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.