Bayesian causal inference with some invalid instrumental variables

In observational studies, instrumental variables estimation is greatly utilized to identify causal effects. One of the key conditions for the instrumental variables estimator to be consistent is the exclusion restriction, which indicates that instruments affect the outcome of interest only via the exposure variable of interest. We propose a likelihood-free Bayesian approach to make consistent inferences about the causal effect when there are some invalid instruments in a way that they violate the exclusion restriction condition. Asymptotic properties of the proposed Bayes estimator, including consistency and normality, are established. A simulation study demonstrates that the proposed Bayesian method produces consistent point estimators and valid credible intervals with correct coverage rates for Gaussian and non-Gaussian data with some invalid instruments. We also demonstrate the proposed method through the real data application.