We study the problem of constructing an estimator of the average treatment effect (ATE) that exhibits doubly-robust asymptotic linearity (DRAL). This is a stronger requirement than doubly-robust consistency. A DRAL estimator can yield asymptotically valid Wald-type confidence intervals even when the propensity score or the outcome model is inconsistently estimated. On the contrary, the celebrated doubly-robust, augmented-IPW (AIPW) estimator generally requires consistent estimation of both nuisance functions for standard root-n inference. We make three main contributions. First, we propose a new hybrid class of distributions that consists of the structure-agnostic class introduced in Balakrishnan et al (2023) with additional smoothness constraints. While DRAL is generally not possible in the pure structure-agnostic class, we show that it can be attained in the new hybrid one. Second, we calculate minimax lower bounds for estimating the ATE in the new class, as well as in the pure structure-agnostic one. Third, building upon the literature on doubly-robust inference (van der Laan, 2014, Benkeser et al, 2017, Dukes et al 2021), we propose a new estimator of the ATE that enjoys DRAL. Under certain conditions, we show that its rate of convergence in the new class can be much faster than that achieved by the AIPW estimator and, in particular, matches the minimax lower bound rate, thereby establishing its optimality. Finally, we clarify the connection between DRAL estimators and those based on higher-order influence functions (Robins et al, 2017) and complement our theoretical findings with simulations.
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Stress and material deformation field predictions are among the most important tasks in computational mechanics. These predictions are typically made by solving the governing equations of continuum mechanics using finite element analysis, which can become computationally prohibitive considering complex microstructures and material behaviors. Machine learning (ML) methods offer potentially cost effective surrogates for these applications. However, existing ML surrogates are either limited to low-dimensional problems and/or do not provide uncertainty estimates in the predictions. This work proposes an ML surrogate framework for stress field prediction and uncertainty quantification for diverse materials microstructures. A modified Bayesian U-net architecture is employed to provide a data-driven image-to-image mapping from initial microstructure to stress field with prediction (epistemic) uncertainty estimates. The Bayesian posterior distributions for the U-net parameters are estimated using three state-of-the-art inference algorithms: the posterior sampling-based Hamiltonian Monte Carlo method and two variational approaches, the Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A systematic comparison of the predictive accuracy and uncertainty estimates for these methods is performed for a fiber reinforced composite material and polycrystalline microstructure application. It is shown that the proposed methods yield predictions of high accuracy compared to the FEA solution, while uncertainty estimates depend on the inference approach. Generally, the Hamiltonian Monte Carlo and Bayes by Backprop methods provide consistent uncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the other hand, are more difficult to interpret and depend strongly on the method's design.
We introduce an algorithm that simplifies the construction of efficient estimators, making them accessible to a broader audience. 'Dimple' takes as input computer code representing a parameter of interest and outputs an efficient estimator. Unlike standard approaches, it does not require users to derive a functional derivative known as the efficient influence function. Dimple avoids this task by applying automatic differentiation to the statistical functional of interest. Doing so requires expressing this functional as a composition of primitives satisfying a novel differentiability condition. Dimple also uses this composition to determine the nuisances it must estimate. In software, primitives can be implemented independently of one another and reused across different estimation problems. We provide a proof-of-concept Python implementation and showcase through examples how it allows users to go from parameter specification to efficient estimation with just a few lines of code.
After decades of attention, emergence continues to lack a centralized mathematical definition that leads to a rigorous emergence test applicable to physical flocks and swarms, particularly those containing both deterministic elements (eg, interactions) and stochastic perturbations like measurement noise. This study develops a heuristic test based on singular value curve analysis of data matrices containing deterministic and Gaussian noise signals. The minimum detection criteria are identified, and statistical and matrix space analysis developed to determine upper and lower bounds. This study applies the analysis to representative examples by using recorded trajectories of mixed deterministic and stochastic trajectories for multi-agent, cellular automata, and biological video. Examples include Cucker Smale and Vicsek flocking, Gaussian noise and its integration, recorded observations of bird flocking, and 1D cellular automata. Ensemble simulations including measurement noise are performed to compute statistical variation and discussed relative to random matrix theory noise bounds. The results indicate singular knee analysis of recorded trajectories can detect gradated levels on a continuum of structure and noise. Across the eight singular value decay metrics considered, the angle subtended at the singular value knee emerges with the most potential for supporting cross-embodiment emergence detection, the size of noise bounds is used as an indication of required sample size, and the presence of a large fraction of singular values inside noise bounds as an indication of noise.
We study the problem of parametric estimation for continuously observed stochastic differential equation driven by fractional Brownian motion. Under some assumptions on drift and diffusion coefficients, we construct maximum likelihood estimator and establish its the asymptotic normality and moment convergence of the drift parameter when a small dispersion coefficient vanishes.
We study the numerical approximation of advection-diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method. The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problem-dependent basis set, itself pre-computed in an offline stage. The approach is implemented here using basis functions that locally resolve both the diffusion and the advection terms. Variants with additional bubble functions and possibly weak inter-element continuity are proposed. Some theoretical arguments and a comprehensive set of numerical experiments allow to investigate and compare the stability and the accuracy of the approaches. The best approach constructed is shown to be adequate for both the diffusion- and advection-dominated regimes, and does not rely on an auxiliary stabilization parameter that would have to be properly adjusted.
Difference-in-differences (DID) is a popular approach to identify the causal effects of treatments and policies in the presence of unmeasured confounding. DID identifies the sample average treatment effect in the treated (SATT). However, a goal of such research is often to inform decision-making in target populations outside the treated sample. Transportability methods have been developed to extend inferences from study samples to external target populations; these methods have primarily been developed and applied in settings where identification is based on conditional independence between the treatment and potential outcomes, such as in a randomized trial. We present a novel approach to identifying and estimating effects in a target population, based on DID conducted in a study sample that differs from the target population. We present a range of assumptions under which one may identify causal effects in the target population and employ causal diagrams to illustrate these assumptions. In most realistic settings, results depend critically on the assumption that any unmeasured confounders are not effect measure modifiers on the scale of the effect of interest (e.g., risk difference, odds ratio). We develop several estimators of transported effects, including g-computation, inverse odds weighting, and a doubly robust estimator based on the efficient influence function. Simulation results support theoretical properties of the proposed estimators. As an example, we apply our approach to study the effects of a 2018 US federal smoke-free public housing law on air quality in public housing across the US, using data from a DID study conducted in New York City alone.
Low-Rank Tensor Completion, a method which exploits the inherent structure of tensors, has been studied extensively as an effective approach to tensor completion. Whilst such methods attained great success, none have systematically considered exploiting the numerical priors of tensor elements. Ignoring numerical priors causes loss of important information regarding the data, and therefore prevents the algorithms from reaching optimal accuracy. Despite the existence of some individual works which consider ad hoc numerical priors for specific tasks, no generalizable frameworks for incorporating numerical priors have appeared. We present the Generalized CP Decomposition Tensor Completion (GCDTC) framework, the first generalizable framework for low-rank tensor completion that takes numerical priors of the data into account. We test GCDTC by further proposing the Smooth Poisson Tensor Completion (SPTC) algorithm, an instantiation of the GCDTC framework, whose performance exceeds current state-of-the-arts by considerable margins in the task of non-negative tensor completion, exemplifying GCDTC's effectiveness. Our code is open-source.
The study of intelligent systems explains behaviour in terms of economic rationality. This results in an optimization principle involving a function or utility, which states that the system will evolve until the configuration of maximum utility is achieved. Recently, this theory has incorporated constraints, i.e., the optimum is achieved when the utility is maximized while respecting some information-processing constraints. This is reminiscent of thermodynamic systems. As such, the study of intelligent systems has benefited from the tools of thermodynamics. The first aim of this thesis is to clarify the applicability of these results in the study of intelligent systems. We can think of the local transition steps in thermodynamic or intelligent systems as being driven by uncertainty. In fact, the transitions in both systems can be described in terms of majorization. Hence, real-valued uncertainty measures like Shannon entropy are simply a proxy for their more involved behaviour. More in general, real-valued functions are fundamental to study optimization and complexity in the order-theoretic approach to several topics, including economics, thermodynamics, and quantum mechanics. The second aim of this thesis is to improve on this classification. The basic similarity between thermodynamic and intelligent systems is based on an uncertainty notion expressed by a preorder. We can also think of the transitions in the steps of a computational process as a decision-making procedure. In fact, by adding some requirements on the considered order structures, we can build an abstract model of uncertainty reduction that allows to incorporate computability, that is, to distinguish the objects that can be constructed by following a finite set of instructions from those that cannot. The third aim of this thesis is to clarify the requirements on the order structure that allow such a framework.
Agglomeration techniques are important to reduce the computational costs of numerical simulations and stand at the basis of multilevel algebraic solvers. To automatically perform the agglomeration of polyhedral grids, we propose a novel Geometrical Deep Learning-based algorithm that can exploit the geometrical and physical information of the underlying computational domain to construct the agglomerated grid and simultaneously guarantee the agglomerated grid's quality. In particular, we propose a bisection model based on Graph Neural Networks (GNNs) to partition a suitable connectivity graph of computational three-dimensional meshes. The new approach has a high online inference speed and can simultaneously process the graph structure of the mesh, the geometrical information of the mesh (e.g. elements' volumes, centers' coordinates), and the physical information of the domain (e.g. physical parameters). Taking advantage of this new approach, our algorithm can agglomerate meshes of a domain composed of heterogeneous media in an automatic way. The proposed GNN techniques are compared with the k-means algorithm and METIS: standard approaches for graph partitioning that are meant to process only the connectivity information on the mesh. We demonstrate that the performance of our algorithms outperforms available approaches in terms of quality metrics and runtimes. Moreover, we demonstrate that our algorithm also shows a good level of generalization when applied to more complex geometries, such as three-dimensional geometries reconstructed from medical images. Finally, the capabilities of the model in performing agglomeration of heterogeneous domains are tested in the framework of problems containing microstructures and on a complex geometry such as the human brain.
We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constrains and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non computable.
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