Statistical thinking: From Tukey to Vardi and beyond

Data miners (minors?) and neural networkers tend to eschew modelling, misled perhaps by misinterpretation of strongly expressed views of John Tukey. I discuss Vardi’s views of these issues as well as other aspects of Vardi’s work in emision tomography and in sampling bias. Statistics is not in my main skill set but I take this opportunity to record a few thoughts about it I have accumulated over the years. In the ’60’s, John Tukey and his followers brought exploratory data analysis into statistics, partly as a revolt against what was then perceived as an overly rigid and brittle mathematical modelling philosophy that held sway at that time. Some problems seemed to demand such a purely data-driven approach where data mining methods in the absence of mathematical modelling is the driving philosophical methodology. One did not want to be biased by preconceived ideas about the origin of the data by formulating a model but instead allowed the data to “speak for itself”. Vardi liked mathematical modelling and was very good at it. He also promoted data mining, depending on the problem and thus straddled both philosophies. He and I often debated these issues, and were often in friendly disagreement. I will try to argue with concrete examples of work of Vardi and others in statistics that the pendulum should again swing back a bit towards encouraging more mathematical modelling to obtain maximal benefit from the use of statistical procedures by allowing physics, biology, and other fields of science to enter the statistical problem formulation via mathematical modelling of the specific statistical problem at hand. I would argue that the solution to a specific problem ought to somehow depend on the problem itself, which is not the case with neural-nets and other data-driven approaches that live mostly or entirely within the data or training set of the problem. Data-driven statistics has the danger of isolating statistics from the rest of the scientific and mathematical communities by not allowing valuable cross-pollination of ideas from other fields. To illustrate these ideas I will discuss among other concrete examples of statistics problems: emission tomography, machine learning, sampling bias. These topics were debated frequently by Vardi and me. I will do my best to give Vardi’s side as honestly as possible. Needless to say, I wish he were here to continue the debate. I will quote Tukey and/or Vardi on issues I will raise, and you should be aware that people who quote absentees don’t allow the quotees to modify the positions they are being quoted on unless they are in agreement with the ∗Research partially supported by National Science Foundation Grant DMS-0504387. Department of Statistics and Biostatistics, Rutgers University, Hill Center, Busch Campus, Piscataway NJ 08854, USA, e-mail: [email protected]