Rolf Sundberg: "Shaved dice" inference — Two contrasting points of view of a simple situation.
Tid: On 2017-05-17 kl 15.15 - 16.15
Plats: Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University
Medverkande: Rolf Sundberg, Department of Mathematics, Stockholm University
Abstract:
Two dice are rolled repeatedly, but only the sum of each roll is registered. Have the two dice been "shaved", so two of the six sides appear more frequently?
In an entertaining and instructive article in The American Statistician 2010, Pavlides & Perlman consider the following (artificial) inference problem. A pair of dice used at a casino are possibly "shaved", such that two known, mutually opposite sides of the die have a higher probability than the other four sides. This would be a trivial example of inference from binomial data, but for one complication: the two dice are rolled together and the statistics collected "unfortunately" do not represent the individual dice outcomes, but only the sum of the two outcomes. Pavlides and Perlman demonstrate how this complication is represented by a curved multinomial exponential family, with a scalar parameter but a minimal sufficient statistic of higher dimension. One consequence, typical for curved families, is that the likelihood equation does not have an explicit solution. Pavlides and Perlman calculate and illustrate the Fisher information and other statistical characteristics of the model.
The purpose of my talk is to contrast their approach by the point of view to see data as incomplete data from a simple exponential family. This supplementary approach is in some respects simpler, it provides additional insight about the likelihood equation and the Fisher information, it opens up for the EM algorithm, and it illustrates the information content in ancillary statistics. The discussion will appear in The American Statistician this year.
The two contrasting approaches are (of course) possible in wider generality, for example in genetics, observing phenotypes of underlying genotypes.
For the audience, a basic exposure to exponential families is a desirable prerequisite. Our course Statistical Models is excellent background, so students of that course are warmly welcomed.