Location Reparameterization and Default Priors for Statistical Analysis

This paper develops default priors for Bayesian analysis that reproduce familiar frequentist and Bayesian analyses for models that are exponential or location. For the vector parameter case there is an information adjustment that avoids the Bayesian marginalization paradoxes and properly targets the prior on the parameter of interest thus adjusting for any complicating nonlinearity the details of this vector Bayesian issue will be investigated in detail elsewhere. As in wide generality a statistical model has an inference component structure that is approximately exponential or approximately location to third order, this provides general default prior procedures that can be described as reweighting likelihood in accord with a Jeffreys’ prior based on observed information. Two asymptotic models, that have variable and parameter of the same dimension and agree at a data point to first derivative conditional on an approximate ancillary, produce the same p-values to third order for inferences concerning scalar interest parameters. With some given model of interest there is then the opportunity to choose some second model to best assist the calculations or best achieve certain inference objectives. Exponential models are useful for obtaining accurate approximations while location models present possible parameter values in a direct measurement or location manner. We derive the general construction of the location reparameterization that gives the natural parameter of the location model coinciding with the given model to first derivative at a data point the derivation is in algorithmic form that is suitable for computer algebra. We then define a general default prior based on this location reparameterization this gives third order agreement between frequentist p-values and Bayesian survivor values in the vector case however, an adjustment factor is needed for component parameters that are not linear in the location parameterization. The general default prior can be difficult to calculate. But if we choose to work only to the secondorder, a Jeffreys’ prior based on the observed information function gives second order agreement between the frequentist p-values and Bayesian survivor values again adjustments are needed for parameters nonlinear in the vector location parameter the adjustment is a ratio of two nuisance information determinants, one for the nuisance parameter as given and one for the locally equivalent linear nuisance parameter.