I’ve just started reading Jaynes on prior formation, and I’d love to see more posts here on the topic. Maybe I’ll write one if I ever have the chance to get some reading done.
As far as this problem goes, I agree we have some information about other colors. I want to know what Omega counts as “red” though, because that will go a long way in determining what sort of prior we’d assign.
Based on my limited understanding of physics, if we assume the bead only reflects a single wavelength, then it would be red if the wavelength were between 620 and 750nm. The visible spectrum goes from 380 to 750nm, so a uniform distribution over wavelength gives a probability of 0.34.
Most objects reflect multiple wavelengths though. In this case, the color of an object could be characterized as a distribution over the visible spectrum of the reflected light. To count as red, the distribution would need a mean between 620 and 750nm. We might need an additional constraint on the shape of the distribution so it isn’t too spread out. I don’t know how you’d begin calculating the measure of distributions that meet these constraints though.
I think drawing a red bead will marginally increase our probability of red.
In the end, I think our estimate is much more likely to depend on our expectation of Omega’s motives than knowledge about colors, though.
I’ve just started reading Jaynes on prior formation, and I’d love to see more posts here on the topic. Maybe I’ll write one if I ever have the chance to get some reading done.
As far as this problem goes, I agree we have some information about other colors. I want to know what Omega counts as “red” though, because that will go a long way in determining what sort of prior we’d assign.
Based on my limited understanding of physics, if we assume the bead only reflects a single wavelength, then it would be red if the wavelength were between 620 and 750nm. The visible spectrum goes from 380 to 750nm, so a uniform distribution over wavelength gives a probability of 0.34.
Most objects reflect multiple wavelengths though. In this case, the color of an object could be characterized as a distribution over the visible spectrum of the reflected light. To count as red, the distribution would need a mean between 620 and 750nm. We might need an additional constraint on the shape of the distribution so it isn’t too spread out. I don’t know how you’d begin calculating the measure of distributions that meet these constraints though.
I think drawing a red bead will marginally increase our probability of red.
In the end, I think our estimate is much more likely to depend on our expectation of Omega’s motives than knowledge about colors, though.