# Using prior expansions for prior-data conflict checking

## TL;DR

This paper introduces a method for detecting prior-data conflicts in Bayesian analysis by expanding the prior into a larger family and using a marginal likelihood score, with applications in regression and quantum state estimation.

## Contribution

The paper proposes a novel approach to prior-data conflict checking using prior expansions and score statistics, extending to hierarchical priors and complex models.

## Key findings

- Effective detection of prior-data conflicts in linear regression with LASSO penalties
- Application of prior expansion method in quantum state estimation with physical constraints
- Comparison with existing methods shows improved sensitivity to conflicts

## 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 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 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.