Several methods have been proposed for performing variable selection for the goal of estimating the average treatment effect of a point treatment on an outcome. Much less work has focused on developing methods for performing variable selection when the goal is instead to estimate the causal effect of a longitudinal treatment on an outcome. We introduce a Bayesian model averaging approach to estimate counterfactual means under longitudinal treatment regimes as well as the parameters of longitudinal marginal structural models. The Bayesian model averaging framework provides a principled way to produce valid post-selection inference. We further devise an informative prior distribution that favors the selection of optimal adjustment sets, in the sense that the resulting estimator has the lowest possible variance among all possible adjustment sets. We show through simulation results that our proposed method outperforms fully adjusted estimators and even performs similarly to an oracle estimator in several scenarios. In a real-data illustration, our method yields estimates with markedly reduced standard errors for the effect of cumulative exposure to statins for the prevention of a first cardiovascular disease or death among older adults.

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