Bayesian inference and relative belief, theory and applications

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This research is concerned with developing methodology for the elicitation of priors, measuring the bias induced by a prior, model checking, checking for prior-data conflict and methods for statistical inference based upon a measure of evidence known as the relative belief ratio. ******While statistical methodology is becoming increasingly important in scientific, technological and social applications, there is little agreement among practitioners as to what constitutes the correct formulation of a statistical problem and furthermore, even when the ingredients are agreed upon, there may be disagreement concerning the solution. Controversies such as the recent concerns over the use of p-values provide some evidence for this. Given that the role of a statistician is to provide guidance on how one is to reason in contexts where data variation results in uncertainty, it is necessary for the discipline to do as much as possible to resolve the ambiguities arising from such conflicts.******One of the central points of disagreement focuses on the use of prior probability distributions as these represent clear subjective inputs into statistical analyses. These inputs would seem to run contrary to the scientific tradition of the objectivity of the investigator. This concern is undoubtedly valid, but at the same it ignores the obvious fact that scientific investigations always involve subjective aspects resulting from the choices made by the scientist. The role of this research is to develop tools for dealing with this subjectivity in a scientifically valid fashion. This will be accomplished in part through the development of elicitation arguments for a broad variety of contexts so that the choices are based on clearly articulated criteria. Furthermore, methodology will be developed that allows for the measurement and control over the biases introduced by these choices and allows for the assessment of whether or not the choices made are in conflict with the objective data.******Once the ingredients of a statistical analysis have been chosen, and at least partially validated as described, the next step is to apply the rules of a theory of statistical inference to these ingredients together with the data to derive inferences. A relevant theory of inference should be based on a clear definition of how to measure the statistical evidence relevant to questions of interest. For many approaches to statistical theory, the lack of a clear definition of such a measure of evidence produces yet another ambiguity. The theory of relative belief inference has been developed to provide an approach that is based on a valid measure of statistical evidence. This research proposal is also concerned with the further development of the associated theory and the application of this to a broad variety of statistical problems with practical relevance. ******This research will also require the development of suitable computational tools for approximations.*****