The posterior covariance matrix W defined by the log-likelihood of each observation plays important roles both in the sensitivity analysis and frequencist's evaluation of the Bayesian estimators. This study focused on the matrix W and its principal space; we term the latter as an essential subspace. First, it is shown that they appear in various statistical settings, such as the evaluation of the posterior sensitivity, assessment of the frequencist's uncertainty from posterior samples, and stochastic expansion of the loss; a key tool to treat frequencist's properties is the recently proposed Bayesian infinitesimal jackknife approximation (Giordano and Broderick (2023)). In the following part, we show that the matrix W can be interpreted as a reproducing kernel; it is named as W-kernel. Using the W-kernel, the essential subspace is expressed as a principal space given by the kernel PCA. A relation to the Fisher kernel and neural tangent kernel is established, which elucidates the connection to the classical asymptotic theory; it also leads to a sort of Bayesian-frequencist's duality. Finally, two applications, selection of a representative set of observations and dimensional reduction in the approximate bootstrap, are discussed. In the former, incomplete Cholesky decomposition is introduced as an efficient method to compute the essential subspace. In the latter, different implementations of the approximate bootstrap for posterior means are compared.
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We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.
Building robust, interpretable, and secure AI system requires quantifying and representing uncertainty under a probabilistic perspective to mimic human cognitive abilities. However, probabilistic computation presents significant challenges for most conventional artificial neural network, as they are essentially implemented in a deterministic manner. In this paper, we develop an efficient probabilistic computation framework by truncating the probabilistic representation of neural activation up to its mean and covariance and construct a moment neural network that encapsulates the nonlinear coupling between the mean and covariance of the underlying stochastic network. We reveal that when only the mean but not the covariance is supervised during gradient-based learning, the unsupervised covariance spontaneously emerges from its nonlinear coupling with the mean and faithfully captures the uncertainty associated with model predictions. Our findings highlight the inherent simplicity of probabilistic computation by seamlessly incorporating uncertainty into model prediction, paving the way for integrating it into large-scale AI systems.
Ensuring that refugees and asylum seekers thrive (e.g., find employment) in their host countries is a profound humanitarian goal, and a primary driver of employment is the geographic location within a host country to which the refugee or asylum seeker is assigned. Recent research has proposed and implemented algorithms that assign refugees and asylum seekers to geographic locations in a manner that maximizes the average employment across all arriving refugees. While these algorithms can have substantial overall positive impact, using data from two industry collaborators we show that the impact of these algorithms can vary widely across key subgroups based on country of origin, age, or educational background. Thus motivated, we develop a simple and interpretable framework for incorporating group fairness into the dynamic refugee assignment problem. In particular, the framework can flexibly incorporate many existing and future definitions of group fairness from the literature (e.g., maxmin, randomized, and proportionally-optimized within-group). Equipped with our framework, we propose two bid-price algorithms that maximize overall employment while simultaneously yielding provable group fairness guarantees. Through extensive numerical experiments using various definitions of group fairness and real-world data from the U.S. and the Netherlands, we show that our algorithms can yield substantial improvements in group fairness compared to an offline benchmark fairness constraints, with only small relative decreases ($\approx$ 1%-5%) in global performance.
Scattering networks yield powerful and robust hierarchical image descriptors which do not require lengthy training and which work well with very few training data. However, they rely on sampling the scale dimension. Hence, they become sensitive to scale variations and are unable to generalize to unseen scales. In this work, we define an alternative feature representation based on the Riesz transform. We detail and analyze the mathematical foundations behind this representation. In particular, it inherits scale equivariance from the Riesz transform and completely avoids sampling of the scale dimension. Additionally, the number of features in the representation is reduced by a factor four compared to scattering networks. Nevertheless, our representation performs comparably well for texture classification with an interesting addition: scale equivariance. Our method yields superior performance when dealing with scales outside of those covered by the training dataset. The usefulness of the equivariance property is demonstrated on the digit classification task, where accuracy remains stable even for scales four times larger than the one chosen for training. As a second example, we consider classification of textures.
Physical activity plays a significant role in the well-being of individuals with Chronic obstructive Pulmonary Disease (COPD). Specifically, it has been directly associated with changes in hospitalization rates for these patients. However, previous investigations have primarily been conducted in a cross-sectional or longitudinal manner and have not considered a continuous perspective. Using the telEPOC program we use telemonitoring data to analyze the impact of physical activity adopting a functional data approach. However, Traditional functional data methods, including functional regression models, typically assume a consistent data domain. However, the data in the telEPOC program exhibits variable domains, presenting a challenge since the majority of functional data methods, are based on the fact that data are observed in the same domain. To address this challenge, we introduce a novel fully functional methodology tailored to variable domain functional data, eliminating the need for data alignment, which can be computationally taxing. Although models designed for variable domain data are relatively scarce and may have inherent limitations in their estimation methods, our approach circumvents these issues. We substantiate the effectiveness of our methodology through a simulation study, comparing our results with those obtained using established methodologies. Finally, we apply our methodology to analyze the impact of physical activity in COPD patients using the telEPOC program's data. Software for our method is available in the form of R code on request at \url{//github.com/Pavel-Hernadez-Amaro/V.D.F.R.M-new-estimation-approach.git}.
We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem.
Predictions under interventions are estimates of what a person's risk of an outcome would be if they were to follow a particular treatment strategy, given their individual characteristics. Such predictions can give important input to medical decision making. However, evaluating predictive performance of interventional predictions is challenging. Standard ways of evaluating predictive performance do not apply when using observational data, because prediction under interventions involves obtaining predictions of the outcome under conditions that are different to those that are observed for a subset of individuals in the validation dataset. This work describes methods for evaluating counterfactual performance of predictions under interventions for time-to-event outcomes. This means we aim to assess how well predictions would match the validation data if all individuals had followed the treatment strategy under which predictions are made. We focus on counterfactual performance evaluation using longitudinal observational data, and under treatment strategies that involve sustaining a particular treatment regime over time. We introduce an estimation approach using artificial censoring and inverse probability weighting which involves creating a validation dataset that mimics the treatment strategy under which predictions are made. We extend measures of calibration, discrimination (c-index and cumulative/dynamic AUCt) and overall prediction error (Brier score) to allow assessment of counterfactual performance. The methods are evaluated using a simulation study, including scenarios in which the methods should detect poor performance. Applying our methods in the context of liver transplantation shows that our procedure allows quantification of the performance of predictions supporting crucial decisions on organ allocation.
This paper proposes a new approach to fit a linear regression for symbolic internal-valued variables, which improves both the Center Method suggested by Billard and Diday in \cite{BillardDiday2000} and the Center and Range Method suggested by Lima-Neto, E.A. and De Carvalho, F.A.T. in \cite{Lima2008, Lima2010}. Just in the Centers Method and the Center and Range Method, the new methods proposed fit the linear regression model on the midpoints and in the half of the length of the intervals as an additional variable (ranges) assumed by the predictor variables in the training data set, but to make these fitments in the regression models, the methods Ridge Regression, Lasso, and Elastic Net proposed by Tibshirani, R. Hastie, T., and Zou H in \cite{Tib1996, HastieZou2005} are used. The prediction of the lower and upper of the interval response (dependent) variable is carried out from their midpoints and ranges, which are estimated from the linear regression models with shrinkage generated in the midpoints and the ranges of the interval-valued predictors. Methods presented in this document are applied to three real data sets cardiologic interval data set, Prostate interval data set and US Murder interval data set to then compare their performance and facility of interpretation regarding the Center Method and the Center and Range Method. For this evaluation, the root-mean-squared error and the correlation coefficient are used. Besides, the reader may use all the methods presented herein and verify the results using the {\tt RSDA} package written in {\tt R} language, that can be downloaded and installed directly from {\tt CRAN} \cite{Rod2014}.
In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.
Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.
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