  In this paper we provide a theoretical analysis of counterfactual invariance.
We present a variety of existing definitions, study how they relate to each
other and what their graphical implications are. We then turn to the current
major question surrounding counterfactual invariance, how does it relate to
conditional independence? We show that whilst counterfactual invariance implies
conditional independence, conditional independence does not give any
implications about the degree or likelihood of satisfying counterfactual
invariance. Furthermore, we show that for discrete causal models
counterfactually invariant functions are often constrained to be functions of
particular variables, or even constant.
