Theory 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 this has a central role in Bayesian and frequentist approaches to inference.  There continues to be an ongoing debate about the use of these different modes of inference in scientific advances. Careful study of the basic principles of statistical inference can help to inform this debate.  This 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 shed light on 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 in particular involve very large numbers of parameters, relative to the number of observations we can collect, new theory is needed to inform for statistical summaries, inferences and predictions. A major focus of this research program is contributing to these developments.  Data that is very large, and/or complex, is often used to design algorithms that give good predictions; this is the focus of much work in machine learning and some branches of artificial intelligence. Statistical best practices around the collection, and protection, of data can be very useful in assessing the reliability of these methods for widespread use in the population. Statistical concepts relevant to inference can also be very useful to advance the explainability of these algorithms. This  research program will emphasize the importance of statistical thinking in learning from data.