Validating uncertainty propagation approaches for two-stage Bayesian spatial models using simulation-based calibration

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.