Causal Inference in Metric Spaces via Distance Profiles
Abstract
Modern interventions increasingly produce object-valued outcomes, such as distributions, covariance matrices, networks, and trees. Existing causal frameworks often exploit special geometry such as one-dimensional Wasserstein quantiles or geodesic Fréchet means, whereas metric statistics studies random objects without confounding adjustment. We propose the profile causal effect Δ(ω,t), which describes where and at what scale a treatment shifts probability mass, including changes in dispersion, tails, or structure that leave mean-level summaries unchanged. Formally, Δ(ω,t)=Pr{d(ω,Y(1))≤ t}−Pr{d(ω,Y(0))≤ t} contrasts the potential-outcome probabilities of the same anchor-centered metric ball and requires no structure beyond the metric. On grids fixed by a deterministic or auxiliary design, we combine cross-fitted doubly robust scores with high-dimensional Gaussian approximation and the multiplier bootstrap to obtain simultaneous confidence bands and a global test under regularity conditions. The bands bound treatment-induced changes in ball and shell probabilities simultaneously and, at fixed anchors, extend to all radii. A protocol guides the choice of anchors and radii. Across the experiments, the global test detects concentration effects that mean-quantile and Fréchet-mean summaries miss. Its empirical size is near or below the nominal level, even under confounding, where naive metric tests are substantially oversized.
Related
- Realising atomless laws as distance distributions on metric measure spaces
- Minimum Specification Perturbation: Robustness as Distance-to-Falsification in Causal Inference
- Default-Distance Entropy and Metric Dimension in Finite Geometries
- Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance
- Repairing covariance functions on metric trees through a Hausdorff-type distance
BibTeX
@misc{song2026causal,
title = {{Causal Inference in Metric Spaces via Distance Profiles}},
author = {Zheye Song and Zhiheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33491},
eprint = {2610.33491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33491},
}
RIS
TY - UNPB
TI - Causal Inference in Metric Spaces via Distance Profiles
AU - Zheye Song
AU - Zhiheng Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33491
UR - https://arxiv.org/abs/2610.33491
AB - Modern interventions increasingly produce object-valued outcomes, such as distributions, covariance matrices, networks, and trees. Existing causal frameworks often exploit special geometry such as one-dimensional Wasserstein quantiles or geodesic Fréchet means, whereas metric statistics studies random objects without confounding adjustment. We propose the profile causal effect Δ(ω,t), which describes where and at what scale a treatment shifts probability mass, including changes in dispersion, tails, or structure that leave mean-level summaries unchanged. Formally, Δ(ω,t)=Pr{d(ω,Y(1))≤ t}−Pr{d(ω,Y(0))≤ t} contrasts the potential-outcome probabilities of the same anchor-centered metric ball and requires no structure beyond the metric. On grids fixed by a deterministic or auxiliary design, we combine cross-fitted doubly robust scores with high-dimensional Gaussian approximation and the multiplier bootstrap to obtain simultaneous confidence bands and a global test under regularity conditions. The bands bound treatment-induced changes in ball and shell probabilities simultaneously and, at fixed anchors, extend to all radii. A protocol guides the choice of anchors and radii. Across the experiments, the global test detects concentration effects that mean-quantile and Fréchet-mean summaries miss. Its empirical size is near or below the nominal level, even under confounding, where naive metric tests are substantially oversized.
ER -