On the Interpretation of do(x)

1 Introduction

The structural causal modeling (SCM) framework described in [1], [2], [3] defines and computes quantities of the form Q=E[Y|do(X=x)] which are interpreted as the causal effect of X on Y. The computation of Q simulates a minimally invasive intervention that sets the value of X to x, and leaves all other relationships unaltered. Several critics of SCM have voiced concerns about this interpretation of Q when X is non-manipulable; that is, X is a variable whose value cannot be controlled directly by an experimenter [4], [5], [6], [7], [8]. Indeed, asking for the effect of setting X to a constant x makes perfect sense when X is a treatment, say “drug-1” or “diet-2,” but how can we imagine an action do(X=x) when X is non-manipulable, like gender, race, or even a state of a variable such as blood-pressure or cholesterol level?^[1]

Mathematically, the expression Q=E[Y|do(x)] (short for Q=E[Y|do(X=x)]), is perfectly well-defined when X is part of a causal model M, for it can be computed using the surgical procedure of the do-operator [6, p. 24]. Yet conceptually, Q raises two questions when X is a state of a variable. The first question is semantical: What information does Q convey aside from being a mathematical property of our model? Since one cannot translate Q into a prediction about the effect of an executable action, what does Q tell us about reality which is not just an artifact of the model? Take for example the proposition: “The number of variables in the model is a prime number”; it is undeniably a property of M, but would hardly qualify as a feature of reality. The second question raised is empirical: Even assuming that Q conveys an important feature of reality, how can we test it empirically? And if we cannot test it, is it part of science? I will address these two questions in the following sections.

2 The semantics of Q

Assume we are conducting an observational study guided by model M in which Q is identifiable, and is evaluated to be

where q(x) is some function of x, computed from the joint distribution of observed variables in the model. To what use can one put this information? I will discuss three distinct uses.

1. Q represents a theoretical limit on the causal effects of manipulable interventions that might become available in the future.

2. Q imposes constraints on the causal effects of currently manipulable variables.

3. Q serves as an auxiliary mathematical operation in the derivation of causal effects of manipulable variables.

3 Testing do(x) claims

We are now ready to tackle the final question posed in the introduction: Granted that Q=q conveys useful information to policy makers, how can we test it empirically?

Since X is non-manipulable, we must forgo verification of Q through the direct control of X, and settle instead on indirect tests as is commonly done in observational studies. This calls for devising observational or experimental studies capable of refuting the claim Q(x)=q(x) and ascertaining that our data do not clash with this claim.

Since the claim Q(x)=q(x) is a product of both the data and the modeling assumptions embedded in M, confirming the testable implications of those assumptions constitutes a test for the equality Q(x)=q(x).

Not all models have testable implications, but those that do advertise those implications in the model’s graph and invite standard statistical tests for verification. Typical are conditional independence tests and equality constraints. For example, if Q(x) is identifiable through the back-door criterion and there are several sets of covariates that satisfy the criterion, then equating the adjustment formulae generated by each of those sets provides a test for M, and hence a test for Q.

If the model contains manipulable variables, then randomized controls over the manipulable variables provide additional tests for the structure of M, and hence for the validity of Q. To illustrate, consider the front-door model of Fig. 1, where I is manipulable, X non-manipulable, and U an unobserved confounder. The model has no testable implication in observational studies. However, randomizing I yields an estimate of P(y|do(I)), which should be equal to the estimand of P(y|do(I)) obtained through the front-door formula. [6, pp. 81–83]. Equating the two provides a refutable test for the assumptions embedded in the model, and hence for Q(x), where

Figure 1

A model in which equating the effect of I on Y in an RCT with that obtained through the front-door formula produces a test for Q(x).

We see that, whereas direct tests of Q(x) are infeasible, indirect tests are available, thus affirming the empirical content of Q. Metaphorically, these tests can be likened to the way planet Neptune was discovered (1845)—not by direct observation, but through the anomaly it caused in the trajectory of Uranus.

4 Non-manipulability and reinforcement learning

The role of models in handling a non-manipulable variable has interesting parallels in machine learning applications, especially in its reinforce learning (RL) variety [14], [15]. Skipping implementational details, a RL algorithm is given a set of actions or interventions, say I={I1,I2,…,In), and is required to find, for every observed state s of the environment an action Ik that maximizes the long-term reward achievable by acting Ik at state s. This reward function can be written as E[Y|do(Ik),s], with Y the stream of future payoffs received by acting Ik.

Through trial and error training of a neural network, the RL algorithm constructs a functional mapping between each state s and the next action to be taken. In the course of this construction, however, the algorithm evaluates a huge number of reward functions of the form E[Y|do(Ik),s] which, for a given s are very similar to the function Q(x) that has been the focus of our discussion in this paper.

A question often asked about the RL framework is whether it is equivalent in power to SCM in terms of its ability to predict the effects of interventions.

The answer is a qualified YES. By deploying interventions in the training stage, RL allows us to infer the consequences of those interventions, but ONLY those interventions. It cannot go beyond and predict the effects of actions not tried in training. To do that, a causal model is required [16]. This limitation is equivalent to the one faced by researchers who deny legitimacy to Q(x) when X is non-manipulable. In the RL context, however, the prohibition extends to manipulable variables as well, in case they were not activated in the training phase.

A simple example illustrating this point is shown in Fig. 1, which depicts the causal structure of the environment prior to learning. X and Z are manipulable, while U1 and U2 are unobserved. Suppose we train a machine to learn the effect of manipulating Z on both Y and X. We now wish to infer the effect of action do(X=x) that was not accessible during training. Having a causal model, as in Fig. 2(a), the task can be accomplished through do-calculus [17], [18], giving:

Thus, the freedom to manipulate Z and estimate its effects on X and Y enables us to evaluate the effect of an action do(X=x) which was never tried before.

Figure 2

Model (a) permits us to learn the effect of X on Y by manipulating Z, instead of X. In Model (b) learning the effect of X on Y requires that X itself be manipulated.

To see the critical role that causal modeling plays in this exercise, note that the model in Fig. 2(b) does not permit such evaluation by any algorithm whatsoever, a fact verifiable from the model structure [17]. This means that a model-blind RL algorithm would be unable to tell whether the optimal choice of untried actions can be computed from those tried.

5 Conclusions

We have shown that causal effects associated with non-manipulable variables have empirical semantics along several dimensions. They provide theoretical limits, as well as valuable constraints over causal effects of manipulable variables. They facilitate the derivation of causal effects of manipulable variables and, finally, they can be tested for validity, albeit indirectly.

Doubts and trepidations concerning the effects of non-manipulable variables and their empirical content should give way to appreciating the important roles that these effects play in causal inference.

Turning attention to machine learning, we have shown parallels between estimating the effects of non-manipulable variables and learning the effect of feasible yet untried actions. The role of causal modeling was shown to be critical in both frameworks.

Armed with these clarifications, researchers need not be concerned with the distinction between manipulable and non-manipulative variables, except of course in the design of actual experiments. In the analytical stage, including model specification, identification and estimation, all variables can be treated equally, and are therefore equally eligible to receive the do-operator and to deliver the ramifications in its effect.