I currently try to understand Likelihood Principle and I frankly don't get it at all. So, I will write all my question as a list, even if those might be pretty basic questions.
- What exactly does "all of the information" phrase mean in the context of this principle? (as in all of the information in a sample is contained in the likelihood function.)
- Is the principle somehow connected to the very provable fact, that $p(x|y)\propto p(y|x)p(x)$? Is the "likelihood" in the principle the same thing, as $p(y|x)$, or not?
- How can a mathematical theorem be "controversial"? My (weak) understanding of math is that a theorem is either proven, or is not proven. To what category does Likelihood Principle fall?
- How is the Likelihood Principle important for Bayesian inference, which is based on $p(x|y)\propto p(y|x)p(x)$ formula?