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The effects of treatments may differ between persons with different characteristics. Addressing such treatment heterogeneity is crucial to investigate whether patients with specific characteristics are likely to benefit from a new treatment. The current paper presents a novel Bayesian method for superiority decision-making in the context of randomized controlled trials with multivariate binary responses and heterogeneous treatment effects. The framework is based on three elements: a) Bayesian multivariate logistic regression analysis with a P\'olya-Gamma expansion; b) a transformation procedure to transfer obtained regression coefficients to a more intuitive multivariate probability scale (i.e., success probabilities and the differences between them); and c) a compatible decision procedure for treatment comparison with prespecified decision error rates. Procedures for a priori sample size estimation under a non-informative prior distribution are included. A numerical evaluation demonstrated that decisions based on a priori sample size estimation resulted in anticipated error rates among the trial population as well as subpopulations. Further, average and conditional treatment effect parameters could be estimated unbiasedly when the sample was large enough. Illustration with the International Stroke Trial dataset revealed a trend towards heterogeneous effects among stroke patients: Something that would have remained undetected when analyses were limited to average treatment effects.

We consider assessing causal mediation in the presence of a post-treatment event (examples include noncompliance, a clinical event, or a terminal event). We identify natural mediation effects for the entire study population and for each principal stratum characterized by the joint potential values of the post-treatment event. We derive efficient influence functions for each mediation estimand, which motivate a set of multiply robust estimators for inference. The multiply robust estimators are consistent under four types of misspecifications and are efficient when all nuisance models are correctly specified. We illustrate our methods via simulations and two real data examples.

A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.

We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

Timely implementation of interventions to slow cognitive decline among older adults requires accurate monitoring to detect changes in cognitive function. Data gathered using wearable devices that can continuously monitor factors known to be associated with cognition could be used to train machine learning models and develop wearable-based cognitive monitoring systems. Using data from over 2,400 older adults in the National Health and Nutrition Examination Survey (NHANES) we developed prediction models to differentiate older adults with normal cognition from those with poor cognition based on outcomes from three cognitive tests measuring different domains of cognitive function. During repeated cross-validation, CatBoost, XGBoost, and Random Forest models performed best when predicting cognition based on processing speed, working memory, and attention (median AUCs >0.82) compared to immediate and delayed recall (median AUCs >0.72) and categorical verbal fluency (median AUC >0.68). Activity and sleep parameters were also more strongly associated with processing speed, working memory, and attention compared to other cognitive subdomains. Our work provides proof of concept that wearable-based cognitive monitoring systems may be a viable alternative to traditional methods for monitoring processing speeds, working memory, and attention. We further identified novel metrics that could be targets in future causal studies seeking to better understand how sleep and activity parameters influence cognitive function among older adults.

The timely detection of disease outbreaks through reliable early warning signals (EWSs) is indispensable for effective public health mitigation strategies. Nevertheless, the intricate dynamics of real-world disease spread, often influenced by diverse sources of noise and limited data in the early stages of outbreaks, pose a significant challenge in developing reliable EWSs, as the performance of existing indicators varies with extrinsic and intrinsic noises. Here, we address the challenge of modeling disease when the measurements are corrupted by additive white noise, multiplicative environmental noise, and demographic noise into a standard epidemic mathematical model. To navigate the complexities introduced by these noise sources, we employ a deep learning algorithm that provides EWS in infectious disease outbreak by training on noise-induced disease-spreading models. The indicator's effectiveness is demonstrated through its application to real-world COVID-19 cases in Edmonton and simulated time series derived from diverse disease spread models affected by noise. Notably, the indicator captures an impending transition in a time series of disease outbreaks and outperforms existing indicators. This study contributes to advancing early warning capabilities by addressing the intricate dynamics inherent in real-world disease spread, presenting a promising avenue for enhancing public health preparedness and response efforts.

We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.

Arthroplasty is commonly performed to treat joint osteoarthritis, reducing pain and improving mobility. While arthroplasty has known several technical improvements, a significant share of patients are still unsatisfied with their surgery. Personalised arthroplasty improves surgical outcomes however current solutions require delays, making it difficult to integrate in clinical routine. We propose a fully automated workflow to design patient-specific implants, presented for total knee arthroplasty, the most widely performed arthroplasty in the world nowadays. The proposed pipeline first uses artificial neural networks to segment the proximal and distal extremities of the femur and tibia. Then the full bones are reconstructed using augmented statistical shape models, combining shape and landmarks information. Finally, 77 morphological parameters are computed to design patient-specific implants. The developed workflow has been trained using 91 CT scans of lower limb and evaluated on 41 CT scans manually segmented, in terms of accuracy and execution time. The workflow accuracy was $0.4\pm0.2mm$ for the segmentation, $1.2\pm0.4mm$ for the full bones reconstruction, and $2.8\pm2.2mm$ for the anatomical landmarks determination. The custom implants fitted the patients' anatomy with $0.6\pm0.2mm$ accuracy. The whole process from segmentation to implants' design lasted about 5 minutes. The proposed workflow allows for a fast and reliable personalisation of knee implants, directly from the patient CT image without requiring any manual intervention. It establishes a patient-specific pre-operative planning for TKA in a very short time making it easily available for all patients. Combined with efficient implant manufacturing techniques, this solution could help answer the growing number of arthroplasties while reducing complications and improving the patients' satisfaction.

Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.