Abstract
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 an appropriate choice of expansion can provide more sensitive checks for conflicts of certain types. Extensions to hierarchically specified priors and connections with other approaches to prior-data conflict checking are considered, and 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 prior expansion for constructing our check. A second application concerns a problem in quantum state estimation, where a multinomial model is considered with physical constraints on the model parameters. In this example, the usefulness of different prior expansions is demonstrated for obtaining checks which are sensitive to different aspects of the prior.
Highlights
A common approach to checking the likelihood in a statistical analysis is to consider model expansions motivated by thinking about plausible departures from the assumed model
The method we suggest here is different, because the choice of a particular prior expansion provides the flexibility to design checks which are sensitive to conflicts of certain kinds
One of the main advantages of the check we propose is that by appropriate choices of the prior expansion we can obtain checks of conflict which are sensitive to different aspects of the prior
Summary
A common approach to checking the likelihood in a statistical analysis is to consider model expansions motivated by thinking about plausible departures from the assumed model. The purpose of this work is to consider model expansion for checking for prior-data conflict, rather than for checking the likelihood. Our approach to prior-data conflict checking considers embedding the original prior into a larger family, which we write as g(θ|γ), where γ is some expansion parameter and the original prior is g(θ|γ0) for some value γ0. One of the main advantages of the check we propose is that by appropriate choices of the prior expansion we can obtain checks of conflict which are sensitive to different aspects of the prior. We discuss prior-data conflict checking and how this differs from checking the likelihood component in a model-based statistical analysis.
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