Causal inference for spatiotemporal point processes in the presence of outcome spillover and carryover

1. Problem Statement

The paper addresses the challenge of performing causal inference on continuous spatiotemporal point processes (e.g., earthquake locations, disease infections, financial transactions) when outcome spillover (interference) and carryover (temporal dependence) are present.

• The Core Difficulty: Standard causal inference methods often assume independent units or rely on coarse aggregation (binning data into spatial/temporal cells). Aggregation discards exact event locations and times, obscuring the internal spatiotemporal structure necessary to model how an event in one region "excites" events in another.
• The Spillover Mechanism: In many real-world scenarios (e.g., contagious diseases or seismic activity), an event in a treated area can trigger events in a control area, and vice versa. This violates the Stable Unit Treatment Value Assumption (SUTVA).
• The Data Challenge: Post-treatment, the observed process is an unlabelled superposition of latent control and treatment components. It is unknown whether a specific observed event was generated by the treatment process or the control process, even if the event occurred in a treated cell.

2. Methodology

The authors propose a model-based framework that retains exact event times and locations while estimating causal effects.

A. Framework and Potential Outcomes

• Tessellation: The post-treatment spatiotemporal window is partitioned into a finite set of cells ($I_j$). These cells serve as the units for treatment assignment ($Z_j \in \{0, 1\}$).
• Latent Superposition: The observed process $N$ is modeled as $N = N_0 + N_1$, where $N_0$ is the latent control process and $N_1$ is the latent treatment process.
• Unlabelled Events: An event in a treated cell is not necessarily a "treatment event"; it could be a control event triggered by spillover. The goal is to infer the latent labels ($r_i \in \{0, 1\}$) for each event.
• Causal Estimands: The paper defines several estimands based on expected event counts under different treatment allocations ($z$):
  • Individual Treatment Effect (ITE): Difference in expected counts for a specific cell under treatment vs. control.
  • Average Individual Treatment Effect (AITE): Average of ITEs across cells.
  • Difference under Fixed Allocations (DAITE): Total count difference between two specific allocation regimes (e.g., all-treated vs. all-untreated).

B. Identification

Identification relies on standard causal assumptions adapted for interference:

1. Consistency: The observed process matches the potential process under the realized treatment.
2. Conditional Exchangeability: Treatment assignment is independent of potential outcomes given pre-treatment history and covariates.
3. Positivity: All relevant treatment allocations have a non-zero probability.
4. Regime-Stable Structural Model: For allocations outside the observed support (off-support), identification relies on a parametric model (e.g., Hawkes process) that remains stable across different treatment regimes.

C. Estimation: Stochastic EM (SEM)

Since the latent labels are unobserved, the authors employ a Stochastic Expectation-Maximization (SEM) algorithm:

1. E-step (Labeling): Instead of calculating the intractable expectation over all $2^N$ label configurations, the algorithm uses a discrepancy-guided proposal. It samples label configurations that focus on regions where the current fitted intensities disagree most with the observed data (e.g., where the control model under-predicts counts in a treated cell).
2. M-step (Parameter Estimation): Given the sampled labels, the algorithm maximizes the complete-data log-likelihood to update the intensity parameters ($\theta$).
3. Iteration: The process alternates between relabeling events and updating parameters until convergence.

3. Key Theoretical Contributions

The paper provides rigorous finite-sample theory to justify the SEM approach, which is non-trivial due to the feedback loops in history-dependent processes (like Hawkes processes).

• Contraction to a Statistical Floor: The authors analyze a "predictable blockwise hard-EM" surrogate to prove that the estimation error contracts geometrically to a statistical floor.
  • This floor is governed by locally ambiguous regions (where the likelihood ratio between treatment and control is close to zero) and martingale fluctuations.
  • Unlike standard EM, where label errors might not propagate, in Hawkes processes, a single early label error can perturb future intensities. The theory shows that under specific conditions, the algorithm can still recover the true parameters.
• Conditions for Consistency: The paper derives "compatibility conditions" (Proposition 5.2) required for the statistical floor to vanish as the sample size grows. This involves balancing the "decisiveness" threshold ($b$) against the alignment of scores and the mass of the ambiguous band.
• Plug-in Guarantees: The authors prove that if the parameter estimator converges, the plug-in causal estimands (ITE, AITE, etc.) also converge. They distinguish between fixed-volume contrasts (which converge if parameters do) and unnormalized growing-window totals (which require additional rate conditions).

4. Results

Simulation Study

• Setup: Simulated spatiotemporal Hawkes processes with a 10x10 grid, where treatment reduces the "branching ratio" (excitation strength).
• Findings:
  • Naive Approach: Treating events in treated cells as purely "treatment" events significantly underestimates the treatment effect due to spillover inflating the fitted treated intensity.
  • SEM Approach: The proposed SEM method substantially reduces bias and tracks the "oracle" (true labels known) benchmark closely.
  • Pre-treatment Data: Using pre-treatment history to estimate baseline parameters is crucial; relying solely on post-treatment data degrades performance.

Application: Oklahoma Seismic Activity

• Context: Analyzing the causal effect of a 2015 directive reducing wastewater injection volumes in Oklahoma on seismic activity.
• Challenge: Seismicity exhibits strong temporal carryover and spatial spillover. Treatment was not randomized.
• Findings:
  • Naive Fit: Suggested ~341 earthquakes were prevented by the policy.
  • SEM Fit: Suggested -118 prevented events (i.e., potentially more events than expected, or no short-term reduction).
  • Interpretation: The naive model failed to account for the fact that earthquakes in treated areas were likely triggered by the long memory of untreated areas (spillover). The SEM model, by disentangling the latent processes, revealed that the short-term causal effect was negligible or negative, cautioning against early conclusions about policy efficacy when carryover is long-range.

5. Significance

• Methodological Advancement: This is one of the first frameworks to perform causal inference on continuous spatiotemporal point processes without resorting to coarse aggregation, which preserves the rich structure of the data.
• Handling Interference: It provides a rigorous solution to the "spillover" problem in continuous time and space, moving beyond discrete-time approximations.
• Practical Utility: The application to Oklahoma seismicity demonstrates the critical importance of correctly modeling latent processes. Ignoring spillover can lead to qualitatively wrong policy conclusions (e.g., claiming a policy worked when it did not, or vice versa).
• Theoretical Depth: The derivation of non-asymptotic contraction rates and the identification of the "statistical floor" provides a new theoretical understanding of the limits of causal inference in complex, history-dependent systems.

In summary, the paper offers a robust, likelihood-based framework for disentangling causal effects in complex spatiotemporal data, validated by both theory and real-world application, showing that ignoring outcome spillover can lead to severe biases in causal estimation.