Danny Uncanny7 wrote:Edit: without letting the problem sprawl out of hand, I guess what I really want to figure out is this: for an arbitrary image (or any data set really), what is the probability that each pixel was independently selected from the same probability distribution? I think that the answer will in some way come down to the autocorrelation, or some form of it.
A1) Reading this as "what is the probability that a given probability distribution would produce this arbitrary image?"
Nearly zero, assuming your probability distribution isn't boring.
Any fixed image is very, very unlikely to be produced by a particular probability distribution. If your per-pixel distribution was uniform, then the odds of a pure white image are equal to the odds that your exact image would be produced.
A2) Lets say you actually want an answer to the question asked -- then it depends on your priors. Suppose you have a belief that a certain kind of systematic error could occur -- local smears, creases, circles or what have you. You build a model where a family of those have a certain chance of occurring, and there is a certain chance of a uniform uncorrelated image. Then from this, we could work out what the probability that your image was produced by each model in particular.
However, this depends hugely on what your priors are. And, in fact, you need to build a complete set of models of alternative explanations you want to examine (or a statistical model that effectively contains them) if you want an exact answer.
Your priors, however, are highly important. Usually these priors are implicit in the statistical analysis you end up doing -- maybe you sort your images implicitly into "more ordered" and "less ordered" categories, use that to implicitly inform your priors...
A3) You actually don't want an answer to the question. Rather, you want a tool that you can use to say "well, this image is pretty darn random" or "this image seems to have some kind of pattern", and it outputs some value that gets bigger if there seems to be some kind of pattern that the tool recognizes.