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CCC comments on Probability is in the Mind - Less Wrong
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So, I've been on this site for awhile. When I first came here, I had never had a formal introduction to Bayes' theorem, but it sounded a lot like ideas that I had independently worked out in my high school and college days (I was something of an amateur mathematician and game theorist).
A few days ago I was reading through one of your articles - I don't remember which one - and it suddenly struck me that I may not actually understand priors as well as I think I do.
After re-reading some fo the series, and then working through the math, I'm now reasonably convinced that I don't properly understand priors at all - at least, not intuitively, which seems to be an important aspect for actually using them.
I have a few weird questions that I'm hoping someone can answer, that will help point me back towards the correct quadrant of domain space. I'll start with a single question, and then see if I can claw my way towards understanding from there based on the answers:
Imagine there is a rational, Bayesian AI named B9 which has been programmed to visually identify and manipulate geometric objects. B9's favorite object is a blue ball, but B9 has no idea that it is blue: B9 sees the world through a black and white camera, and has always seen the world through a black and white camera. Until now, B9 has never heard of "colors" - no one has mentioned "colors" to B9, and B9 has certainly never experienced them. Today, unbeknownst to B9, B9's creator is going to upgrade its camera to a full-color system, and see how long it takes B9 to adapt to the new inputs.
The camera gets switched in 5 seconds. Before the camera gets switched, what prior probability does B9 assign to the possibility that its favorite ball is blue?
I'd imagine something like <error in world model: concept 'blue': no definition found>. It would be like asking whether or not the ball is supercalifragilisticexpialidocious.
If B9 has recently been informed that 'blue' is a property, then the prior would be very low. Can balls even be blue? If balls can be blue, then what percentage of balls are blue? There is also a possibility that, if some balls can be blue, all balls are blue; so the probability distribution would have a very low mean but a very high standard deviation.
Any further refinement requires B9 to obtain additional information; if informed that balls can be blue, the odds go up; if informed that some balls are blue, the odds go up further; if further informed that not all balls are blue, the standard deviation drops somewhat. If presented with the luminance formula, the odds may go up significantly (it can't be used to prove blueness, but it can be used to limit the number of possible colours the ball can be, based on the output of the black-and-white camera).