Let's say I have an experiment which yields discrete results between 1 and $N$. I am modelling the results using a number of statistical models and want to use Akaike (corrected) or Bayesian Information Criterion to choose the best model. How can I derive AICc or BIC if the predicted variables are discrete and bounded? Is there a difference in what we mean by "sample size" in AICc or BIC formula?
"I am shocked that there is so much ambiguity about this issues. Isn't statistics a part of mathematics? Shouldn't there be a clear proof when X works and when it doesn't?"
My answer: one of the most important things I ever learned was that while statistics does heavily involve math, it is not just math. It shares a lot with fields like law and politics. Some of the most important questions in statistics come down to value judgments: What's your goal? What's your data? What are you willing to assume? What can you get out of a model that is very likely to be much simpler than our incredibly complex reality.
But, yes, it would be nice to have a clear answer to the question, "When my data are discrete (in my case, binary or categorical), is the traditional AICc appropriate?"