20.8 Putting It All Together: Comparing Objectives

As an overarching illustration, let $\hat{f}$ be any trained predictor (ML model, regression, etc.) and let $\hat{\beta}$ be a parameter estimator from a structural or causal model. Their respective tasks differ:

• Form of Output
  • $\hat{f}$ is a function from $\mathcal{X} \to \mathcal{Y}$.
  • $\hat{\beta}$ is a vector of parameters with theoretical meaning.
• Criterion
  • Prediction: Minimizes predictive loss $\mathbb{E}[L(Y,\hat{f}(X))]$.
  • Causal Inference: Seeks $\beta$ such that $Y = m_\beta(X)$ is a correct structural representation. Minimizes bias in $\beta$, or satisfies orthogonality conditions in method-of-moments style, etc.
• Validity
  • Prediction: Usually validated by out-of-sample experiments or cross-validation.
  • Estimation: Validated by theoretical identification arguments, assumptions about exogeneity, randomization, or no omitted confounders.
• Interpretation
  • Prediction: “$\hat{f}(x)$ is our best guess of $Y$ for new $x$.”
  • Causal Inference: “$\beta$ measures how $Y$ changes if we intervene on $X$.”

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