Theory and Methods of Statistical Inference

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Modern technology has simplified the collection of large and complex sets of data, which are being used to answer important research questions in many fields of science and engineering.  Statistical models and methods are an essential part of this research, and understanding these methods requires progress on the theory of statistical modelling and inference. The proposed research program is intended to deepen our understanding of the intellectual foundations of the field of statistics and to provide a framework for developing new methods of analysis. Research in statistical theory looks for commonalities underlying a wide range of scientific problems. The feedback cycle between theory and applications of statistical science is one of the most interesting and important aspects of the subject.***Particular emphasis will be placed on developing methods of inference based on the likelihood function, as it plays a central role in both Bayesian and frequentist inference.  There continues to be an ongoing debate about the role of these two modes of inference in scientific advances; a very accessible overview of the debate was featured in the New York Times (September 29, 2014). Careful study of the basic principles of statistical inference can help to inform this debate.  My research program also emphasizes the study of mathematical properties of inference methods using asymptotic expansions, a technique that studies how methods depend on the size of the data set being analysed.  With infinite amounts of data, Bayesian and frequentist methods agree, but it turns out that their disagreement in finite samples can be pinpointed with the help of asymptotic expansions.  ***In the current technological landscape, the amount of data available to scientists and engineers is nearly unlimited, but as the size of a set of data increases, so does the complexity of the mathematical models used to help us understand the structure in the data.  These models are used to summarize key features of a problem, to make inferences about scientific hypotheses under study, and to make predictions for what we might expect to see in similar circumstances.  When the models become very complex, and particularly involve complex dependencies among measurements, statistical inference faces challenges both computationally and theoretically.  Computationally, we may not be able to construct the likelihood function, and inferentially we may not be able to assess the properties of estimated quantities based on the likelihood function. As a result a number of simplifications of likelihood functions have been designed for particular applications. A major focus of the proposed research program is understanding the theoretical properties of these, thus illuminating how computational needs interact with scientific needs for accurate and efficient inference. **