Bayesian applications in auditory research (McMillan & Cannon, 2019)

<b>Purpose: </b>This article presents a basic exploration of Bayesian inference to inform researchers unfamiliar to this type of analysis of the many advantages this readily available approach provides.<b>Method: </b>First, we demonstrate the development of Bayes’ theorem, the cornerstone of Bayesian statistics, into an iterative process of updating priors. Working with a few assumptions, including normalcy and conjugacy of prior distribution, we express how one would calculate the posterior distribution using the prior distribution and the likelihood of the parameter. Next, we move to an example in auditory research by considering the effect of sound therapy for reducing the perceived loudness of tinnitus. In this case, as well as most real-world settings, we turn to Markov chain simulations because the assumptions allowing for easy calculations no longer hold. Using Markov chain Monte Carlo methods, we can illustrate several analysis solutions given by a straightforward Bayesian approach.<b>Conclusion: </b>Bayesian methods are widely applicable and can help scientists overcome analysis problems, including how to include existing information, run interim analysis, achieve consensus through measurement, and, most importantly, interpret results correctly.<br><b>Supplemental Material S1. </b>SAS code for running the analysis described in the article.<br><b>Supplemental Material S2.</b> MS Excel workbook, allowing the reader to experiment with the model described in the article. <br>McMillan, G. P., &amp; Cannon, J. B. (2019). Bayesian applications in auditory research. <i>Journal of Speech, Language, and Hearing Research, 62,</i> 577–586. https://doi.org/10.1044/2018_JSLHR-L-ASTM-18-0250<br><b>Publisher Note: </b>This article is part of the Research Forum: Advancing Statistical Methods in Speech, Language, and Hearing Sciences.<br>

**研究目的**：本文对贝叶斯推断（Bayesian inference）展开基础探究，旨在为不熟悉此类分析方法的研究者介绍这一易获取的分析路径所具备的诸多优势。
**研究方法**：首先，我们将贝叶斯统计（Bayesian statistics）的基石——贝叶斯定理（Bayes’ theorem）推演为先验更新的迭代流程。在假设先验分布满足正态性与共轭性的前提下，阐述如何基于先验分布与参数似然计算后验分布。随后，我们以听觉研究领域的案例为例，探讨声音疗法对降低耳鸣主观响度的作用效果。在此场景及多数现实情境中，由于满足简易计算的假设不再成立，我们转而采用马尔可夫链模拟方法。借助马尔可夫链蒙特卡洛（Markov chain Monte Carlo）方法，我们可展示通过简洁的贝叶斯分析路径所能得到的若干分析方案。
**研究结论**：贝叶斯方法应用范围广泛，可帮助研究者解决诸多分析难题，包括纳入已有信息、开展中期分析、通过测量达成共识，以及最为关键的正确解读研究结果。
**补充材料S1**：用于运行本文所述分析的SAS代码。
**补充材料S2**：可供读者实操本文所述模型的Microsoft Excel工作簿。
McMillan, G. P., & Cannon, J. B. (2019). 听觉研究中的贝叶斯应用。《言语、语言与听力研究杂志》, 62, 577–586. https://doi.org/10.1044/2018_JSLHR-L-ASTM-18-0250
**出版说明**：本文隶属于「推进言语、语言与听力科学的统计方法」研究论坛专栏。