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Context. The best precision that can be achieved to estimate the location ofa stellar-like object is a topic of permanent interest in the astrometriccommunity.  Aims. We analyse bounds for the best position estimation of a stellar-likeobject on a CCD detector array in a Bayesian setting where the position isunknown, but where we have access to a prior distribution. In contrast to aparametric setting where we estimate a parameter from observations, theBayesian approach estimates a random object (i.e., the position is a randomvariable) from observations that are statistically dependent on the position.  Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimummean square error (MMSE) of the best estimator of the position of a pointsource on a linear CCD-like detector, as a function of the properties ofdetector, the source, and the background.  Results. We quantify and analyse the increase in astrometric performance fromthe use of a prior distribution of the object position, which is not availablein the classical parametric setting. This gain is shown to be significant forvarious observational regimes, in particular in the case of faint objects orwhen the observations are taken under poor conditions. Furthermore, we presentnumerical evidence that the MMSE estimator of this problem tightly achieves theBayesian CR bound. This is a remarkable result, demonstrating that all theperformance gains presented in our analysis can be achieved with the MMSEestimator.  Conclusions The Bayesian CR bound can be used as a benchmark indicator of theexpected maximum positional precision of a set of astrometric measurements inwhich prior information can be incorporated. This bound can be achieved throughthe conditional mean estimator, in contrast to the parametric case where nounbiased estimator precisely reaches the CR bound.

Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object