Bayes factor sensitivity analysis examines how the evidence for one hypothesis over another depends on the prior distribution. In complex models, the standard approach refits the model at each hyper-parameter value, and the total computational cost scales linearly in the grid size. We propose a method that recovers the entire sensitivity curve from a single additional model fit. The key identity decomposes the Bayes factor at any hyper-parameter value $γ_x$ into an ``anchor'' Bayes factor at a fixed reference $γ_0$ and a Savage--Dickey density ratio in an extended model that places a hyper-prior on $γ$. Once this extended model is fit, the Bayes factor at any $γ_x$ follows from the anchor value and a ratio of two posterior density ordinates. To approximate this ratio, we employ the importance-weighted marginal density estimator (IWMDE). Because the sensitivity parameter enters the model only through the prior distribution on the model parameters, the data likelihood cancels in the IWMDE, reducing it to a simple ratio of prior density evaluations on the MCMC draws, without any additional likelihood computation. The resulting estimator is fast, remains accurate even with small MCMC samples, and substantially outperforms kernel density estimation across the full sensitivity range. The method extends naturally to simultaneous sensitivity over multiple hyper-parameters and to Bayesian model averaging. We illustrate it on a univariate Bayesian $t$-test with exact Bayes factors for validation, a bivariate informed $t$-test, and a Bayesian model-averaged meta-analysis, obtaining accurate sensitivity curves at a fraction of the brute-force cost.

翻译：贝叶斯因子敏感性分析考察证据支持某一假设相对于另一假设的程度如何依赖于先验分布。在复杂模型中，标准方法需在每个超参数值处重新拟合模型，总计算成本随网格规模线性增长。我们提出一种方法，仅需一次额外模型拟合即可恢复完整敏感性曲线。核心恒等式将任意超参数值$γ_x$处的贝叶斯因子分解为固定参考点$γ_0$处的“锚定”贝叶斯因子与扩展模型（对$γ$施加超先验）中Savage-Dickey密度比。一旦拟合该扩展模型，任意$γ_x$处的贝叶斯因子均可由锚定值与两个后验密度纵坐标的比值得出。为近似该比值，我们采用重要性加权边际密度估计量（IWMDE）。由于敏感性参数仅通过模型参数的先验分布进入模型，数据似然在IWMDE中相互抵消，简化为对MCMC采样点上先验密度评估的简单比值，无需额外似然计算。所得估计量计算快速，即使在小MCMC样本下仍保持精度，并在整个敏感性范围内显著优于核密度估计。该方法可自然扩展至多个超参数的同步敏感性分析及贝叶斯模型平均。我们通过单变量贝叶斯$t$检验（使用精确贝叶斯因子进行验证）、双变量信息性$t$检验及贝叶斯模型平均元分析对其加以验证，以暴力求解法的一小部分成本获得精确的敏感性曲线。