Abstract
The features of a logically sound approach to a theory of statistical reasoning are discussed. A particular approach that satisfies these criteria is reviewed. This is seen to involve selection of a model, model checking, elicitation of a prior, checking the prior for bias, checking for prior-data conflict and estimation and hypothesis assessment inferences based on a measure of evidence. A long-standing anomalous example is resolved by this approach to inference and an application is made to a practical problem of considerable importance, which, among other novel aspects of the analysis, involves the development of a relevant elicitation algorithm.
Highlights
IntroductionIt is relevant to ask what characteristics should be required of a theory of statistical reasoning
It is relevant to ask what characteristics should be required of a theory of statistical reasoning.The phrase statistical reasoning is used here, as opposed to statistical inference, because there is a logical separation between how the ingredients to a statistical problem are chosen and checked for their validity, and the inference step that involves the application of the rules of a theory of inference to the ingredients
The desiderata for a theory of statistical reasoning include the following: a methodology for choosing a model, an elicitation algorithm for selecting a prior, methodology for assessing the bias in the ingredients chosen, model checking and checking for prior-data conflict procedures and a theory of inference based upon a measure of statistical evidence
Summary
It is relevant to ask what characteristics should be required of a theory of statistical reasoning. Noteworthy is the Jeffreys–Lindley paradox where an increasingly diffuse prior typically leads to overwhelming evidence in favor of a hypothesis even when it seems contradicted by the data The discussion of this paradox in [1] shows that the relative belief approach to inference leads to a clear resolution. As far as we know, there have been no attempts to use this quantity as the central concept in derivation of a theory of statistical inference The work of [14] is noteworthy in this regard
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