Model-based clustering of moderate or large dimensional data is notoriously difficult. We propose a model for simultaneous dimensionality reduction and clustering by assuming a mixture model for a set of latent scores, which are then linked to the observations via a Gaussian latent factor model. This approach was recently investigated by Chandra et al. (2023). The authors use a factor-analytic representation and assume a mixture model for the latent factors. However, performance can deteriorate in the presence of model misspecification. Assuming a repulsive point process prior for the component-specific means of the mixture for the latent scores is shown to yield a more robust model that outperforms the standard mixture model for the latent factors in several simulated scenarios. The repulsive point process must be anisotropic to favor well-separated clusters of data, and its density should be tractable for efficient posterior inference. We address these issues by proposing a general construction for anisotropic determinantal point processes. We illustrate our model in simulations as well as a plant species co-occurrence dataset.
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Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.
We introduce a new observational setting for Positive Unlabeled (PU) data where the observations at prediction time are also labeled. This occurs commonly in practice -- we argue that the additional information is important for prediction, and call this task "augmented PU prediction". We allow for labeling to be feature dependent. In such scenario, Bayes classifier and its risk is established and compared with a risk of a classifier which for unlabeled data is based only on predictors. We introduce several variants of the empirical Bayes rule in such scenario and investigate their performance. We emphasise dangers (and ease) of applying classical classification rule in the augmented PU scenario -- due to no preexisting studies, an unaware researcher is prone to skewing the obtained predictions. We conclude that the variant based on recently proposed variational autoencoder designed for PU scenario works on par or better than other considered variants and yields advantage over feature-only based methods in terms of accuracy for unlabeled samples.
In contemporary problems involving genetic or neuroimaging data, thousands of hypotheses need to be tested. Due to their high power, and finite sample guarantees on type-1 error under weak assumptions, Monte-Carlo permutation tests are often considered as gold standard for these settings. However, the enormous computational effort required for (thousands of) permutation tests is a major burden. Recently, Fischer and Ramdas (2024) constructed a permutation test for a single hypothesis in which the permutations are drawn sequentially one-by-one and the testing process can be stopped at any point without inflating the type I error. They showed that the number of permutations can be substantially reduced (under null and alternative) while the power remains similar. We show how their approach can be modified to make it suitable for a broad class of multiple testing procedures. In particular, we discuss its use with the Benjamini-Hochberg procedure and illustrate the application on a large dataset.
Recently, data depth has been widely used to rank multivariate data. The study of the depth-based $Q$ statistic, originally proposed by Liu and Singh (1993), has become increasingly popular when it can be used as a quality index to differentiate between two samples. Based on the existing theoretical foundations, more and more variants have been developed for increasing power in the two sample test. However, the asymptotic expansion of the $Q$ statistic in the important foundation work of Zuo and He (2006) currently has an optimal rate $m^{-3/4}$ slower than the target $m^{-1}$, leading to limitations in higher-order expansions for developing more powerful tests. We revisit the existing assumptions and add two new plausible assumptions to obtain the target rate by applying a new proof method based on the Hoeffding decomposition and the Cox-Reid expansion. The aim of this paper is to rekindle interest in asymptotic data depth theory, to place Q-statistical inference on a firmer theoretical basis, to show its variants in current research, to open the door to the development of new theories for further variants requiring higher-order expansions, and to explore more of its potential applications.
We prove an abstract convergence result for a family of dual-mesh based quadrature rules on tensor products of simplical meshes. In the context of the multilinear tensor-product finite element discretization of reaction-drift-diffusion equations, our quadrature rule generalizes the mass-lump rule, retaining its most useful properties; for a nonnegative reaction coefficient, it gives an $O(h^2)$-accurate, nonnegative diagonalization of the reaction operator. The major advantage of our scheme in comparison with the standard mass lumping scheme is that, under mild conditions, it produces an $O(h^2)$ consistency error even when the integrand has a jump discontinuity. The finite-volume-type quadrature rule has been stated in a less general form and applied to systems of reaction-diffusion equations related to particle-based stochastic reaction-diffusion simulations (PBSRD); in this context, the reaction operator is \textit{required} to be an $M$-matrix and a standard model for bimolecular reactions has a discontinuous reaction coefficient. We apply our convergence results to a finite element discretization of scalar drift-diffusion-reaction model problem related to PBSRD systems, and provide new numerical convergence studies confirming the theory.
The discretization of fluid-poromechanics systems is typically highly demanding in terms of computational effort. This is particularly true for models of multiphysics flows in the brain, due to the geometrical complexity of the cerebral anatomy - requiring a very fine computational mesh for finite element discretization - and to the high number of variables involved. Indeed, this kind of problems can be modeled by a coupled system encompassing the Stokes equations for the cerebrospinal fluid in the brain ventricles and Multiple-network Poro-Elasticity (MPE) equations describing the brain tissue, the interstitial fluid, and the blood vascular networks at different space scales. The present work aims to rigorously derive a posteriori error estimates for the coupled Stokes-MPE problem, as a first step towards the design of adaptive refinement strategies or reduced order models to decrease the computational demand of the problem. Through numerical experiments, we verify the reliability and optimal efficiency of the proposed a posteriori estimator and identify the role of the different solution variables in its composition.
Electromagnetic stimulation probes and modulates the neural systems that control movement. Key to understanding their effects is the muscle recruitment curve, which maps evoked potential size against stimulation intensity. Current methods to estimate curve parameters require large samples; however, obtaining these is often impractical due to experimental constraints. Here, we present a hierarchical Bayesian framework that accounts for small samples, handles outliers, simulates high-fidelity data, and returns a posterior distribution over curve parameters that quantify estimation uncertainty. It uses a rectified-logistic function that estimates motor threshold and outperforms conventionally used sigmoidal alternatives in predictive performance, as demonstrated through cross-validation. In simulations, our method outperforms non-hierarchical models by reducing threshold estimation error on sparse data and requires fewer participants to detect shifts in threshold compared to frequentist testing. We present two common use cases involving electrical and electromagnetic stimulation data and provide an open-source library for Python, called hbMEP, for diverse applications.
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization which results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading therefore to a corrective advective term which cancels out non-reciprocity at the leading order but not at the second order.
A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.
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