### Identifying conditional plans

Section 4.2 of the book (p. 113) gives an identification condition and estimation formula for the effect of a conditional action, namely, the effect of an action *do*(*X=g*(*z*)) where *Z* is a measurement taken prior to the action. Is this equation generalizable to the case of several actions, i.e., conditional plan?

The difficulty seen is that this formula was derived on the assumption that *X* does not change the value of *Z*. However, in a multi-action plan, some actions in *X* could change observations *Z* that are used to guide future actions. We do not have notation for distinguishing post-intevention from pre-intevention observations.

The need for notational distinction between post-intevention from pre-inter-vention observations is valid, and will be satisfied in Chapter 7 where we deal with counterfactuals. The case of conditional plans, however, can be handled without resorting to richer notation. The reason is that the observations which dictate the choice of an action are not changed by that action, while those that have changed by previous actions are well captured by the

P(y|do(x),z) notation.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 valueYattains if we set variablesXto valuesx.' Similarly,Y_{Xg}is taken to mean 'the valueYattains if we set variablesXto whatever values they would have attained under the stochastic policyg.' Note thatY_{x}andY_{Xg}are both random variables, just as the original variableY.Say we have a set of

Kaction variablesXthat occur in some temporal order. We will indicate the time at which a given variable is acted on by a superscript, so a variableXoccurs before^{ i}Xif^{ j}i<j. For a givenX, we denote^{ i}X^{< i}to be the set of action variables precedingX.^{ i}We are interested in the probability distribution of a set of

outcome variablesY, under a policy that sets the values of eachX^{ i}^{ }Xto the output of functionsg(known in advance) which pay attention to some set of prior variables_{i}Z^{i}, as well as the previous interventions onX^{< i}. At the same time, the variablesZ^{i}are themselves affected by previous interventions. To define this recursion appropriately, we use an inductive definition. The base case isX^{1}_{g}=g_{1}(Z^{1}). The inductive case is . Here the subscriptgrepresents the policy we use, in other words,g= {g|_{i}i= 1, 2, …,K}. We can now write the quantity of interest:Let . The key observation here is that if we observe

Zto take on particular values,_{g}Xcollapse to unique values as well because_{g}Xis a function of_{g}Z. We let_{g}x= {_{z}x^{1}_{z},…,x^{K}_{z}} be the values attained byXin the situation where_{g}Zhas been observed to equal_{g}z= {z^{1},…,z^{K}}. We note here that if we knowz, we can computex_{z}in advance, because the functionsgare fixed in advance and known to us. However, we don't know what values_{i}Zmight obtain, so we use case analysis to consider all possible value combinations. We then obtain:_{g}Here we note that

Z^{i}cannot depend on subsequent interventions. So we obtainNow we note that the subscripts in the first and second terms are redundant, since the

do(x) already implies such subscripts for all variables in the expression. Thus we can rewrite the target quantity as_{z}or, more succinctly,

We see that we can compute this expression from

P(y|do(x)),z) andP(z|do(x)), whereY,X,Zare disjoint sets. Complete conditions for identifying these quantities from a joint distribution in a given graphGare given in [2], [1].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 withzbeing 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).Comment by Judea — May 8, 2006 @ 12:00 am