关联函数的约束概率分布
Constrained probability distributions of correlation functions
摘要：
背景：两点关联函数在整个宇宙学中被用作随机场统计的度量。当用于贝叶斯参数估计时，它们的似然函数通常由高斯近似代替。然而，这已被证明是不够的。目的：对于高斯随机场的情况，我们寻找一个精确的概率分布的关联函数，这可以提高未来的数据分析的准确性。方法：我们使用一个完全解析的方法，首先展开的随机场在其傅立叶模式，然后计算的特征函数。最后，我们推导出的概率分布函数，使用积分留数。我们使用一个完整的解析公式的数值实现来讨论这个函数的行为。结果如下：我们推导了高斯随机场的关联函数的单变量和双变量概率分布函数，并概述了如何计算更高的联合分布。我们给出的结果的形式的模式展开，但在一个特殊的情况下，我们也发现了一个封闭的形式的表达。我们计算的分布的时刻，在单变量的情况下，我们讨论的Edgeworth展开近似。我们还评论的困难，在一个快速和准确的数值实现我们的结果，并在未来可能的应用。
英文摘要：
Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation. However, this has been shown to be insufficient. Aims: For the case of Gaussian random fields, we search for an exact probability distribution of correlation functions, which could improve the accuracy of future data analyses. Methods: We use a fully analytic approach, first expanding the random field in its Fourier modes, and then calculating the characteristic function. Finally, we derive the probability distribution function using integration by residues. We use a numerical implementation of the full analytic formula to discuss the behaviour of this function. Results: We derive the univariate and bivariate probability distribution function of the correlation functions of a Gaussian random field, and outline how higher joint distributions could be calculated. We give the results in the form of mode expansions, but in one special case we also find a closed-form expression. We calculate the moments of the distribution and, in the univariate case, we discuss the Edgeworth expansion approximation. We also comment on the difficulties in a fast and exact numerical implementation of our results, and on possible future applications.
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