A place for philosophy? the rise of modeling in statistical science
Large statistical models seem to have reached epidemic proportions. I will document the harm they are currently causing and contrast it with some success stories of smaller modeling efforts. A case study involving a model for flipping coins shows how "adding a few bells and whistles" can lead to nonsense. Finally, an effort to list the "real uses" of models is offered as a start to making sense of current modeling efforts. 1. Two ideal examples. Example 1. The mathematical formulation of statistical problems. A clear description of mathematical statistics can be simply put: One is given a family of probability measures {Pg(dx)}ge@. One observes x & X drawn from Pg(dx) and is required to make a guess 9{x) e © for the true 6. Usually, the quality of a guess 0 is quantified by a loss function L(0,6). Then, one tries to choose 9 so that / L(9(x), 9)Pg{dx) is small. In a Bayesian version of the set up, an apriori distribution n(dO) is specified and one chooses 9 so that / L(0(x),6)Pg(dx)-K(d6) is small. With {Pg}, L, and 7r specified, this is usually straight forward. Of course, in any real problem, Pg(dx) is only an approximation and there may be problems in specifying L and it. This is further grist for the mathematical mill: How does 9 vary with changes in {Pe}eeo> L, and 7r? The above formulation has had great success at codifying statistical practice and in suggesting new estimates (and new math problems). The clearest treatment of modern mathematical statistics is in Lehmann (1983, 1986). Example 2. Gauss and the linear model. Here is a more applied example of mathematical statistics: Gauss' discovery of the asteroid Ceres. The year is 1801. The 21-year-old Gauss has just finished his monumental Disquisitiones Mathematica; he notices he does not have a job. At this time people believed there were only five planets; then, the astronomer Patuzzi spotted the asteroid Ceres and took readings on it for 60 days. Finally, he lost Ceres in the sun. His observations caused great excitement in Received March 2, 1998. 1991 Mathematics Subject Classification. Primary 62A99; Secondary 62J99. ©1998 Brown University