Title: Smooth Sailing: Lipschitz-Driven Uncertainty Quantification for Spatial Associations
Venue: NeurIPS 2025
Accepted: yes
Keywords: Spatial

Confidence Intervals

Linear Regression

Lipschitz

Uncertainty Quantification
Abstract: Estimating associations between spatial covariates and responses — rather than merely predicting responses — is central to environmental science, epidemiology, and economics. For instance, public health officials might be interested in whether air pollution has a strictly positive association with a health outcome, and the magnitude of any effect. Standard machine learning methods often provide accurate predictions but offer limited insight into covariate-response relationships. And we show that existing methods for constructing confidence (or credible) intervals for associations can fail to provide nominal coverage in the face of model misspecification and nonrandom locations — despite both being essentially always present in spatial problems. We introduce a method that constructs valid frequentist confidence intervals for associations in spatial settings. Our method requires minimal assumptions beyond a form of spatial smoothness and a homoskedastic Gaussian error assumption. In particular, we do not require model correctness or covariate overlap between training and target locations. Our approach is the first to guarantee nominal coverage in this setting and outperforms existing techniques in both real and simulated experiments. Our confidence intervals are valid in finite samples when the noise of the Gaussian error is known, and we provide an asymptotically consistent estimation procedure for this noise variance when it is unknown.


Peer reviews:
comments: SUMMARY:
This paper proposes a new method for constructing valid confidence intervals for spatial associations under model misspecification and distribution shift. By assuming only that the response is Lipschitz-continuous in space, the method avoids strong modeling assumptions and achieves nominal coverage where existing methods fail. Experiments on simulated and real data show superior coverage and efficiency compared to standard approaches.

STRENGTHS AND WEAKNESSES:
**Strengths**:
- **Clarity and Accessibility**: The paper is very clearly written and easy to follow. Even for readers outside the immediate subfield, the key ideas and contributions are accessible.  
- **Strong Motivation**: The motivation is compelling and well-grounded in real-world challenges of spatial inference under model misspecification and distribution shift.  
- **Theoretical Rigor**: The analysis is mathematically solid, with well-justified assumptions and detailed, carefully presented proofs.

**Weakness**:
- **Title Meaning**: Why is the paper titled *"Smooth Sailing"*? How does it relate to the method and its content?

QUESTIONS:
1. The paper considers the setting where $Y$ is an $N$-long column vector. How would the proposed method extend to cases where $Y$ is high-dimensional (i.e., multivariate responses)?

2. From a mathematical perspective, what are the fundamental reasons that make the proposed method better than Gaussian Processes (GP BCI), particularly under model misspecification or distribution shift?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-27

comments: SUMMARY:
The paper concerns estimating associations between covariates and responses with the correct frequentist coverage of the confidence interval. With this aim, the authors proposed a new method for associations in spatial settings.

STRENGTHS AND WEAKNESSES:
The authors described the ideas and demonstrated them using simulation studies. I have some concerns about the novelty, although I might overlook it.
I struggled to understand the logic behind Assumptions 1 and 2, the existence of independent maps from spatial locations to the response and covariates, respectively. i.e., Two data-generating processes. It would be more natural to consider a parametric model with covariates which are mapped from locations to account for the association between response and spatial location. In the simulation studies, the authors generated data in this way. I think it is good to clarify this point.

Good estimation of a map between location and covariate influences the parameter estimation; the authors assume that they are smooth, indicated by the Lipschitz constant. To me, the smoothness assumption of this map sounds overly simplified, and there is a bias-variance trade-off and more thorough discussion and studies about its impact on parameter estimation.

Although the authors discussed variance parameter estimation in the paper, a confidence interval with the nominal coverage challenges, as the authors noted in the simulation study. Simple linear models are considered in the paper. Although the authors plan to extend this to GLM, the current version seems to present limited studies.

QUESTIONS:
please see the strength and weakness section.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes confidence intervals for association in spatial setting where the linear model is misspecified. Additive Gaussian noise assumption and covariates are fixed function of spatial location assumption are imposed on the data-generating process. Further, under the assumption that the response is a Lipschitz function of the spatial location is assumed, and the authors provide confidence intervals with valid coverage based on these conditions.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is well-organized, with clear logic between different sections. I didn’t check the mathematical details of the paper, but it seems solid at first glance. I do not work in the spatial association area, so it is hard for me to judge the contribution of the paper given the assumptions made in the paper. \n\nI found the introduction poorly written. The second to the fourth paragraph talks about possible relationships between source and spatial location, and lacks focus. It is unclear what the authors are trying to convey. I would recommend the authors to focus directly on the linear model and discuss other possible models either in related work or discussion section. \n\nI found the claim on the contribution of the paper in accurate: the authors claim that their result hold under non parametric assumptions on the data-generating distributions. However, as stated in their assumptions 1 and 2, additive Gaussian noise assumption is clearly imposed on their data-generating process. Additionally, they require smoothness assumptions on the DGP. The statement is very misleading.\n\nQUESTIONS:\nIn addition to the questions raised in above, I found the Assumption 1 very restrictive. It effectively washes away the need for covariates in this setting. Assumption 1 states that for fixed spatial location parameter in the model, the corresponding covariate value is always the same. I found this assumption unreasonable across many applications that the authors mentioned (e.g., air pollution). The explanation that the authors provided in the paper does not directly justify why this assumption holds in applications.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers constructing confidence intervals for a linear regression problem under covariate shift, where both training and test data is associated with a \"spatial information\". \n\nConcretely, with $\\mathcal{S}$ being some metric space (which should be thought of as low-dimensional - e.g. earth surface), the authors consider a model where there is an (unkown) Lipschitz function $\\chi : \\mathcal{S} \\to \\mathbb{R}^D$, mapping the spatial location into covariates, and the (noisy) observations are given as $Y_i = f(S_i) + \\varepsilon$ for some Lipschitz function $f: \\mathcal{S} \\to \\mathbb{R}$. The algorithm observes $N$ tuples $(S_i, X_i, Y_i)$ for $S_i \\sim \\mathcal{D}_1$, such that $X_i = \\chi(S_i)$, followed by $M$ tuples $(S_i\\^*, X\\_i\\^\\*)$ where $S_i$ is drawn from a potentially different distribution $\\mathcal{D}_2$. \n\nThe relation between $Y_i$ and $X_i$ is not assumed to be linear -- the authors nevertheless suggest fitting a linear model in this misspecified case, attempting to approximate $Y_i$ as $\\langle \\theta, X_i\\rangle$ as well as possible - as they claim even if it does not necessairly yield the best possible predictions for $Y^*$, it is more interpretable than alternatives, and often might provide qualitative understanding of which covariates $X_i$ have impact on $Y$.\n\nAs such, they attempt to provide a confidence intervals for coordinates of the least square linear regression parameter at the \"target locations\" $S_i^\\*$\n$\\theta_{OLS} := \\mathrm{arg min}\\_{\\theta} \\sum\\_{i \\leq M} \\mathbb{E} \\left[(Y\\_i^\\* - \\theta^T X\\_i^\\*)^2 | S\\_i^\\*\\right],$ \nwhere, again $Y_i^\\*$ are not observed by the algorithm, (and averaged over in the formula above)\n\nTo this end, they approximate $f(S\\_i^\\*) = \\mathbb{E}[Y\\_i^\\* | S\\_i^\\*]$ as a weighted average $Y\\_i$ for nearby points $S\\_i$ in the training set (for experiments, they used simply putting weight $1$ on the nearest neighbor and $0$ on all remaining entries), producing estimated vector of responses $\\hat{Y}\\_i^\\*$. They use this to compute $\\hat{\\theta}\\_{OLS} = (X^{\\*T} X^\\*)^{-1} X^{\\*T} \\hat{Y}^\\*$, and calculate the confidence interval (valid under the relatively weak assumption that the response $f(S)$ is Lipschitz in $S$) for coordinates $\\hat{\\theta}\\_{OLS, p}$, by bounding bias introduced by approximating $Y\\_i^\\*$ with $\\hat{Y}\\_i^\\*$. Thanks to the fact that the $\\hat{Y}\\_i^\\*$ was taken as weighted averages of nearby points, obtaining bound on a bias reduces to a specific Wasserstein distance calculation.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is extremely well motivated, with both natural and relevant setting - authors explicitly provide a number of applications across various sciences for which the data is associated with spatial information, and the distributions of the spatial variable differs on the training and test data. The linear regression is already ubiquitous in sciences, in particular due to being more interpretable than more complex models. As such, having reliable confidence interval for regression coefficients in the misspecified case is particularly helpful.\n\nAssumption that the covariates are just a (deterministic) Lipschitz function of the spatial variable is a relatively strong limitation, but it has been explicitly acknowledged by authors - this work is a step in the right direction, and as they mention, there is a room for further research. It seems reasonable to assume, on the other hand, that in many scenarios the conditional expectation of the response variable on the spatial variable is Lipschitz in the spatial variable.\n\nIt is unclear to me, how to pick a specific Lipschitz constant for the dependence of the expected response on the spatial variable, in a given practical situation, and moreover - how often one should expect that the space $\\mathcal{S}$ is covered by a training data within the radius much smaller than the chosen Lipschitz constant - this already shows up in Figure 3 (the comparison of this method with other methods for the tree cover experiment): while the proposed method, is the only one of the compared ones that achieves a nominal coverage (i.e. confidence intervals are reliable), the width of the confidence intervals is an order of magnitude larger than the one produced by, for example, GP-BCI, and the coverage is in fact way larger than one would intuitively deem necessary - which might lead to unnecesairly rejecting many potential scientific results as statistically insignificant - arguably, the problem less prevalent than the opposite problem of false-positives in science.\n\nThe proposed framework allows for approximating $\\mathbb{E}[Y^\\* | S^\\*]$ as a weighted linear combination of nearby training points, but in most calculations a very simple weight distribution is used, specifically the \"nearest neighbor\" weight $1$ put on the nearest neighbor of $S^\\*$ in the dataset. Slightly more discussion on why this choice was made, and how it affects the results, particularly depending on how densely space $\\mathcal{S}$ is covered by the training data. This seems like a somewhat unnatural choice, as it approximates a Lipschitz function $f$ by essentially piecewise constant function on the Voroni diagram given by the data points $S_i$ -- which is far from Lipschitz. The authors briefly mention $K$-nearest neighbors as a different choice; another natural choice would be to use weights proportional to, say, gaussian kernel i. e. $w\\_i \\propto \\exp(-d(S^\\*, S\\_i)^2 / L^2)$; it is slightly more discussion (and potentially a follow-up work) on how this choice affects the results would be great.\n\nQUESTIONS:\nSee \"Strengths and weaknesses\"; Are there some guidelines for choosing the Lipschitz constant for $f$? How likely is it in practice that the space is covered densely enough by the training data to provide useful confidence intervals? What is the effect on the mechanism for the choice of weights for the approximation of $\\mathbb{E}[Y^* | S^*]$, on the results (and especially, how it depends on the density of the sampled points $S_i$ in the space $\\mathcal{S}$?)\n\nLIMITATIONS:\nSome limitations have been addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-10"}