Analysis of the Bayesian Cramér-Rao lower bound in astrometry

Abstract

Context. The best precision that can be achieved to estimate the location of a stellar-like object is a topic of permanent interest in the astrometric community. Aims. We analyze bounds for the best position estimation of a stellar-like object on a CCD detector array in a Bayesian setting where the position is unknown, but where we have access to a prior distribution. In contrast to a parametric setting where we estimate a parameter from observations, the Bayesian approach estimates a random object (i.e., the position is a random variable) from observations that are statistically dependent on the position. Methods. We characterize the Bayesian Cramér-Rao (CR) that bounds the minimum mean square error (MMSE) of the best estimator of the position of a point source on a linear CCD-like detector, as a function of the properties of detector, the source, and the background.Results. We quantify and analyze the increase in astrometric performance from the use of a prior distribution of the object position, which is not available in the classical parametric setting. This gain is shown to be significant for various observational regimes, in particular in the case of faint objects or when the observations are taken under poor conditions. Furthermore, we present numerical evidence that the MMSE estimator of this problem tightly achieves the Bayesian CR bound. This is a remarkable result, demonstrating that all the performance gains presented in our analysis can be achieved with the MMSE estimator. Conclusions. The Bayesian CR bound can be used as a benchmark indicator of the expected maximum positional precision of a set of astrometric measurements in which prior information can be incorporated. This bound can be achieved through the conditional mean estimator, in contrast to the parametric case where no unbiased estimator precisely reaches the CR bound.

Optimality of the maximum likelihood estimator in astrometry

“…The use of the CR bound on other applications is also of interest, e.g., in assessing the performance of star trackers to guide satellites with demanding pointing constraints (Zhang et al 2016), or to meaningfully compare positional differences from different catalogues (for an example involving the SDSS and Gaia see Lemon et al (2017)). Finally, a formulation of the (non-parametric) Bayesian CR bound in astrometry, using the so-called " Van-Trees inequality" (Van Trees 2004), has been presented by our group in Echeverria et al (2016): This approach is particularly well suited for objects at the edge of detectability, and where some prior information is available, and has been proposed for the analysis of Gaia data for faint sources, or for those with a poor observational history (Michalik et al 2015;Michalik & Lindegren 2016).…”