13
$\begingroup$

I have seen the $do(x)$ operator everywhere in some literature review I am doing on Causality (see, for instance this wikipedia entry). However, I cannot find a formal and general definition of this operator.

Can someone point me to a good reference on this? I am interested in a general definition rather than its interpretation in a particular experiment.

$\endgroup$
11
$\begingroup$

That is $do$-calculus. They explain it here:

Interventions and counterfactuals are defined through a mathematical operator called $do(x)$, which simulates physical interventions by deleting certain functions from the model, replacing them with a constant $X = x$, while keeping the rest of the model unchanged. The resulting model is denoted $M_x$.

$\endgroup$
11
$\begingroup$

A probabilistic Structural Causal Model (SCM) is defined as a tuple $M = \langle U, V, F, P(U) \rangle$ where $U$ is a set of exogeneous variables, $V$ a set of endogenous variables, $F$ is a set of structural equations that determines the values of each endogenous variable and $P(U)$ a probability distribution over the domain of $U$.

In a SCM we represent the effect of an intervention on a variable $X$ by a submodel $M_x = \langle U, V, F_x, P(U) \rangle$ where $F_x$ indicates that the structural equation for $X$ is replaced by the new interventional equation. For example, the atomic intervention of setting the variable $X$ to a specific value $x$ --- usually denoted by $do(X = x)$ --- consists of replacing the equation for $X$ with the equation $X = x$.

To make ideas clear, imagine a nonparametric structural causal model $M$ defined by the following structural equations:

$$ Z = U_z\\ X = f(Z, U_x)\\ Y = g(X,Z, U_y) $$

Where the disturbances $U$ have some probability distribution $P(U)$. This induces a probability distribution over the endogenous variables $P_M(Y, Z, X)$, and in particular a conditional distribution of $Y$ given $X$, $P_M(Y|X)$.

But notice $P_M(Y|X)$ is the "observational" distribution of $Y$ given $X$ in the context of model $M$. What would be the effect on the distribution of $Y$ if we intervened on $X$ setting it to $x$? This is nothing more than the probability distribution of $Y$ induced by the modified model $M_x$:

$$ Z = U_z\\ X = x\\ Y = g(X, Z, U_y) $$

That is, the interventional probability of $Y$ if we set $X= x$ is given by the the probability induced in submodel $M_x$, that is, $P_{M_x}(Y|X=x)$ and it's usually denoted by $P(Y|do(X = x))$. The $do(X= x)$ operator makes it clear we are computing the probability of $Y$ in a submodel where there is an intervention setting $X$ equal to $x$, which corresponds to overriding the structural equation of $X$ with the equation $X =x$.

The goal of many analyses is to find how to express the interventional distribution $P(Y|do(X))$ in terms of the joint probability of the observational (pre-intervention) distribution.

do-calculus

The do-calculus is not the same thing as the $do(\cdot)$ operator. The do-calculus consists of three inference rules to help "massage" the post-intervention probability distribution and get $P(Y|do(X))$ in terms of the observational (pre-intervention) distribution. Hence, instead of doing derivations by hand, such as in this question, you can let an algorithm perform the derivations and automatically give you a nonparametric expression for identifying your causal query of interest (and the do-calculus is complete for recursive nonparametric structural causal models).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.