天体测量中Bayesian Cramer-Rao下限的分析：研究先验信息对物体位置的影响
Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object
摘要：
上下文如何以最佳的精确度来估计类星物体的位置是天体测量界长期关注的一个话题。目标。我们分析的最佳位置估计的一个恒星状的物体上的CCD探测器阵列在贝叶斯设置的位置是未知的，但我们有机会获得先验分布的界限。与我们根据观察估计参数的参数设置相反，贝叶斯方法估计随机对象（即，该位置是随机变量）从统计上依赖于该位置的观测中。方法.我们描述了贝叶斯克拉默-拉奥（CR）的特征，该CR限制了线性CCD类检测器上点源位置的最佳估计器的最小均方误差（MMSE），作为检测器、源和属性的函数。背景。结果我们量化和分析的天体测量性能的增加，从使用的对象位置的先验分布，这是不可用的经典参数设置。这种增益被证明是显着的各种观测制度，特别是在微弱的对象的情况下，或在恶劣的条件下进行的意见。此外，我们提出的数值证据表明，这个问题的MMSE估计紧紧实现贝叶斯CR界。这是一个显著的结果，表明我们的分析中提出的所有性能增益都可以用MMSE估计器实现。结论贝叶斯CR界可以作为一个基准指标的预期最大的位置精度的一组天体测量，先验信息可以纳入。这个界限可以通过条件均值估计来实现，与参数情况相反，在参数情况下没有无偏估计精确地达到CR界限。
英文摘要：
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 analyse 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 Cramer-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 analyse 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.
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