16.3 Causation versus Prediction
Understanding the relationship between causation and prediction is crucial in statistical modeling.
Let \(Y\) be an outcome variable dependent on \(X\), and our aim is to manipulate \(X\) to maximize a payoff function \(\pi(X, Y)\) (Kleinberg et al. 2015). The decision on \(X\) hinges on:
\[ \begin{aligned} \frac{d\pi(X, Y)}{d X} &= \frac{\partial \pi}{\partial X} (Y) + \frac{\partial \pi}{\partial Y} \frac{\partial Y}{\partial X} \\ &= \frac{\partial \pi}{\partial X} \text{(Prediction)} + \frac{\partial \pi}{\partial Y} \text{(Causation)} \end{aligned} \]
Empirical work is essential for estimating the derivatives in this equation:
\(\frac{\partial Y}{\partial X}\) is required for causal inference to determine \(X\)’s effect on \(Y\),
\(\frac{\partial \pi}{\partial X}\) is required for prediction of \(Y\).
