Bayes assisted frequentist model assessment and statistical inference

点击翻译按钮获取中文摘要

Statistical analysis of data generally requires the analyst to make assumptions about the way the data were generated. My research is largely directed towards developing statistical techniques to check these assumptions. The assumptions data analysts make can generally be wrong in many ways. As a result many of the methods that have been suggested for assessing the assumptions have been ad hoc; they have focused on one particular direction in which the assumptions might be wrong and suggested some method for detecting departures from assumptions in that particular direction. The methods themselves are suggested largely because they sound reasonable or because they seem to lend themselves to mathematical analysis. In the next five years I, with my students and colleagues, will be trying to systematize these ad hoc approaches. I will use so-called Bayesian assisted approaches to describe the possible directions of departure from the assumptions and then try to find methods that are approximately the best possible for the particular Bayesian description chosen (in technical words, best possible for a given “prior distribution” on the nature of the failure of the model assumptions). Such a principled approach leaves much work to be done to evaluate the resulting analysis techniques. In complex statistical problems there is a great deal of flexibility in the choice of the Bayesian prior description and it can be hard to see whether or not a particular choice has buried within it unreasonable real world assumptions; evaluation of this danger for the particular suggestions we make will be one focus of my work. Another focus will arise from the fact that it will usually not be possible to find a computationally feasible “best” procedure; usually approximately best is as good as we can do. We then need to evaluate the quality of the approximation. It is also sensible to remember that some of the ad hoc procedures work very well. A further focus of my work will be to evaluate how far short of best possible such procedures lie. Often an easily computable, nearly best, method will be more happily received than a complex, hard to compute, best possible solution. In addition to this basic work in the area of model assessment, I will be pursuing a variety of problems in other areas. The most important of these considers searches for new physics such as the search for the Higgs boson at CERN. This particular search involved scanning a range of possible masses for the sought-for particle and at each mass looking, statistically, for a signal in the relevant part of a data set. The current strategy for searches of this kind then either declares the particle found, if the statistical evidence is sufficiently overwhelming or, if it is not found, runs through the range of possible masses excluding as many masses as the data permit from being possible candidates of the true mass if the particle really does exist. With students I have been, and will be, working on using the Bayesian ideas mentioned above to combine these two steps into a one-step procedure that achieves the best possible long run behaviour.