On the foundations of likelihood principle

According to the likelihood principle, if the designs produce proportional likelihood functions, one could make an identical inference about a parameter from the data irrespective of the design, which yields the data. If it comes to that, there are several counter-examples, and/or paradoxical consequences to likelihood principle. Besides, as we will see, contrary to a widely held opinion, such a principle is not a direct consequence of Bayes theorem. In particular, the piece of information about the design is one part of the evidence, and it is relevant for the prior. Later on, Jeffreys non-informative prior is used to show how different designs result in different priors. Another basic idea of the present paper is that (apart from other information) the equiprobability assumption is to be linked to the idea of the impartiality of design with respect to the parameter under consideration. The whole paper has remarkable implications on the foundations of statistics from the notion of sufficiency, the relevance of the stopping rule and of the randomization in survey sampling and in the experimental design, the difference between ignorable and non-ignorable designs, until a reconciliation of different approaches to the inductive reasoning in statistical inference.