Modification of Bayesian Updating where Continuous Parameters have Differing Relationships with New and Existing Data

Abstract: Bayesian analyses are often performed using so-called noninformative priors, with a view to achieving objective inference about unknown parameters on which available data depends. Noninformative priors depend on the relationship of the data to the parameters over the sample space. Combining Bayesian updating - multiplying an existing posterior density for parameters being estimated by a likelihood function derived from independent new data that depend on those parameters and renormalizing - with use of noninformative priors gives rise to inconsistency where existing and new data depend on continuous parameters in different ways. In such cases, noninformative priors for inference from only the existing and from only the new data would differ, so Bayesian updating would give different final posterior densities depending on which set of data was used to derive an initial posterior and which was used to update that posterior. I propose a revised Bayesian updating method, which resolves this inconsistency by updating the prior as well as the likelihood function, and involves only a single application of Bayes' theorem. The revised method is also applicable where actual prior information as to parameter values exists and inference that objectively reflects the existing information as well as new data is sought. I demonstrate by numerical testing the probability-matching superiority of the proposed revised updating method, in two cases.