Causal inference necessarily relies upon untestable assumptions; hence, it is crucial to assess the robustness of obtained results to violations of identification assumptions. However, such sensitivity analysis is only occasionally undertaken in practice, as many existing methods only apply to relatively simple models and their results are often difficult to interpret. We take a more flexible approach to sensitivity analysis and view it as a constrained stochastic optimization problem. We focus on linear models with an unmeasured confounder and a potential instrument. We show how the $R^2$-calculus - a set of algebraic rules that relates different (partial) $R^2$-values and correlations - can be applied to identify the bias of the $k$-class estimators and construct sensitivity models flexibly. We further show that the heuristic "plug-in" sensitivity interval may not have any confidence guarantees; instead, we propose a boostrap approach to construct sensitivity intervals which perform well in numerical simulations. We illustrate the proposed methods with a real study on the causal effect of education on earnings and provide user-friendly visualization tools.
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The August 2022 special election for U.S. House Representative in Alaska featured three main candidates and was conducted by by single-winner ranked choice voting method known as ``instant runoff voting." The results of this election displayed a well-known but relatively rare phenomenon known as the ``center squeeze:" The most centrist candidate, Mark Begich, was eliminated in the first round despite winning an overwhelming majority of second-place votes. In fact, Begich was the {\em Condorcet winner} of this election: Based on the cast vote record, he would have defeated both of the other two candidates in head-to-head contests, but he was eliminated in the first round of ballot counting due to receiving the fewest first-place votes. The purpose of this paper is to use the data in the cast vote record to explore the range of likely outcomes if this election had been conducted under two alternative voting methods: Approval Voting and STAR (``Score Then Automatic Runoff") Voting. We find that under the best assumptions available about voter behavior, the most likely outcomes are that Peltola would still have won the election under Approval Voting, while Begich would have won under STAR Voting.
Subjective image quality measures based on deep neural networks are very related to models of visual neuroscience. This connection benefits engineering but, more interestingly, the freedom to optimize deep networks in different ways, make them an excellent tool to explore the principles behind visual perception (both human and artificial). Recently, a myriad of networks have been successfully optimized for many interesting visual tasks. Although these nets were not specifically designed to predict image quality or other psychophysics, they have shown surprising human-like behavior. The reasons for this remain unclear. In this work, we perform a thorough analysis of the perceptual properties of pre-trained nets (particularly their ability to predict image quality) by isolating different factors: the goal (the function), the data (learning environment), the architecture, and the readout: selected layer(s), fine-tuning of channel relevance, and use of statistical descriptors as opposed to plain readout of responses. Several conclusions can be drawn. All the models correlate better with human opinion than SSIM. More importantly, some of the nets are in pair of state-of-the-art with no extra refinement or perceptual information. Nets trained for supervised tasks such as classification correlate substantially better with humans than LPIPS (a net specifically tuned for image quality). Interestingly, self-supervised tasks such as jigsaw also perform better than LPIPS. Simpler architectures are better than very deep nets. In simpler nets, correlation with humans increases with depth as if deeper layers were closer to human judgement. This is not true in very deep nets. Consistently with reports on illusions and contrast sensitivity, small changes in the image environment does not make a big difference. Finally, the explored statistical descriptors and concatenations had no major impact.
Different statistical samples (e.g., from different locations) offer populations and learning systems observations with distinct statistical properties. Samples under (1) 'Unconfounded' growth preserve systems' ability to determine the independent effects of their individual variables on any outcome-of-interest (and lead, therefore, to fair and interpretable black-box predictions). Samples under (2) 'Externally-Valid' growth preserve their ability to make predictions that generalize across out-of-sample variation. The first promotes predictions that generalize over populations, the second over their shared exogeneous factors. We illustrate these theoretic patterns in the full American census from 1840 to 1940, and samples ranging from the street-level all the way to the national. This reveals sample requirements for generalizability over space, and new connections among the Shapley value, U-Statistics (Unbiased Statistics), and Hyperbolic Geometry.
The spectral density matrix is a fundamental object of interest in time series analysis, and it encodes both contemporary and dynamic linear relationships between component processes of the multivariate system. In this paper we develop novel inference procedures for the spectral density matrix in the high-dimensional setting. Specifically, we introduce a new global testing procedure to test the nullity of the cross-spectral density for a given set of frequencies and across pairs of component indices. For the first time, both Gaussian approximation and parametric bootstrap methodologies are employed to conduct inference for a high-dimensional parameter formulated in the frequency domain, and new technical tools are developed to provide asymptotic guarantees of the size accuracy and power for global testing. We further propose a multiple testing procedure for simultaneously testing the nullity of the cross-spectral density at a given set of frequencies. The method is shown to control the false discovery rate. Both numerical simulations and a real data illustration demonstrate the usefulness of the proposed testing methods.
Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonparametric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covariates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.
Stochastic gradient descent (SGD) is a scalable and memory-efficient optimization algorithm for large datasets and stream data, which has drawn a great deal of attention and popularity. The applications of SGD-based estimators to statistical inference such as interval estimation have also achieved great success. However, most of the related works are based on i.i.d. observations or Markov chains. When the observations come from a mixing time series, how to conduct valid statistical inference remains unexplored. As a matter of fact, the general correlation among observations imposes a challenge on interval estimation. Most existing methods may ignore this correlation and lead to invalid confidence intervals. In this paper, we propose a mini-batch SGD estimator for statistical inference when the data is $\phi$-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, we show that the proposed estimator is asymptotically normal, and its limiting distribution can be effectively approximated by the bootstrap procedure. The proposed method is memory-efficient and easy to implement in practice. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.
Non-asymptotic statistical analysis is often missing for modern geometry-aware machine learning algorithms due to the possibly intricate non-linear manifold structure. This paper studies an intrinsic mean model on the manifold of restricted positive semi-definite matrices and provides a non-asymptotic statistical analysis of the Karcher mean. We also consider a general extrinsic signal-plus-noise model, under which a deterministic error bound of the Karcher mean is provided. As an application, we show that the distributed principal component analysis algorithm, LRC-dPCA, achieves the same performance as the full sample PCA algorithm. Numerical experiments lend strong support to our theories.
Grid-free Monte Carlo methods such as \emph{walk on spheres} can be used to solve elliptic partial differential equations without mesh generation or global solves. However, such methods independently estimate the solution at every point, and hence do not take advantage of the high spatial regularity of solutions to elliptic problems. We propose a fast caching strategy which first estimates solution values and derivatives at randomly sampled points along the boundary of the domain (or a local region of interest). These cached values then provide cheap, output-sensitive evaluation of the solution (or its gradient) at interior points, via a boundary integral formulation. Unlike classic boundary integral methods, our caching scheme introduces zero statistical bias and does not require a dense global solve. Moreover we can handle imperfect geometry (e.g., with self-intersections) and detailed boundary/source terms without repairing or resampling the boundary representation. Overall, our scheme is similar in spirit to \emph{virtual point light} methods from photorealistic rendering: it suppresses the typical salt-and-pepper noise characteristic of independent Monte Carlo estimates, while still retaining the many advantages of Monte Carlo solvers: progressive evaluation, trivial parallelization, geometric robustness, \etc{}\ We validate our approach using test problems from visual and geometric computing.
Models with high-dimensional parameter spaces are common in many applications. Global sensitivity analyses can provide insights on how uncertain inputs and interactions influence the outputs. Many sensitivity analysis methods face nontrivial challenges for computationally demanding models. Common approaches to tackle these challenges are to (i) use a computationally efficient emulator and (ii) sample adaptively. However, these approaches still involve potentially large computational costs and approximation errors. Here we compare the results and computational costs of four existing global sensitivity analysis methods applied to a test problem. We sample different model evaluation time and numbers of model parameters. We find that the emulation and adaptive sampling approaches are faster than Sobol' method for slow models. The Bayesian adaptive spline surface method is the fastest for most slow and high-dimensional models. Our results can guide the choice of a sensitivity analysis method under computational resources constraints.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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