  This paper provides a critical review of the Bayesian perspective of causal
inference based on the potential outcomes framework. We review the causal
estimands, identification assumptions, the general structure of Bayesian
inference of causal effects, and sensitivity analysis. We highlight issues that
are unique to Bayesian causal inference, including the role of the propensity
score, definition of identifiability, the choice of priors in both low and high
dimensional regimes. We point out the central role of covariate overlap and
more generally the design stage in Bayesian causal inference. We extend the
discussion to two complex assignment mechanisms: instrumental variable and
time-varying treatments. Throughout, we illustrate the key concepts via
examples.
