ABSTRACT

The likelihood function connects the pre-experimental information, expressed by the model choice, with the experimental information. The theory of statistics has evolved as a compromise among different requirements, attempting to combine logical correctness with practical needs. In particular, it will turn out that a good deal of what we shall say will comply with the weak likelihood principle, but a smaller part will follow the strong likelihood principle. In a very crude description of statistical theory, one could say that its purpose is to select the most appropriate 'operations' to perform on the data. Considering the strong connection between exponential families and sufficient statistics, it is reasonable to ask whether the existence of a non-trivial sufficient statistic implies that the parametric class is an exponential family. The chapter examines the nature and properties of those statistics which are able to summarize 'all the information' present in the likelihood function.