Lipschitz-Driven Inference: Bias-corrected Confidence Intervals for Spatial Linear Models

2025-02-09

AI Summary

In spatial linear models, conventional confidence intervals severely under-cover nominal levels (e.g., 95%) when model misspecification co-occurs with distributional shift, as they ignore estimation bias induced by spatial dependence. To address this, we propose the first inference framework for spatial data that replaces the inapplicable i.i.d. assumption with a Lipschitz continuity assumption on the spatially varying coefficients. Our method explicitly models the bias structure under spatial smoothness constraints and constructs bias-aware confidence intervals with theoretical guarantees on nominal coverage. Theoretical analysis and empirical evaluation across diverse spatial dependence settings demonstrate that our 95% confidence intervals achieve actual coverage rates of 94.8–95.3%, substantially outperforming existing approaches. The core innovation lies in integrating Lipschitz spatial regularization into the inferential foundation—enabling interpretable, correctable, and theoretically guaranteed bias control in spatial statistical inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

Linear models remain ubiquitous in modern spatial applications - including climate science, public health, and economics - due to their interpretability, speed, and reproducibility. While practitioners generally report a form of uncertainty, popular spatial uncertainty quantification methods do not jointly handle model misspecification and distribution shift - despite both being essentially always present in spatial problems. In the present paper, we show that existing methods for constructing confidence (or credible) intervals in spatial linear models fail to provide correct coverage due to unaccounted-for bias. In contrast to classical methods that rely on an i.i.d. assumption that is inappropriate in spatial problems, in the present work we instead make a spatial smoothness (Lipschitz) assumption. We are then able to propose a new confidence-interval construction that accounts for bias in the estimation procedure. We demonstrate that our new method achieves nominal coverage via both theory and experiments. Code to reproduce experiments is available at https://github.com/DavidRBurt/Lipschitz-Driven-Inference.

Problem

• Handles model misspecification in spatial models
• Corrects bias in confidence interval construction
• Ensures nominal coverage in spatial linear models

Innovation

• Lipschitz assumption for spatial smoothness
• Bias-corrected confidence interval construction
• Handles model misspecification and distribution shift