Data-Driven Influence Functions for Optimization-Based Causal Inference
- URL: http://arxiv.org/abs/2208.13701v4
- Date: Thu, 15 Jun 2023 18:14:40 GMT
- Title: Data-Driven Influence Functions for Optimization-Based Causal Inference
- Authors: Michael I. Jordan, Yixin Wang, Angela Zhou
- Abstract summary: We study a constructive algorithm that approximates Gateaux derivatives for statistical functionals by finite differencing.
We study the case where probability distributions are not known a priori but need to be estimated from data.
- Score: 105.5385525290466
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We study a constructive algorithm that approximates Gateaux derivatives for
statistical functionals by finite differencing, with a focus on functionals
that arise in
causal inference. We study the case where probability distributions are not
known a priori but need to be estimated from data. These estimated
distributions lead to empirical Gateaux derivatives, and we study the
relationships between empirical, numerical, and analytical Gateaux derivatives.
Starting with a case study of the interventional mean (average potential
outcome), we delineate the relationship between finite differences and the
analytical Gateaux derivative. We then derive requirements on the rates of
numerical approximation in perturbation and smoothing that preserve the
statistical benefits of one-step adjustments, such as rate double robustness.
We then study more complicated functionals such as dynamic treatment regimes,
the linear-programming formulation for policy optimization in infinite-horizon
Markov decision processes, and sensitivity analysis in causal inference. More
broadly, we study optimization-based estimators, since this begets a class of
estimands where identification via regression adjustment is straightforward but
obtaining influence functions under minor variations thereof is not. The
ability to approximate bias adjustments in the presence of arbitrary
constraints illustrates the usefulness of constructive approaches for Gateaux
derivatives. We also find that the statistical structure of the functional
(rate double robustness) can permit less conservative rates for
finite-difference approximation. This property, however, can be specific to
particular functionals; e.g., it occurs for the average potential outcome
(hence average treatment effect) but not the infinite-horizon MDP policy value.
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