[论文解读] Practical Guidance for Bayesian Inference in Astronomy

Gwendolyn M. Eadie, Joshua S. Speagle+6 more|arXiv (Cornell University)|Feb 9, 2023

Spectroscopy and Chemometric Analyses被引用 8

一句话总结

提供跨学科的贝叶斯记号与在天文学中执行贝叶斯推断的实用指南，以视差为运行示例，强调先验、似然、后验以及后验预测检验。

ABSTRACT

In the last two decades, Bayesian inference has become commonplace in astronomy. At the same time, the choice of algorithms, terminology, notation, and interpretation of Bayesian inference varies from one sub-field of astronomy to the next, which can lead to confusion to both those learning and those familiar with Bayesian statistics. Moreover, the choice varies between the astronomy and statistics literature, too. In this paper, our goal is two-fold: (1) provide a reference that consolidates and clarifies terminology and notation across disciplines, and (2) outline practical guidance for Bayesian inference in astronomy. Highlighting both the astronomy and statistics literature, we cover topics such as notation, specification of the likelihood and prior distributions, inference using the posterior distribution, and posterior predictive checking. It is not our intention to introduce the entire field of Bayesian data analysis -- rather, we present a series of useful practices for astronomers who already have an understanding of the Bayesian "nuts and bolts" and wish to increase their expertise and extend their knowledge. Moreover, as the field of astrostatistics and astroinformatics continues to grow, we hope this paper will serve as both a helpful reference and as a jumping off point for deeper dives into the statistics and astrostatistics literature.

研究动机与目标

• 澄清天文学与统计学之间的术语与记号差异，以促进可重复性与理解。
• 就天文问题中指定先验、似然和后验分布提供实用指导。
• 以视差为运行示例，展示贝叶斯工作流程与距离估算。
• 突出后验预测检验作为模型充分性与计算检查的诊断工具。

提出的方法

• 给出贝叶斯定理及数据 y 与参数 theta 的标准记号，强调 p(theta|y) 与 p(y|theta)。
• 将似然定义为 p(y|d) 或 p(y|varpi)（视差），并讨论相关的抽样分布与模型选择（如高斯 vs 离散）。
• 讨论先验分布（信息型 vs 非信息型），包括截断均匀分布与用于距离/视差的物理动机先验的示例。
• 推导单恒星与团距离问题的后验形式，并在没有闭式解时描述如何计算或近似后验。
• 将后验预测检验作为诊断工具来比较真实数据与从后验预测分布中模拟的数据。

Figure 1: Bayesian inference of the distance $d=1/\varpi$ to a star based on the measured parallax $y$ . Far left: The likelihood for parallax $\varpi$ is normal with variance assumed known. Center left: A transformation of parameters from $\varpi$ to $d$ gives a non-normal PDF. Note that a non-nega

实验结果

研究问题

• RQ1 贝叶斯记号与概念应如何在天文学与统计学之间翻译，以提高清晰度与可重复性？
• RQ2 在天文贝叶斯模型中指定先验与似然的最佳实践是什么？这些选择如何影响后验推断？
• RQ3 如何使用和验证从视差数据得到的距离估计的后验分布（包括星团的情况）？
• RQ4 后验预测检验在评估模型充分性与计算方法方面扮演何种角色？

主要发现

• 清晰的贝叶斯记号翻译有助于跨学科的研究者解读结果。
• 在数据有限或参数化改变先验含义时，先验对后验有显著影响（例如距离与视差的关系）。
• 物理动机的先验（如 d^2 e^{-d/L}）比简单的均匀先验更能反映空间密度。
• 后验分布可能多峰或偏斜，需要可视化与超越简单均值或1-sigma区间的总结。
• 后验预测检验有助于模型评估并诊断抽样或模型不适合的问题。
• 运行中的视差示例展示了如何组合多个测量并将不确定性传播到推导量。

Figure 2: The three posterior distributions corresponding to each prior distribution shown in Figure 1 : uniform in parallax prior (blue curve), uniform in distance prior (red curve), and physically-motivated prior (orange curve). Also shown are one summary statistic for each posterior: the mode (bl