This paper outlines a framework for quantifying the prior’s contribution to posterior inference in the presence of prior-likelihood discordance, a broader concept than the usual notion of prior-likelihood conflict. We achieve this dual purpose by extending the classic notion of prior sample size, M, in three directions: (I) estimating M beyond conjugate families; (II) formulating M as a relative notion that is as a function of the likelihood sample size k, M(k), which also leads naturally to a graphical diagnosis; and (III) permitting negative M, as a measure of prior-likelihood conflict, that is, harmful discordance. Our asymptotic regime permits the prior sample size to grow with the likelihood data size, hence making asymptotic arguments meaningful for investigating the impact of the prior relative to that of likelihood. It leads to a simple asymptotic formula for quantifying the impact of a proper prior that only involves computing a centrality and a spread measure of the prior and the posterior. We use simulated and real data to illustrate the potential of the proposed framework, including quantifying how weak is a ‘weakly informative’ prior adopted in a study of lupus nephritis. Whereas we take a pragmatic perspective in assessing the impact of a prior on a given inference problem under a specific evaluative metric, we also touch upon conceptual and theoretical issues such as using improper priors and permitting priors with asymptotically non-vanishing influence.

中文翻译：

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用于评估先前影响和先前可能性不一致的先前样本量扩展

本文概述了一个框架，用于在存在先验似然不一致的情况下量化先验对后验推理的贡献，这是一个比通常的先验似然冲突概念更广泛的概念。我们通过扩展的经典概念实现此双重目的之前样本大小，中号，在三个方向：（I）估计中号超出共轭的家庭; (II) 将M公式化为一个相对概念，它是似然样本大小k的函数，  M ( k )，这也自然导致图形诊断；(III) 允许负M，作为先验似然冲突的衡量标准，即有害的不协调。我们的渐近机制允许先验样本大小随着似然数据大小而增长，因此使渐近论证对于调查先验相对于似然的影响有意义。它导致了一个简单的渐近公式，用于量化适当先验的影响，该公式仅涉及计算先验和后验的中心性和扩展度量。我们使用模拟和真实数据来说明拟议框架的潜力，包括量化狼疮性肾炎研究中先前采用的“弱信息”的弱程度。而我们在评估先验对特定评估指标下给定推理问题的影响时采取务实的观点，