Double Machine Learning for Causal Inference under Shared-State Interference

We thank the reviewer for their comments. We will incorporate this feedback in our updated manuscript.

• Application-based simulations. Exploration of more complex and real-world scenarios (using either simulated or real data) would be a valuable direction for future work. Our focus in this paper is on the novel theoretical methodological contribution. Our simulation settings were intended to be simple, to demonstrate how naive estimates can be biased even in simple, one-dimensional settings. Thank you for the suggestion to add coverage rates of variance estimates. We will include this in the appendix.
• Notation. Thank you for pointing out the notational inconsistencies. We will fix these and adhere to potential outcome notation where the dependence on $X_t$ is implicit.

We thank the reviewer for their thoughtful feedback. We will incorporate your suggestions, including providing coverage rates in Appendix F, in our updated manuscript. In the coverage rate plots, our consistent variance estimator approaches the target coverage rate as $T$ grows, while the coverage rates for naive estimators is near zero due to their bias.

• Dependence of $X_t$ and $D_t$ on $H_t$. (Copying this from our response to reviewer c4Lj.) This problem formulation would indeed be more general and would be a valuable direction for future work. We made the choice to have $H_t$ affect $Y_t$ but not $X_t$ and $D_t$ for two reasons. First, we found it hard to imagine situations in which $H_t$ exerts significant influence on unit characteristics and treatment assignments: most often, unit characteristics should be innate to the unit (e.g. their price sensitivity or preferences for particular types of content) and treatment assignments should be exogenous (we imagine treatment conditions like new features on a platform or inducements like discounts). Second, our context presents the simplest possible change from canonical iid models and so our work can be directly instructive for comparison with models where no shared-state is present by simply removing the shared-state variable from the outcome structural equations. Assuming dependence of $X_t$ and $D_t$ on $H_t$ would be possible under Theorem 3.1 as long as the data $W_t$ still obeys geometric ergodicity and detailed balance.
• Existence of auxiliary data. Auxiliary data may exist when there are multiple similar systems or markets where unit behavior will be similar. For example, on a social media platform with multiple forums, the data from one forum might be used to construct nuisance estimates for measuring treatment effects in another forum. This is an assumption common to other methods using machine learning for inference or uncertainty quantification (see, e.g., Angelopoulos et al 2023), but of course, this is a limitation on how widely applicable any such method requiring auxiliary data can be. On the other hand, conceptually, the auxiliary sample assumption cleanly separates the machine learning from application of the learned predictors for inference. Relaxations of this requirement may be possible, but we wanted to preserve the conceptual clarity in our paper, so we leave these extensions for future work.
• Real-world validations. Analysis of real-world settings would be a valuable contribution for future work. We wanted to include simulations for the sake of simplicity and brevity.
• Related work. We are not aware of other methods that formalize causal inference through shared-state interference as we do. Several other papers formalize interference through markets or through recommender systems as we note in the related work, but these are context-specific. For example, Munro 2024 allows for causal inference under shared-state interference in settings like auctions where the shared-states are prices and allocations of goods exhibit a cutoff structure. Our work is complementary by offering a context-agnostic approach that does not require, e.g., knowledge of the allocation mechanism if the data generating process satisfies our Markov chain assumption.

A. N. Angelopoulos, S. Bates, C. Fannjiang, M. I. Jordan, and T. Zrnic. Prediction-Powered Inference, Nov. 2023. URL http://arxiv.org/abs/2301.09633. arXiv:2301.09633 [cs, q-bio, stat].

E. Munro. Causal Inference under Interference through Designed Markets. 2024.

We thank the reviewer for their feedback, as well as their comments about the soundness of theoretical results and strength of foundation for the work. We will incorporate your comments into our work. We clarify several points and answer questions below.

• Data generating process details. We provide specific explanations of the data generating process in Appendix F, as we note in Line 300, column 2.
• Naive Horvitz-Thompson (HT) versus switchback HT estimators. The HT estimator in figure 2 is a naive HT estimator which merely takes a difference in means across treatment and control observations. The SB estimator in figure 3 is a different estimator which averages only over observations where the last $m$ have been all treatment or all control. The naive HT estimator, in general, may be biased since it does not account for the shared state, but the SB estimator is unbiased. We will clarify this distinction, especially since the SB estimator is a HT-style estimator but is different from the naive HT estimator.
• $H_t$ only affects $Y_t$, not $X_t$ and $D_t$. This problem formulation would indeed be more general and would be a valuable direction for future work. We made the choice to have $H_t$ affect $Y_t$ but not $X_t$ and $D_t$ for two reasons. First, we found it hard to imagine situations in which $H_t$ exerts significant influence on unit characteristics and treatment assignments: most often, unit characteristics should be innate to the unit (e.g. their price sensitivity or preferences for particular types of content) and treatment assignments should be exogenous (we imagine treatment conditions like new features on a platform or inducements like discounts). Second, our context presents the simplest possible change from canonical iid models and so our work can be directly instructive for comparison with models where no shared-state is present by simply removing the shared-state variable from the outcome structural equations. Assuming dependence of $X_t$ and $D_t$ on $H_t$ would be possible under Theorem 3.1 as long as the data $W_t$ still obeys geometric ergodicity and detailed balance.
• Notation. We use standard potential outcome notation with exposure mapping where potential outcomes are typically not written as a function of covariates (even though outcomes depend on covariates). In other words, $Y_t$ does depend on $X_t$, but this dependence is omitted from notation. We note that we inadvertently included $X_t$ in the potential outcome notation of some equations, like 4.5, and we will fix this. Thanks for the question.
• Independence of $W_{aux}$. $W_{aux}$ need not be iid as long as the nuisance parameter learners converge at appropriate rates. The tradeoff between quality (say, independence) and quantity (number of observations) of data is implicit in the nuisance learner rates assumptions, but exploration of this question would be a valuable direction for future work.
• Choices of $\gamma$. You can assume $\gamma = 0$ as a special case of Theorem 3.1. We include the high-probability bounds because it follows the results in Chernozhoukov and because it allows for the possibility that eta falls outside of the nuisance realization set with some probability. $\gamma$ might be useful if, as we note in our response to Reviewer YLJC, $m$ may be mis-specified with some probability.

Thank you for your constructive feedback! We also thank you for noting the expressivity of our formality, relevance to practical settings and gap in research on causal inference in algorithmic systems and markets. We respond to each of your questions below:

• Estimating $m$ in Practice: We note that for the procedure to be valid, we just need an upper bound on $m$, so that the switch period can be chosen to be greater than this upper bound. The literature on switchback experiments proposes procedures to estimating $m$ if it is not known. See, e.g., Bojinov et al 2022, Section 4.4. Informally, the procedure involves computing the estimator using different candidate values $m$ and running a series of hypothesis tests to see whether the results are the same for different values. If the results are different, then the smaller candidate for $m$ can be shown to be less than its true value, with high probability. The uncertainty introduced by unknown $m$ can be incorporated in our high-probability bound by choosing $\gamma$ to account for the Type 2 error of estimating $m$ is smaller than it actually is.
• Auxiliary Data requirement: We will add more discussion of this point to the paper. It would be a worthwhile direction for future research to explore how approximately independent data (such as blocks observed far away) may be used to train nuisance predictors that are approximately independent from the data they are evaluated on.
• Applicability to Recommender Systems: Application of our framework to real-world empirical contexts would be a valuable direction for future work. Given the theoretical focus of the paper and the constraints of space, we don’t do this. We strongly believe that our framework is applicable to relevant empirical settings.

I. Bojinov, D. Simchi-Levi, and J. Zhao. Design and Analysis of Switchback Experiments, Apr. 2022. URL http://arxiv.org/abs/2009.00148. arXiv:2009.00148 [stat].