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wuncidunci 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?
Your question is not well specified. Event though you might think that the proposition "its favorite ball is blue" is something that has a clear meaning, it is highly dependent on to which precision it will be able to see colours, how wide the interval defined as blue is, and how it considers multicoloured objects. If we suppose it would categorise the observed wavelength into one of 27 possible colours (one of those being blue), and further suppose that it knew the ball to be of a single colour and not patterned, and further not have any background information about the relative frequencies of different colours of balls or other useful prior knowledge, the prior probability would be 1/27. If we suppose that it had access to internet and had read this discussion on LW about the colourblind AI, it would increase its probability by doing an update based on the probability of this affecting the colour of its own ball.