Kleinberg et al. (2015) introduce the concept of prediction policy problems, and claim to unite prediction problems and causal inference problems in a single framework.
In section I of the paper, right column, page 1, they set up the problem:
- $Y$ is an outcome variable, which depends on $X_0$ and $X$, so I'm guessing $Y(X_0, X)$
- $X_0$ is a decision made by a policymaker (so it's an intervention/treatment variable
- $X$ is not really defined, but I'm guessing it is the values of some set of predictors of outcome $Y$
- $\pi(X_0, Y)$ is a payoff function that depends on the outcome, as well as on the treatment variable indirectly through its effect on $Y$
Then they write:
$$ \frac{\partial \pi(X_0,Y)}{\partial X_0} = \frac{\partial \pi}{\partial X_0} (Y) + \frac{\partial \pi}{\partial Y} \frac{\partial Y}{\partial X_0} $$
I am confused by this statement for three reasons.
First, I tried to expand the notation on the l.h.s. as:
$$ \frac{\partial \pi(X_0,Y(X_0))}{\partial X_0} $$
But that can't be right, because I'm pretty sure the r.h.s. would then be:
$$ \frac{\partial \pi(X_0,Y)}{\partial X_0} = \frac{\partial \pi}{\partial X_0} \frac{\partial X_0}{\partial X_0} + \frac{\partial \pi}{\partial Y} \frac{\partial Y}{\partial X_0} = \frac{\partial \pi}{\partial X_0} + \frac{\partial \pi}{\partial Y} \frac{\partial Y}{\partial X_0} $$
Note the absence of the $(Y)$ to the left of the plus sign by comparison to the paper.
But then I remembered that, at least according to the words in the paper, $Y$ depends on both $X_0$ and the mysterious $X$.
Second, what role does $X$ play here and why does it not show up anywhere in the equation (at least not explicitly)?
Third, is there something notationally special about the parenthesis around $Y$ in the equation that I am missing that could help me understand the derivation better? For example, could this notation just be a weird way of making explicit that the partial derivative $\partial z / \partial X_0$ is potentially a function of $Y$ (say, because $z$ is nonlinear in $Y$), that is, if $\partial z / \partial X_0$ we’re given as $f$, then $f(Y)$?
