The generalized linear mixed model (GLMM) is a popular statistical approach for handling correlated data, and is used extensively in applications areas where big data is common, including biomedical data settings. The focus of this paper is scalable statistical inference for the GLMM, where we define statistical inference as: (i) estimation of population parameters, and (ii) evaluation of scientific hypotheses in the presence of uncertainty. Artificial intelligence (AI) learning algorithms excel at scalable statistical estimation, but rarely include uncertainty quantification. In contrast, Bayesian inference provides full statistical inference, since uncertainty quantification results automatically from the posterior distribution. Unfortunately, Bayesian inference algorithms, including Markov Chain Monte Carlo (MCMC), become computationally intractable in big data settings. In this paper, we introduce a statistical inference algorithm at the intersection of AI and Bayesian inference, that leverages the scalability of modern AI algorithms with guaranteed uncertainty quantification that accompanies Bayesian inference. Our algorithm is an extension of stochastic gradient MCMC with novel contributions that address the treatment of correlated data (i.e., intractable marginal likelihood) and proper posterior variance estimation. Through theoretical and empirical results we establish our algorithm's statistical inference properties, and apply the method in a large electronic health records database.
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We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.
We investigate one/two-sample mean tests for high-dimensional compositional data when the number of variables is comparable with the sample size, as commonly encountered in microbiome research. Existing methods mainly focus on max-type test statistics which are suitable for detecting sparse signals. However, in this paper, we introduce a novel approach using sum-type test statistics which are capable of detecting weak but dense signals. By establishing the asymptotic independence between the max-type and sum-type test statistics, we further propose a combined max-sum type test to cover both cases. We derived the asymptotic null distributions and power functions for these test statistics. Simulation studies demonstrate the superiority of our max-sum type test statistics which exhibit robust performance regardless of data sparsity.
Generalization to new samples is a fundamental rationale for statistical modeling. For this purpose, model validation is particularly important, but recent work in survey inference has suggested that simple aggregation of individual prediction scores does not give a good measure of the score for population aggregate estimates. In this manuscript we explain why this occurs, propose two scoring metrics designed specifically for this problem, and demonstrate their use in three different ways. We show that these scoring metrics correctly order models when compared to the true score, although they do underestimate the magnitude of the score. We demonstrate with a problem in survey research, where multilevel regression and poststratification (MRP) has been used extensively to adjust convenience and low-response surveys to make population and subpopulation estimates.
Determining which parameters of a non-linear model could best describe a set of experimental data is a fundamental problem in science and it has gained much traction lately with the rise of complex large-scale simulators (a.k.a. black-box simulators). The likelihood of such models is typically intractable, which is why classical MCMC methods can not be used. Simulation-based inference (SBI) stands out in this context by only requiring a dataset of simulations to train deep generative models capable of approximating the posterior distribution that relates input parameters to a given observation. In this work, we consider a tall data extension in which multiple observations are available and one wishes to leverage their shared information to better infer the parameters of the model. The method we propose is built upon recent developments from the flourishing score-based diffusion literature and allows us to estimate the tall data posterior distribution simply using information from the score network trained on individual observations. We compare our method to recently proposed competing approaches on various numerical experiments and demonstrate its superiority in terms of numerical stability and computational cost.
In a regression model with multiple response variables and multiple explanatory variables, if the difference of the mean vectors of the response variables for different values of explanatory variables is always in the direction of the first principal eigenvector of the covariance matrix of the response variables, then it is called a multivariate allometric regression model. This paper studies the estimation of the first principal eigenvector in the multivariate allometric regression model. A class of estimators that includes conventional estimators is proposed based on weighted sum-of-squares matrices of regression sum-of-squares matrix and residual sum-of-squares matrix. We establish an upper bound of the mean squared error of the estimators contained in this class, and the weight value minimizing the upper bound is derived. Sufficient conditions for the consistency of the estimators are discussed in weak identifiability regimes under which the difference of the largest and second largest eigenvalues of the covariance matrix decays asymptotically and in "large $p$, large $n$" regimes, where $p$ is the number of response variables and $n$ is the sample size. Several numerical results are also presented.
Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over half a century of research on high-dimensional approximation, this remains a challenging problem. Yet, significant advances have been made in the last decade towards efficient methods for doing this, commencing with so-called sparse polynomial approximation methods and continuing most recently with methods based on Deep Neural Networks (DNNs). In tandem, there have been substantial advances in the relevant approximation theory and analysis of these techniques. In this work, we survey this recent progress. We describe the contemporary motivations for this problem, which stem from parametric models and computational uncertainty quantification; the relevant function classes, namely, classes of infinite-dimensional, Banach-valued, holomorphic functions; fundamental limits of learnability from finite data for these classes; and finally, sparse polynomial and DNN methods for efficiently learning such functions from finite data. For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning. Aiming to narrow this gap, we develop the topic of practical existence theory, which asserts the existence of dimension-independent DNN architectures and training strategies that achieve provably near-optimal generalization errors in terms of the amount of training data.
We consider the computation of model-free bounds for multi-asset options in a setting that combines dependence uncertainty with additional information on the dependence structure. More specifically, we consider the setting where the marginal distributions are known and partial information, in the form of known prices for multi-asset options, is also available in the market. We provide a fundamental theorem of asset pricing in this setting, as well as a superhedging duality that allows to transform the maximization problem over probability measures in a more tractable minimization problem over trading strategies. The latter is solved using a penalization approach combined with a deep learning approximation using artificial neural networks. The numerical method is fast and the computational time scales linearly with respect to the number of traded assets. We finally examine the significance of various pieces of additional information. Empirical evidence suggests that "relevant" information, i.e. prices of derivatives with the same payoff structure as the target payoff, are more useful that other information, and should be prioritized in view of the trade-off between accuracy and computational efficiency.
Many statistical analyses, in both observational data and randomized control trials, ask: how does the outcome of interest vary with combinations of observable covariates? How do various drug combinations affect health outcomes, or how does technology adoption depend on incentives and demographics? Our goal is to partition this factorial space into ``pools'' of covariate combinations where the outcome differs across the pools (but not within a pool). Existing approaches (i) search for a single ``optimal'' partition under assumptions about the association between covariates or (ii) sample from the entire set of possible partitions. Both these approaches ignore the reality that, especially with correlation structure in covariates, many ways to partition the covariate space may be statistically indistinguishable, despite very different implications for policy or science. We develop an alternative perspective, called Rashomon Partition Sets (RPSs). Each item in the RPS partitions the space of covariates using a tree-like geometry. RPSs incorporate all partitions that have posterior values near the maximum a posteriori partition, even if they offer substantively different explanations, and do so using a prior that makes no assumptions about associations between covariates. This prior is the $\ell_0$ prior, which we show is minimax optimal. Given the RPS we calculate the posterior of any measurable function of the feature effects vector on outcomes, conditional on being in the RPS. We also characterize approximation error relative to the entire posterior and provide bounds on the size of the RPS. Simulations demonstrate this framework allows for robust conclusions relative to conventional regularization techniques. We apply our method to three empirical settings: price effects on charitable giving, chromosomal structure (telomere length), and the introduction of microfinance.
When building statistical models for Bayesian data analysis tasks, required and optional iterative adjustments and different modelling choices can give rise to numerous candidate models. In particular, checks and evaluations throughout the modelling process can motivate changes to an existing model or the consideration of alternative models to ultimately obtain models of sufficient quality for the problem at hand. Additionally, failing to consider alternative models can lead to overconfidence in the predictive or inferential ability of a chosen model. The search for suitable models requires modellers to work with multiple models without jeopardising the validity of their results. Multiverse analysis offers a framework for transparent creation of multiple models at once based on different sensible modelling choices, but the number of candidate models arising in the combination of iterations and possible modelling choices can become overwhelming in practice. Motivated by these challenges, this work proposes iterative filtering for multiverse analysis to support efficient and consistent assessment of multiple models and meaningful filtering towards fewer models of higher quality across different modelling contexts. Given that causal constraints have been taken into account, we show how multiverse analysis can be combined with recommendations from established Bayesian modelling workflows to identify promising candidate models by assessing predictive abilities and, if needed, tending to computational issues. We illustrate our suggested approach in different realistic modelling scenarios using real data examples.
Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since Hall, M\"uller and Wang (2006)'s result for a fixed number of eigenfunction estimates. In this work, we aim to establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the $\mathcal{L}^2$ and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the $\mathcal{L}^{2}$ bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. We also develop a double truncation technique to handle the uniform convergence of estimated covariance and eigenfunctions. The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed functional data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.
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