Since Fisher's formulation in 1922 of a framework for theoretical statistics, statistical theory has been concerned primarily with the derivation and properties of suitable statistical procedures on the basis of an assumed statistical model (including sensitivity to deviations from this model). Until relatively recently, the theory has paid little attention to the question of how such a model should be chosen. In the present paper, we consider first what Fisher and Neyman had to say about this problem and in Section 2 survey some contributions statistical theory has made to it. In Section 3 we study a distinction between two types of models (empirical and explanatory) which has been discussed by Neyman, Box, and others. A concluding section considers some lines of further work.

A number of distinct roles are identified for probability models used in the analysis of data. Examples are outlined. Some general issues arising in the formulation of such models are discussed.

A strategy for online multivariate exploratory graphical analysis is presented and illustrated, motivated by the need for a routine procedure for searching for structure in multivariate data sets arising in the context of a major pharmaceutical, dyestuffs and agrochemical company.

The concept of predictive likelihood is reviewed and studied. The emphasis is on comparing and clarifying some of the more important predictive likelihoods suggested in the literature. A unified modification and simplification of the sufficiency-based predictive likelihoods is suggested. Other predictive likelihoods discussed include the profile predictive likelihood and various modifications of it.