  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.
