Savje (2023) recommends misspecified exposure effects as a way to avoid strong assumptions about interference when analyzing the results of an experiment. In this discussion, we highlight a key limitation of Savje's recommendation. Exposure effects are not generally useful for evaluating social policies without the strong assumptions that Savje seeks to avoid. Our discussion is organized as follows. Section 2 summarizes our position, section 3 provides a concrete example, and section 4 concludes. Proof of claims are in an appendix.
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We study parameterized and approximation algorithms for a variant of Set Cover, where the universe of elements to be covered consists of points in the plane and the sets with which the points should be covered are segments. We call this problem Segment Set Cover. We also consider a relaxation of the problem called $\delta$-extension, where we need to cover the points by segments that are extended by a tiny fraction, but we compare the solution's quality to the optimum without extension. For the unparameterized variant, we prove that Segment Set Cover does not admit a PTAS unless $\mathsf{P}=\mathsf{NP}$, even if we restrict segments to be axis-parallel and allow $\frac{1}{2}$-extension. On the other hand, we show that parameterization helps for the tractability of Segment Set Cover: we give an FPT algorithm for unweighted Segment Set Cover parameterized by the solution size $k$, a parameterized approximation scheme for Weighted Segment Set Cover with $k$ being the parameter, and an FPT algorithm for Weighted Segment Set Cover with $\delta$-extension parameterized by $k$ and $\delta$. In the last two results, relaxing the problem is probably necessary: we prove that Weighted Segment Set Cover without any relaxation is $\mathsf{W}[1]$-hard and, assuming ETH, there does not exist an algorithm running in time $f(k)\cdot n^{o(k / \log k)}$. This holds even if one restricts attention to axis-parallel segments.
It has been argued that the models used to analyze data from crossover designs are not appropriate when simple carryover effects are assumed. In this paper, the estimability conditions of the carryover effects are found, and a theoretical result that supports them, additionally, two simulation examples are developed in a non-linear dose-response for a repeated measures crossover trial in two designs: the traditional AB/BA design and a Williams square. Both show that a semiparametric model can detect complex carryover effects and that this estimation improves the precision of treatment effect estimators. We concluded that when there are at least five replicates in each observation period per individual, semiparametric statistical models provide a good estimator of the treatment effect and reduce bias with respect to models that assume either, the absence of carryover or simplex carryover effects. In addition, an application of the methodology is shown and the richness in analysis that is gained by being able to estimate complex carryover effects is evident.
In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.
Motivated by optimization with differential equations, we consider optimization problems with Hilbert spaces as decision spaces. As a consequence of their infinite dimensionality, the numerical solution necessitates finite dimensional approximations and discretizations. We develop an approximation framework and demonstrate criticality measure-based error estimates. We consider criticality measures inspired by those used within optimization methods, such as semismooth Newton and (conditional) gradient methods. Furthermore, we show that our error estimates are order-optimal. Our findings augment existing distance-based error estimates, but do not rely on strong convexity or second-order sufficient optimality conditions. Moreover, our error estimates can be used for code verification and validation. We illustrate our theoretical convergence rates on linear, semilinear, and bilinear PDE-constrained optimization.
Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. We propose a novel procedure relying on the general concept of retractions to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. We analyze the well-posedness of the method by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. We extend to the manifold setting a classical result on the asymptotic interpolation error of Hermite interpolation. We finally illustrate these results and the effectiveness of the method with numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns.
This essay provides a comprehensive analysis of the optimization and performance evaluation of various routing algorithms within the context of computer networks. Routing algorithms are critical for determining the most efficient path for data transmission between nodes in a network. The efficiency, reliability, and scalability of a network heavily rely on the choice and optimization of its routing algorithm. This paper begins with an overview of fundamental routing strategies, including shortest path, flooding, distance vector, and link state algorithms, and extends to more sophisticated techniques.
The ability to extract material parameters of perovskite from quantitative experimental analysis is essential for rational design of photovoltaic and optoelectronic applications. However, the difficulty of this analysis increases significantly with the complexity of the theoretical model and the number of material parameters for perovskite. Here we use Gaussian process to develop an analysis platform that can extract up to 8 fundamental material parameters of an organometallic perovskite semiconductor from a transient photoluminescence experiment, based on a complex full physics model that includes drift-diffusion of carriers and dynamic defect occupation. An example study of thermal degradation reveals that changes in doping concentration and carrier mobility dominate, while the defect energy level remains nearly unchanged. This platform can be conveniently applied to other experiments or to combinations of experiments, accelerating materials discovery and optimization of semiconductor materials for photovoltaics and other applications.
Living organisms interact with their surroundings in a closed-loop fashion, where sensory inputs dictate the initiation and termination of behaviours. Even simple animals are able to develop and execute complex plans, which has not yet been replicated in robotics using pure closed-loop input control. We propose a solution to this problem by defining a set of discrete and temporary closed-loop controllers, called "tasks", each representing a closed-loop behaviour. We further introduce a supervisory module which has an innate understanding of physics and causality, through which it can simulate the execution of task sequences over time and store the results in a model of the environment. On the basis of this model, plans can be made by chaining temporary closed-loop controllers. The proposed framework was implemented for a real robot and tested in two scenarios as proof of concept.
Mendelian randomization uses genetic variants as instrumental variables to make causal inferences about the effects of modifiable risk factors on diseases from observational data. One of the major challenges in Mendelian randomization is that many genetic variants are only modestly or even weakly associated with the risk factor of interest, a setting known as many weak instruments. Many existing methods, such as the popular inverse-variance weighted (IVW) method, could be biased when the instrument strength is weak. To address this issue, the debiased IVW (dIVW) estimator, which is shown to be robust to many weak instruments, was recently proposed. However, this estimator still has non-ignorable bias when the effective sample size is small. In this paper, we propose a modified debiased IVW (mdIVW) estimator by multiplying a modification factor to the original dIVW estimator. After this simple correction, we show that the bias of the mdIVW estimator converges to zero at a faster rate than that of the dIVW estimator under some regularity conditions. Moreover, the mdIVW estimator has smaller variance than the dIVW estimator.We further extend the proposed method to account for the presence of instrumental variable selection and balanced horizontal pleiotropy. We demonstrate the improvement of the mdIVW estimator over the dIVW estimator through extensive simulation studies and real data analysis.
In observational studies, covariates with substantial missing data are often omitted, despite their strong predictive capabilities. These excluded covariates are generally believed not to simultaneously affect both treatment and outcome, indicating that they are not genuine confounders and do not impact the identification of the average treatment effect (ATE). In this paper, we introduce an alternative doubly robust (DR) estimator that fully leverages non-confounding predictive covariates to enhance efficiency, while also allowing missing values in such covariates. Beyond the double robustness property, our proposed estimator is designed to be more efficient than the standard DR estimator. Specifically, when the propensity score model is correctly specified, it achieves the smallest asymptotic variance among the class of DR estimators, and brings additional efficiency gains by further integrating predictive covariates. Simulation studies demonstrate the notable performance of the proposed estimator over current popular methods. An illustrative example is provided to assess the effectiveness of right heart catheterization (RHC) for critically ill patients.
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