Minimalist review of dual bounds

This section gives a minimal review of the original dual bounds framework from Ji et al. (2023). A more complete review can be found in the first four pages of the paper.

Given i.i.d. outcomes \(Y_i \in \mathcal{Y}\), treatments \(W_i \in \{0,1\}\) and pre-treatment covariates \(X_i \in \mathcal{Y}\), dual bounds allow analysts to use machine learning to perform inference on partially identified estimands of the form

\[\theta = E[f(Y_i(0), Y_i(1), X_i)],\]

where \(Y_i(1), Y_i(0) \in \mathcal{Y}\) denote potential outcomes. Such estimands are partially identified because we never observe the joint law of the potential outcomes, but the data still contains information on the law of \((Y(0), X)\) and \((Y(1), X)\) allowing us to bound \(\theta\). (More generally, dual bounds can provde bounds on the solutions to generic optimization problems, but we defer discussion of this until later in the tutorial for simplicity).

Thus, to bound \(\theta\), one must estimate the laws of \((Y(1), X)\) and \((Y(0), X)\), typically using machine learning techniques. However, it is not clear if the resulting bounds on \(\theta\) will be valid if the learned machine learning models are misspecified or inaccurate. For example, if one models \(Y\) as having a linear relationship with \((X,W)\), but in truth \(Y\) has a highly nonlinear relationship with \((X,W\)), naive approaches can lead to inaccurate inference.

Dual bounds are designed to leverage the benefits of sophisticated ML models without sacrificing validity. This framework has three key properties:

  1. Flexibility: Dual bounds can wrap on top of any ML model (e.g. generalized linear models, random forests, neural networks, etc).

  2. Validity: In randomized experiments, dual bounds yield provably valid confidence intervals on \(\theta\), even if the underlying ML model is arbitrarily misspecified or inaccurate. In observational studies, they have remarkably strong double robustness properties (see Section 3.4 of Ji et al. (2023)).

  3. Power: On the other hand, if the underlying ML model is highly accurate, dual bounds yield tight confidence bounds in a formal sense (see Section 3.2 of Ji et al. (2023)).

We refer the reader to the original paper for more details on how dual bounds work. In the remaining tutorials, we show how to use dual bounds to perform inference on a variety of estimands.