In Statistical Evidence: A likelihood paradigm, Richard Royall distinguishes between three questions that a statistician might ask when he receives a new observation (or set of observations):
- What do I believe, now that I have this observation?
- What should I do, now that I have this observation?
- What does this observation tell me about hypothesis vs hypothesis ?
He claims that confusing these three questions leads to confusion in the foundations of statistics. The Neyman-Pearson paradigm is concerned with question 2. Royall claims that 3 is best answered by the law of likelihood.
For my part, I think statistics isn’t in the business of answering question 1. I don’t believe that there are normative rules for thought which follow from any formal system. A subjective Bayesian, on the other hand, would argue that belief should obey the probability calculus.