 that is, as the difference between the value that \(Y\) would take under an intervention that sets \(X=x\) in circumstances \(B_i\) and the value that \(Y\) would take under an intervention that sets \(X=x'\) in \(B_i\), where the notion of an intervention is now understood in terms of  (M1)–(M4) rather than in the way  recommended by Pearl. In non-deterministic contexts, the  characterization of causal effect is less straightforward, but one  natural proposal is to define this notion in terms of expectations:  If we let  \(E P_{\do x, B_{i}}(Y)\) be the expectation of \(Y\) with  respect to the probability distribution \(P\) if \(X\) is set to  \(X=x\) by means of an intervention, then the causal effect on \(Y\)  of a change in \(X\) from \(X=x''\) to \(X=x\) might be defined as:  \(E P_{\do x, B_{i}}(Y) - E P_{\do x', B_{i}}  (Y)\). 