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.
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When complex Bayesian models exhibit implausible behaviour, one solution is to assemble available information into an informative prior. Challenges arise as prior information is often only available for the observable quantity, or some model-derived marginal quantity, rather than directly pertaining to the natural parameters in our model. We propose a method for translating available prior information, in the form of an elicited distribution for the observable or model-derived marginal quantity, into an informative joint prior. Our approach proceeds given a parametric class of prior distributions with as yet undetermined hyperparameters, and minimises the difference between the supplied elicited distribution and corresponding prior predictive distribution. We employ a global, multi-stage Bayesian optimisation procedure to locate optimal values for the hyperparameters. Three examples illustrate our approach: a cure-fraction survival model, where censoring implies that the observable quantity is a priori a mixed discrete/continuous quantity; a setting in which prior information pertains to $R^{2}$ -- a model-derived quantity; and a nonlinear regression model.
Mediation analysis aims to decipher the underlying causal mechanisms between an exposure, an outcome, and intermediate variables called mediators. Initially developed for fixed-time mediator and outcome, it has been extended to the framework of longitudinal data by discretizing the assessment times of mediator and outcome. Yet, processes in play in longitudinal studies are usually defined in continuous time and measured at irregular and subject-specific visits. This is the case in dementia research when cerebral and cognitive changes measured at planned visits in cohorts are of interest. We thus propose a methodology to estimate the causal mechanisms between a time-fixed exposure ($X$), a mediator process ($\mathcal{M}_t$) and an outcome process ($\mathcal{Y}_t$) both measured repeatedly over time in the presence of a time-dependent confounding process ($\mathcal{L}_t$). We consider three types of causal estimands, the natural effects, path-specific effects and randomized interventional analogues to natural effects, and provide identifiability assumptions. We employ a dynamic multivariate model based on differential equations for their estimation. The performance of the methods are explored in simulations, and we illustrate the method in two real-world examples motivated by the 3C cerebral aging study to assess: (1) the effect of educational level on functional dependency through depressive symptomatology and cognitive functioning, and (2) the effect of a genetic factor on cognitive functioning potentially mediated by vascular brain lesions and confounded by neurodegeneration.
We establish an invariance principle for polynomial functions of $n$ independent high-dimensional random vectors, and also show that the obtained rates are nearly optimal. Both the dimension of the vectors and the degree of the polynomial are permitted to grow with $n$. Specifically, we obtain a finite sample upper bound for the error of approximation by a polynomial of Gaussians, measured in Kolmogorov distance, and extend it to functions that are approximately polynomial in a mean squared error sense. We give a corresponding lower bound that shows the invariance principle holds up to polynomial degree $o(\log n)$. The proof is constructive and adapts an asymmetrisation argument due to V. V. Senatov. As applications, we obtain a higher-order delta method with possibly non-Gaussian limits, and generalise a number of known results on high-dimensional and infinite-order U-statistics, and on fluctuations of subgraph counts.
We demonstrate a comprehensive semiparametric approach to causal mediation analysis, addressing the complexities inherent in settings with longitudinal and continuous treatments, confounders, and mediators. Our methodology utilizes a nonparametric structural equation model and a cross-fitted sequential regression technique based on doubly robust pseudo-outcomes, yielding an efficient, asymptotically normal estimator without relying on restrictive parametric modeling assumptions. We are motivated by a recent scientific controversy regarding the effects of invasive mechanical ventilation (IMV) on the survival of COVID-19 patients, considering acute kidney injury (AKI) as a mediating factor. We highlight the possibility of "inconsistent mediation," in which the direct and indirect effects of the exposure operate in opposite directions. We discuss the significance of mediation analysis for scientific understanding and its potential utility in treatment decisions.
In many circumstances given an ordered sequence of one or more types of elements/ symbols, the objective is to determine any existence of randomness in occurrence of one of the elements,say type 1 element. Such a method can be useful in determining existence of any non-random pattern in the wins or loses of a player in a series of games played. Existing methods of tests based on total number of runs or tests based on length of longest run (Mosteller (1941)) can be used for testing the null hypothesis of randomness in the entire sequence, and not a specific type of element. Additionally, the Runs Test tends to show results contradictory to the intuition visualised by the graphs of say, win proportions over time due to method used in computation of runs. This paper develops a test approach to address this problem by computing the gaps between two consecutive type 1 elements and thereafter following the idea of "pattern" in occurrence and "directional" trend (increasing, decreasing or constant), employs the use of exact Binomial test, Kenall's Tau and Siegel-Tukey test for scale problem. Further modifications suggested by Jan Vegelius(1982) have been applied in the Siegel Tukey test to adjust for tied ranks and achieve more accurate results. This approach is distribution-free and suitable for small sizes. Also comparisons with the conventional runs test shows the superiority of the proposed approach under the null hypothesis of randomness in the occurrence of type 1 elements.
The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.
In logistic regression modeling, Firth's modified estimator is widely used to address the issue of data separation, which results in the nonexistence of the maximum likelihood estimate. Firth's modified estimator can be formulated as a penalized maximum likelihood estimator in which Jeffreys' prior is adopted as the penalty term. Despite its widespread use in practice, the formal verification of the corresponding estimate's existence has not been established. In this study, we establish the existence theorem of Firth's modified estimate in binomial logistic regression models, assuming only the full column rankness of the design matrix. We also discuss other binomial regression models obtained through alternating link functions and prove the existence of similar penalized maximum likelihood estimates for such models.
Principal stratification is a popular framework for causal inference in the presence of an intermediate outcome. While the principal average treatment effects have traditionally been the default target of inference, it may not be sufficient when the interest lies in the relative favorability of one potential outcome over the other within the principal stratum. We thus introduce the principal generalized causal effect estimands, which extend the principal average causal effects to accommodate nonlinear contrast functions. Under principal ignorability, we expand the theoretical results in Jiang et. al. (2022) to a much wider class of causal estimands in the presence of a binary intermediate variable. We develop identification formulas and derive the efficient influence functions of the generalized estimands for principal stratification analyses. These efficient influence functions motivate a set of multiply robust estimators and lay the ground for obtaining efficient debiased machine learning estimators via cross-fitting based on $U$-statistics. The proposed methods are illustrated through simulations and the analysis of a data example.
Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.
Existing schemes for demonstrating quantum computational advantage are subject to various practical restrictions, including the hardness of verification and challenges in experimental implementation. Meanwhile, analog quantum simulators have been realized in many experiments to study novel physics. In this work, we propose a quantum advantage protocol based on single-step Feynman-Kitaev verification of an analog quantum simulation, in which the verifier need only run an $O(\lambda^2)$-time classical computation, and the prover need only prepare $O(1)$ samples of a history state and perform $O(\lambda^2)$ single-qubit measurements, for a security parameter $\lambda$. We also propose a near-term feasible strategy for honest provers and discuss potential experimental realizations.
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