What Is a Statistical Model? 1

ion. The purpose of abstraction is to incorporate into the mathematics those essential relationships that are taken for granted in the domain of application. Yet that is seldom how it is seen from the outside. In my view, necessity is the mother of abstraction, so there can be no virtue in abstraction for its own sake. It follows that this paper should not have been written had I not judged the need to be clear and the proposed solution reasonably compelling. Unless the information WHAT IS A STATISTICAL MODEL? 1301 is suitably encoded, individual plots, pigs, varieties, treatments and subjects are, in mathematical terms, simply elements in a set. We can, and we must, do better than that. A neighbor relationship on a graph is a possible step in this direction, but this paper pushes that idea a little further by expecting each object (graph) to be embedded in other objects in such a way that relationships are preserved. Many of the discussants are clearly more comfortable with algebraic abstraction than I am, others evidently less so. To those floating in the abstract wing it may appear that my hesitant steps have not gone far enough. To others anchored in the concrete it may appear that I have fallen into an abyss of mindless nonsense. The tension between the concrete and the abstract is seldom comfortable, but I believe it is healthy and central to the art of statistical modelling. One does not have to look far in probability, statistics or physics to see that today’s concrete foundations are yesterday’s abstractions. On the other hand, it must ruefully be admitted that most of yesterday’s abstractions are buried elsewhere. Model versus analysis. Statistical models are usually encountered in connection with the statistical analysis of data, and for that reason many statisticians seem to regard the model and the analysis as inextricably intertwined concepts. The situation is quite different in classical applied mathematics, where it is accepted practice to discuss properties of the wave equation or Poisson’s equation independently of any data that might (or might not) be collected. While it is difficult to discuss statistical models without also mentioning analysis, it seems to me that the two notions are logically distinct. As I have used the term in this paper, a model is divorced from any particular area of application in much the same way that the heat equation is not necessarily tied to heat transmission. Although I do not recommend the practice, one might talk of the Bradley–Terry model without indicating whether the intended application is to competition experiments or citation studies or transmission disequilibrium in genetics. In that sense, I am interested in what Tjur calls canonical models. An analysis might well involve the comparison of two or more mutually incompatible models by methods that require numerical approximation of integrals using MCMC. Model checking is an essential component of every analysis, and this might well lead to the consideration of further models. An analysis of financial transaction data might well lead to the conclusion that Brownian motion is not consistent with the data but that an alternative long-tailed model is satisfactory. Thus, in any definition of the term “statistical model,” it is essential to bear in mind that models must exist that are not compatible with any data likely to be collected. By comparison with models, statistical analysis is a complicated topic, and I have tried in the paper to say as little on the matter as I could hope to get away with. As the title suggests, the topic of the paper is narrow. So far as I can tell, it has nothing to say about computation, design or model checking, and very little to say about model selection, or Bayesian versus frequentist analysis.