This work tackles the problem of uncertainty propagation in two-stage Bayesian models, with a focus on spatial applications. A two-stage modeling framework has the advantage of being more computationally efficient than a fully Bayesian approach when the first-stage model is already complex in itself, and avoids the potential problem of unwanted feedback effects. Two ways of doing two-stage modeling are the crude plug-in method and the posterior sampling method. The former ignores the uncertainty in the first-stage model, while the latter can be computationally expensive. This paper validates the two aforementioned approaches and proposes a new approach to do uncertainty propagation, which we call the $\mathbf{Q}$ uncertainty method, implemented using the Integrated Nested Laplace Approximation (INLA). We validate the different approaches using the simulation-based calibration method, which tests the self-consistency property of Bayesian models. Results show that the crude plug-in method underestimates the true posterior uncertainty in the second-stage model parameters, while the resampling approach and the proposed method are correct. We illustrate the approaches in a real life data application which aims to link relative humidity and Dengue cases in the Philippines for August 2018.

翻译：本研究针对两阶段贝叶斯模型中的不确定性传播问题展开探讨，重点关注空间应用场景。当第一阶段模型本身已较为复杂时，两阶段建模框架相比完全贝叶斯方法具有更高的计算效率优势，同时避免了可能存在的非期望反馈效应问题。两阶段建模主要采用两种方法：原始插件法和后验采样法。前者忽略第一阶段模型的不确定性，后者则可能带来较高的计算成本。本文验证了上述两种方法，并提出了一种新的不确定性传播方法——$\mathbf{Q}$不确定性方法，该方法通过集成嵌套拉普拉斯近似（INLA）实现。我们采用基于模拟的校准方法对各方法进行验证，该方法通过检验贝叶斯模型的自洽性进行评估。结果表明：原始插件法会低估第二阶段模型参数的真实后验不确定性，而重采样方法与本文提出的方法均能保持正确性。最后，我们通过实际数据案例演示了这些方法的应用，该案例旨在分析2018年8月菲律宾相对湿度与登革热病例之间的关联。