A Subjective Bayesian Approach to Address Unchecked Assumptions in a Causal Analysis

Abstract

Causal analyses using observational data rely on identifying assumptions that cannot be verified from the observed data alone. When such unchecked assumptions fail, effect estimates may be biased and uncertainty may be understated. We propose the Adjusted Model for Unchecked Assumptions (AMUA), a subjective Bayesian framework that propagates uncertainty about the joint validity of key causal assumptions through a two-component posterior mixture. One component corresponds to the standard observational causal model in which the assumptions hold (referred to as the "assumption-hold component"); the other represents a flexibly defined model that allows the causal assumptions to fail (referred to as the "assumption-fail component"). The mixture weight $π$ is specified by the analyst as the prior probability assigned to the assumption-hold component. This formulation leads to a new diagnostic, $π_{\min}$: the smallest value of $π$ for which the 95\% HPD set for the average treatment effect (ATE) excludes zero. In simulations with a spatially structured unmeasured confounder, we evaluate bias, interval length, and coverage as $π$ varies. The results show that acknowledging assumption uncertainty can widen intervals, and improve coverage relative to analyses that treat the assumptions as certain. In a county-level application relating long-term PM$_{2.5}$ exposure to COVID-19 mortality, AMUA shows how posterior conclusions depend on prior confidence in the identifying assumptions.