A logistic regression model is a specialized model for product-binomial data. When a proper, noninformative prior is placed on the unrestricted model for the product-binomial model, the hypothesis H 0 of a logistic regression model holding can then be assessed by comparing the concentration of the posterior distribution about H 0 with the concentration of the prior about H 0 . This comparison is effected via a relative belief ratio, a measure of the evidence that H 0 is true, together with a measure of the strength of the evidence that H 0 is either true or false. This gives an effective goodness of fit test for logistic regression.
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.
Any Bayesian analysis involves combining information represented through different model components, and when different sources of information are in conflict it is important to detect this. Here we consider checking for prior-data conflict in Bayesian models by expanding the prior used for the analysis into a larger family of priors, and considering a marginal likelihood score statistic for the expansion parameter. Consideration of different expansions can be informative about the nature of any conflict, and extensions to hierarchically specified priors and connections with other approaches to prior-data conflict checking are discussed. Implementation in complex situations is illustrated with two applications. The first concerns testing for the appropriateness of a LASSO penalty in shrinkage estimation of coefficients in linear regression. Our method is compared with a recent suggestion in the literature designed to be powerful against alternatives in the exponential power family, and we use this family as the
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