.
Okay. I thought you were talking about real-valued probability distributions from 0 to 1. But I don’t know if you can claim to draw significant conclusions about epistemic rationality from using the wrong type of probability distribution.
I say this with trepidation, since Peter and Eliezer have both already read this, but...
(If the probability distribution peaked at 1⁄2, it would be not-completely-unreasonable to use a flat distribution, and express a probability as a fixed-point number between 0 and 1. In that case, it would take 60 bits to express 10^-18. With floating point, you’d get a good approximation with 7 bits.)
But you’re not really making a fair comparison. You’re comparing “probability distribution centered on 1/2” with “0, no probability distribution”. If the “centered on 0” choice doesn’t get to have a distribution, neither should the “centered on 1/2″ choice. Then both give you a divergence of infinity.
The KL-divergence comparison assumes use of a probability distribution. The probability distribution that peaks at zero is going to be able to represent 1E-18 with many fewer bits than the one that peaks at 1⁄2. So zero wins in both cases, and there is no demonstrated conflict between epistemic and instrumental rationality.
I was talking about a discrete probability distribution over two possible states: {meteorite, no meteorite}. You seem to be talking about something else.
Okay. I thought you were talking about real-valued probability distributions from 0 to 1. But I don’t know if you can claim to draw significant conclusions about epistemic rationality from using the wrong type of probability distribution.
What do you mean by “the wrong type of probability distribution”?