The role of exchangeability in causal inference

The notion of exchangeability has been recognized in the causal inference literature in various guises, but only rarely in the original Bayesian meaning as a symmetry property between individual units in statistical inference. Since the latter is a standard ingredient in Bayesian inference, we argue that in Bayesian causal inference it is natural to link the causal model, including the notion of confounding and definition of causal contrasts of interest, to the concept of exchangeability. Here we relate the Bayesian notion of exchangeability to alternative conditions for unconfounded inferences, commonly stated using potential outcome variables, and define causal contrasts in the presence of exchangeability in terms of limits of posterior predictive expectations for further exchangeable units. We demonstrate that this reasoning also carries over to longitudinal settings where parametric inferences are susceptible to the so-called null paradox. We interpret the paradox in terms of an exchangeability assumption made on too coarse a scale.