Efficient Bayesian inference for two-stage models

Statistical models often require inputs that are not completely known. This can occur when those inputs are measured with error, indirectly, or when they correspond to an unobservable parameter in another model. A prominent application is environmental epidemiology, where individual air pollution exposure is a key variable for health outcomes, yet it cannot be inferred directly and is estimated by a model. In these cases, the common choice is the two-stage Bayesian statistical modeling approach, where the two levels of the model are written down separately. In this approach, the stage-one model estimates the unknown parameter and those estimates are then incorporated as inputs in the stage-two model. However, to target the correct posterior distributions, two-stage Bayesian models must correctly propagate the uncertainty from the first to the second stages. In practice, researchers often fail to do so and use simplified and incorrect methods. We show both analytically and empirically the negative consequences of failing to correctly account for uncertainty even in a simple setting. Plug-in methods that estimate and fix the inputs are subject to attenuation bias and underestimate uncertainties. Partial posterior methods that propagate uncertainty from the stage-one model without adjusting for the stage-two model fail to correct this bias and overinflate uncertainties. We propose two algorithms for two-stage modeling that propagate the uncertainty across the two stages. The first is a streamlined importance sampling algorithm that performs best when the inputs from the stage-one posterior are approximately independent, while the second provides a correction when this does not occur. We then use analytical and empirical results in a variety of settings to show that, unlike the common competing methods, our algorithms can correctly propagate uncertainties and target the correct distributions when the assumptions are met.

Bayesian inference

Measurement Error

Two-stage modeling

Importance sampling 

Computational Statistics

Symposium on Data Science and Statistics (SDSS) 2025