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Mathematical modelling has served a central role in studying how infectious disease transmission manifests at the population level. These models have demonstrated the importance of population-level factors like social network heterogeneity on structuring epidemic risk and are now routinely used in public health for decision support. One barrier to broader utility of mathematical models is that the existing canon does not readily accommodate the social determinants of health as distinct, formal drivers of transmission dynamics. Given the decades of empirical support for the organizational effect of social determinants on health burden more generally and infectious disease risk more specially, addressing this modelling gap is of critical importance. In this study, we build on prior efforts to integrate social forces into mathematical epidemiology by introducing several new metrics, principally structural causal influence (SCI). Here, SCI leverages causal analysis to provide a measure of the relative vulnerability of subgroups within a susceptible population, which are crafted by differences in healthcare, exposure to disease, and other determinants. We develop our metrics using a general case and apply it to specific one of public health importance: Hepatitis C virus in a population of persons who inject drugs. Our use of the SCI reveals that, under specific parameters in a multi-community model, the "less vulnerable" community may sustain a basic reproduction number below one when isolated, ensuring disease extinction. However, even minimal transmission between less and more vulnerable communities can elevate this number, leading to sustained epidemics within both communities. Summarizing, we reflect on our findings in light of conversations surrounding the importance of social inequalities and how their consideration can influence the study and practice of mathematical epidemiology., Comment: 8 figures, 8 tables

Tools from topological data analysis have been widely used to represent binary images in many scientific applications. Methods that aim to represent grayscale images (i.e., where pixel intensities instead take on continuous values) have been relatively underdeveloped. In this paper, we introduce the Euler-Radon transform, which generalizes the Euler characteristic transform to grayscale images by using o-minimal structures and Euler integration over definable functions. Coupling the Karhunen-Loeve expansion with our proposed topological representation, we offer hypothesis-testing algorithms based on the chi-squared distribution for detecting significant differences between two groups of grayscale images. We illustrate our framework via extensive numerical experiments and simulations., Comment: 85 pages, 9 figures

Randomized experiments often need to be stopped prematurely due to the treatment having an unintended harmful effect. Existing methods that determine when to stop an experiment early are typically applied to the data in aggregate and do not account for treatment effect heterogeneity. In this paper, we study the early stopping of experiments for harm on heterogeneous populations. We first establish that current methods often fail to stop experiments when the treatment harms a minority group of participants. We then use causal machine learning to develop CLASH, the first broadly-applicable method for heterogeneous early stopping. We demonstrate CLASH's performance on simulated and real data and show that it yields effective early stopping for both clinical trials and A/B tests., Comment: NeurIPS 2023 (spotlight)

The ability to interpret machine learning models has become increasingly important as their usage in data science continues to rise. Most current interpretability methods are optimized to work on either (\textit{i}) a global scale, where the goal is to rank features based on their contributions to overall variation in an observed population, or (\textit{ii}) the local level, which aims to detail on how important a feature is to a particular individual in the data set. In this work, a new operator is proposed called the "GlObal And Local Score" (GOALS): a simple \textit{post hoc} approach to simultaneously assess local and global feature variable importance in nonlinear models. Motivated by problems in biomedicine, the approach is demonstrated using Gaussian process regression where the task of understanding how genetic markers are associated with disease progression both within individuals and across populations is of high interest. Detailed simulations and real data analyses illustrate the flexible and efficient utility of GOALS over state-of-the-art variable importance strategies.

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two chi-squared statistic-based algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a data set of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, analysis of variance for functional data, and geometric morphometrics., Comment: 110 pages

Variational Autoencoders (VAEs) have experienced recent success as data-generating models by using simple architectures that do not require significant fine-tuning of hyperparameters. However, VAEs are known to suffer from over-regularization which can lead to failure to escape local maxima. This phenomenon, known as posterior collapse, prevents learning a meaningful latent encoding of the data. Recent methods have mitigated this issue by deterministically moment-matching an aggregated posterior distribution to an aggregate prior. However, abandoning a probabilistic framework (and thus relying on point estimates) can both lead to a discontinuous latent space and generate unrealistic samples. Here we present Hierarchical Empirical Bayes Autoencoder (HEBAE), a computationally stable framework for probabilistic generative models. Our key contributions are two-fold. First, we make gains by placing a hierarchical prior over the encoding distribution, enabling us to adaptively balance the trade-off between minimizing the reconstruction loss function and avoiding over-regularization. Second, we show that assuming a general dependency structure between variables in the latent space produces better convergence onto the mean-field assumption for improved posterior inference. Overall, HEBAE is more robust to a wide-range of hyperparameter initializations than an analogous VAE. Using data from MNIST and CelebA, we illustrate the ability of HEBAE to generate higher quality samples based on FID score than existing autoencoder-based approaches., Comment: 13 pages

While the success of deep neural networks (DNNs) is well-established across a variety of domains, our ability to explain and interpret these methods is limited. Unlike previously proposed local methods which try to explain particular classification decisions, we focus on global interpretability and ask a universally applicable question: given a trained model, which features are the most important? In the context of neural networks, a feature is rarely important on its own, so our strategy is specifically designed to leverage partial covariance structures and incorporate variable dependence into feature ranking. Our methodological contributions in this paper are two-fold. First, we propose an effect size analogue for DNNs that is appropriate for applications with highly collinear predictors (ubiquitous in computer vision). Second, we extend the recently proposed "RelATive cEntrality" (RATE) measure (Crawford et al., 2019) to the Bayesian deep learning setting. RATE applies an information theoretic criterion to the posterior distribution of effect sizes to assess feature significance. We apply our framework to three broad application areas: computer vision, natural language processing, and social science.

The central aim in this paper is to address variable selection questions in nonlinear and nonparametric regression. Motivated by statistical genetics, where nonlinear interactions are of particular interest, we introduce a novel and interpretable way to summarize the relative importance of predictor variables. Methodologically, we develop the "RelATive cEntrality" (RATE) measure to prioritize candidate genetic variants that are not just marginally important, but whose associations also stem from significant covarying relationships with other variants in the data. We illustrate RATE through Bayesian Gaussian process regression, but the methodological innovations apply to other "black box" methods. It is known that nonlinear models often exhibit greater predictive accuracy than linear models, particularly for phenotypes generated by complex genetic architectures. With detailed simulations and two real data association mapping studies, we show that applying RATE enables an explanation for this improved performance., Comment: 28 pages, 5 figures, 1 tables; Supplementary Material

We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that capture the `shape' of data; however, in practice, they have complex structures that make them difficult to use in statistical settings. The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropical geometric representation of barcodes. This makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations. More specifically, we show that exponential family distributions may be assumed, and that likelihood functions for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to human immunodeficiency virus and avian influenza data., Comment: 31 pages, 5 figures

Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining molecular data make other readily-available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to utilize information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic---the smooth Euler characteristic transform (SECT)---that quantifies magnetic resonance images (MRIs) of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus twofold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics., Comment: 30 pages, 9 figures, 1 table

Nonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models. Variable selection for kernel regression models is a challenge partly because, unlike the linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert spaces (RKHS) to define a linear vector space that: (i) captures nonlinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonlinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and linear models, respectively. BAKR is the first method that is competitive in both settings., Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations presented; references added