  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.
