关于天文学中二项总体比例置信区间的估计：贝叶斯方法的简单性和可靠性
On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach
摘要：
我提出了一个关键的审查技术，估计二项人口比例的置信区间推断成功计数在小到中等样本。人口比例经常出现在天文学研究中的感兴趣的数量;例如，在旨在限制棒分数，AGN分数，SMBH分数，合并分数或红色序列分数的研究中，表现出不同的形态特征或恒星星族。然而，两个最广泛使用的技术，用于估计二项式置信区间-“正常近似”和Clopper & Pearson的方法-容易歪曲的程度，统计的不确定性，目前经常遇到的抽样条件下，在天文调查，导致无效使用的实验数据（和，更糟糕的是，低效使用的资源，在获得数据）。因此，我在这里提供了一个二项式统计的基本原理的概述，有两个主要目的：（i）揭示的容易，具有更令人满意的行为的（贝叶斯）二项置信区间可以使用现代数学软件包从beta分布的分位数估计（例如R，matlab，approximatica，IDL，python）;（ii）令人信服地证明了“正常近似”和Clopper & Pearson方法的主要缺陷。
英文摘要：
I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small-to-intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, AGN fraction, SMBH fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals--the 'normal approximation' and the Clopper & Pearson approach--are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (i) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. R, matlab, mathematica, IDL, python); and (ii) to demonstrate convincingly the major flaws of both the 'normal approximation' and the Clopper & Pearson approach for error estimation.
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