On the assumption of Gaussianity for cosmological two-point statistics and parameter dependent covariance matrices
On the assumption of Gaussianity for cosmological two-point statistics and parameter dependent covariance matrices
复制标题
宇宙学两点统计和参数相关协方差矩阵的高斯性假设
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Carron
中科院分区：
文献类型：
--
作者：
J. Carron
在这篇简短的文章中，我们回顾了宇宙学功率谱或高斯场的两点函数的Fisher信息量，以评论高斯估计器的假设，以及在精确宇宙学的背景下使用参数相关协方差矩阵进行参数推断。即使高斯似然的假设是由中心极限定理驱动的，我们也讨论了，如果一致地使用，它会导致违反Cram‘er-Rao界的Fisher信息量，这是由于来自参数相关协方差矩阵的独立但人为的信息。在任何固定的多极子情况下，这一人工项在大量关联场的情况下将成为主导项。虽然估计器的分布确实倾向于具有大量模式的高斯分布，但它的Fisher信息量并不是这样的，因为它们的协方差矩阵从来没有携带独立的信息量，这正是因为分布的非高斯形状。在这一点上，我们讨论了使用参数相关的协方差矩阵和高斯概率来从两点统计中推断参数。作为一个经验法则，高斯似然应该总是与参数空间中固定的协方差矩阵一起使用，因为只有这样才能保证保守信息内容被分配给可观测对象，同时防止出现偏差。
In this brief paper we revisit the Fisher information content of cosmological power spectra or two-point functions of Gaussian fields in order to comment on the assumption of Gaussian estimators and the use of parameter-dependent covariance matrices for parameter inference in the context of precision cosmology. Even though the assumption of a Gaussian likelihood is motivated by the central limit theorem, we discuss that it leads to Fisher information content that violates the Cram\'er-Rao bound if used consistently, owing to independent but artificial information from the parameter-dependent covariance matrix. At any fixed multipole, this artificial term is shown to become dominant in the case of a large number of correlated fields. While the distribution of the estimators does indeed tend to a Gaussian with a large number of modes, it is shown, however, that its Fisher information content does not, in the sense that their covariance matrix never carries independent information content, precisely because of the non-Gaussian shape of the distribution. In this light, we discuss the use of parameter-dependent covariance matrices with Gaussian likelihoods for parameter inference from two-point statistics. As a rule of thumb, Gaussian likelihoods should always be used with a covariance matrix fixed in parameter space, since only this guarantees that conservative information content is assigned to the observables, and at the same time prevents biases appearing.