To see that this is the case, however, we will need to introduce counterfactual notation, and then show how it can be eliminated from our expression. We will use bold letters to denote sets, and normal letters to denote individual elements. Also, capital letters will denote random variables, and small letters will denote possible values these variables could attain. We will write *Y*_{x} to mean 'the value *Y* attains if we set variables **X** to values **x**.' Similarly, *Y*_{Xg} is taken to mean 'the value *Y* attains if we set variables **X** to whatever values they would have attained under the stochastic policy **g**.' Note that *Y*_{x} and *Y*_{Xg} are both random variables, just as the original variable *Y*.

Say we have a set of *K* action variables **X** that occur in some temporal order. We will indicate the time at which a given variable is acted on by a superscript, so a variable *X ^{ i}* occurs before

We are interested in the probability distribution of a set of *outcome variables ***Y**, under a policy that sets the values of each *X ^{ i}*

Let . The key observation here is that if we observe

Here we note that

Now we note that the subscripts in the first and second terms are redundant, since the

or, more succinctly,

We see that we can compute this expression from

To summarize, though conditional plans are represented by complex nested counterfactual expressions, their identification can nevertheless be reduced to identification of conditional interventional distributions of the form *P*(**y** | *do*(**x**), **z**) (possibly with **z** being empty). Moreover, a complete condition for identifying such distributions from evidence exists.

Best wishes,

========Judea Pearl

**References**

[1] Shpitser, I., and Pearl, J. Identification of conditional interventional distributions. In *Uncertainty in Artificial Intelligence* (2006), vol. 22.

[2] Shpitser, I., and Pearl, J. Identification of joint interventional distributions in recursive semi-markovian causal models. In *Twenty-First National Conference on Artificial Intelligence* (2006).