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In the context of a binary treatment, matching is a well-established approach in causal inference. However, in the context of a continuous treatment or exposure, matching is still underdeveloped. We propose an innovative matching approach to estimate an average causal exposure-response function under the setting of continuous exposures that relies on the generalized propensity score (GPS). Our approach maintains the following attractive features of matching: a) clear separation between the design and the analysis; b) robustness to model misspecification or to the presence of extreme values of the estimated GPS; c) straightforward assessment of covariate balance. We first introduce an assumption of identifiability, called local weak unconfoundedness. Under this assumption and mild smoothness conditions, we provide theoretical guarantees that our proposed matching estimator attains point-wise consistency and asymptotic normality. In simulations, our proposed matching approach outperforms existing methods under settings of model misspecification or the presence of extreme values of the estimated GPS. We apply our proposed method to estimate the average causal exposure-response function between long-term PM$_{2.5}$ exposure and all-cause mortality among 68.5 million Medicare enrollees, 2000-2016. We found strong evidence of a harmful effect of long-term PM$_{2.5}$ exposure on mortality. Code for the proposed matching approach is provided in the CausalGPS R package, which is available on CRAN and provides a computationally efficient implementation.

We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signals. Moreover we also show that if the true vector of quantiles is a discrete spline with a few polynomial pieces then both versions attain a near parametric rate of convergence. Corresponding results for the usual trend filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality we also employ our proof techniques to obtain new results for multivariate quantile total variation denoising and high dimensional quantile linear regression.

Machine learning (ML) approaches are used more and more widely in biodiversity monitoring. In particular, an important application is the problem of predicting biodiversity indicators such as species abundance, species occurrence or species richness, based on predictor sets containing, e.g., climatic and anthropogenic factors. Considering the impressive number of different ML methods available in the litterature and the pace at which they are being published, it is crucial to develop uniform evaluation procedures, to allow the production of sound and fair empirical studies. However, defining fair evaluation procedures is challenging: because well-documented, intrinsic properties of biodiversity indicators such as their zero-inflation and over-dispersion, it is not trivial to design good sampling schemes for cross-validation nor good evaluation metrics. Indeed, the classical Mean Squared Error (MSE) fails to capture subtle differences in the performance of different methods, particularly in terms of prediction of very small, or very large values (e.g., zero counts or large counts). In this report, we illustrate this phenomenon by comparing ten statistical and machine learning models on the task of predicting waterbirds abundance in the North-African area, based on geographical, meteorological and spatio-temporal factors. Our results highlight that differnte off-the-shelf evaluation metrics and cross-validation sampling approaches yield drastically different rankings of the metrics, and fail to capture interpretable conclusions.

The network interference model for causal inference places all experimental units at the vertices of an undirected exposure graph, such that treatment assigned to one unit may affect the outcome of another unit if and only if these two units are connected by an edge. This model has recently gained popularity as means of incorporating interference effects into the Neyman--Rubin potential outcomes framework; and several authors have considered estimation of various causal targets, including the direct and indirect effects of treatment. In this paper, we consider large-sample asymptotics for treatment effect estimation under network interference in a setting where the exposure graph is a random draw from a graphon. When targeting the direct effect, we show that -- in our setting -- popular estimators are considerably more accurate than existing results suggest, and provide a central limit theorem in terms of moments of the graphon. Meanwhile, when targeting the indirect effect, we leverage our generative assumptions to propose a consistent estimator in a setting where no other consistent estimators are currently available. We also show how our results can be used to conduct a practical assessment of the sensitivity of randomized study inference to potential interference effects. Overall, our results highlight the promise of random graph asymptotics in understanding the practicality and limits of causal inference under network interference.

Machine unlearning is the task of updating machine learning (ML) models after a subset of the training data they were trained on is deleted. Methods for the task are desired to combine effectiveness and efficiency, i.e., they should effectively "unlearn" deleted data, but in a way that does not require excessive computation effort (e.g., a full retraining) for a small amount of deletions. Such a combination is typically achieved by tolerating some amount of approximation in the unlearning. In addition, laws and regulations in the spirit of "the right to be forgotten" have given rise to requirements for certifiability, i.e., the ability to demonstrate that the deleted data has indeed been unlearned by the ML model. In this paper, we present an experimental study of the three state-of-the-art approximate unlearning methods for linear models and demonstrate the trade-offs between efficiency, effectiveness and certifiability offered by each method. In implementing the study, we extend some of the existing works and describe a common ML pipeline to compare and evaluate the unlearning methods on six real-world datasets and a variety of settings. We provide insights into the effect of the quantity and distribution of the deleted data on ML models and the performance of each unlearning method in different settings. We also propose a practical online strategy to determine when the accumulated error from approximate unlearning is large enough to warrant a full retrain of the ML model.

Unobserved confounding is a fundamental obstacle to establishing valid causal conclusions from observational data. Two complementary types of approaches have been developed to address this obstacle. An extensive line of work is based on taking advantage of fortuitous external aids (such as the presence of an instrumental variable or other proxy), along with additional assumptions to ensure identification. A recent line of work of proximal causal inference (Miao et al., 2018a) has aimed to provide a novel approach to using proxies to deal with unobserved confounding without relying on stringent parametric assumptions. On the other hand, a complete characterization of identifiability of a large class of causal parameters in arbitrary causal models with hidden variables has been developed using the language of graphical models, resulting in the ID algorithm and related extensions (Tian and Pearl, 2002; Shpitser and Pearl, 2006a,b). Celebrated special cases of this approach, such as the front-door model, are able to obtain non-parametric identification in seemingly counter-intuitive situations when a treatment and an outcome share an arbitrarily complicated unobserved common cause. In this paper we aim to develop a synthesis of the proximal and graphical approaches to identification in causal inference to yield the most general identification algorithm in multi- variate systems currently known - the proximal ID algorithm. In addition to being able to obtain non-parametric identification in all cases where the ID algorithm succeeds, our approach allows us to systematically exploit proxies to adjust for the presence of unobserved confounders that would have otherwise prevented identification. In addition, we outline a class of estimation strategies for causal parameters identified by our method in an important special case. We illustration our approach by simulation studies.

We propose straightforward nonparametric estimators for the mean and the covariance functions of functional data. Our setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The definition of our nonparametric estimators depends on the local regularity of the stochastic process generating the functional data. We first propose a simple estimator of this local regularity which takes strength from the replication and regularization features of functional data. Next, we use the "smoothing first, then estimate" approach for the mean and the covariance functions. The new nonparametric estimators achieve optimal rates of convergence. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon a real data example on household power consumption illustrate the effectiveness of the new approach.

In the absence of a randomized experiment, a key assumption for drawing causal inference about treatment effects is the ignorable treatment assignment. Violations of the ignorability assumption may lead to biased treatment effect estimates. Sensitivity analysis helps gauge how causal conclusions will be altered in response to the potential magnitude of departure from the ignorability assumption. However, sensitivity analysis approaches for unmeasured confounding in the context of multiple treatments and binary outcomes are scarce. We propose a flexible Monte Carlo sensitivity analysis approach for causal inference in such settings. We first derive the general form of the bias introduced by unmeasured confounding, with emphasis on theoretical properties uniquely relevant to multiple treatments. We then propose methods to encode the impact of unmeasured confounding on potential outcomes and adjust the estimates of causal effects in which the presumed unmeasured confounding is removed. Our proposed methods embed nested multiple imputation within the Bayesian framework, which allow for seamless integration of the uncertainty about the values of the sensitivity parameters and the sampling variability, as well as use of the Bayesian Additive Regression Trees for modeling flexibility. Expansive simulations validate our methods and gain insight into sensitivity analysis with multiple treatments. We use the SEER-Medicare data to demonstrate sensitivity analysis using three treatments for early stage non-small cell lung cancer. The methods developed in this work are readily available in the R package SAMTx.

Approximate Message Passing (AMP) algorithms have seen widespread use across a variety of applications. However, the precise forms for their Onsager corrections and state evolutions depend on properties of the underlying random matrix ensemble, limiting the extent to which AMP algorithms derived for white noise may be applicable to data matrices that arise in practice. In this work, we study more general AMP algorithms for random matrices $W$ that satisfy orthogonal rotational invariance in law, where $W$ may have a spectral distribution that is different from the semicircle and Marcenko-Pastur laws characteristic of white noise. The Onsager corrections and state evolutions in these algorithms are defined by the free cumulants or rectangular free cumulants of the spectral distribution of $W$. Their forms were derived previously by Opper, \c{C}akmak, and Winther using non-rigorous dynamic functional theory techniques, and we provide rigorous proofs. Our motivating application is a Bayes-AMP algorithm for Principal Components Analysis, when there is prior structure for the principal components (PCs) and possibly non-white noise. For sufficiently large signal strengths and any non-Gaussian prior distributions for the PCs, we show that this algorithm provably achieves higher estimation accuracy than the sample PCs.

Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches.