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Classic Bayesian methods with complex models are frequently infeasible due to an intractable likelihood. Simulation-based inference methods, such as Approximate Bayesian Computing (ABC), calculate posteriors without accessing a likelihood function by leveraging the fact that data can be quickly simulated from the model, but converge slowly and/or poorly in high-dimensional settings. In this paper, we propose a framework for Bayesian posterior estimation by mapping data to posteriors of parameters using a neural network trained on data simulated from the complex model. Posterior distributions of model parameters are efficiently obtained by feeding observed data into the trained neural network. We show theoretically that our posteriors converge to the true posteriors in Kullback-Leibler divergence. Our approach yields computationally efficient and theoretically justified uncertainty quantification, which is lacking in existing simulation-based neural network approaches. Comprehensive simulation studies highlight our method's robustness and accuracy.

Motivated by the need to model the dependence between regions of interest in functional neuroconnectivity for efficient inference, we propose a new sampling-based Bayesian clustering approach for covariance structures of high-dimensional Gaussian outcomes. The key technique is based on a Dirichlet process that clusters covariance sub-matrices into independent groups of outcomes, thereby naturally inducing sparsity in the whole brain connectivity matrix. A new split-merge algorithm is employed to achieve convergence of the Markov chain that is shown empirically to recover both uniform and Dirichlet partitions with high accuracy. We investigate the empirical performance of the proposed method through extensive simulations. Finally, the proposed approach is used to group regions of interest into functionally independent groups in the Autism Brain Imaging Data Exchange participants with autism spectrum disorder and and co-occurring attention-deficit/hyperactivity disorder., Comment: 23 pages, 8 figures, 2 tables

Motivated by the need for computationally tractable spatial methods in neuroimaging studies, we develop a distributed and integrated framework for estimation and inference of Gaussian process model parameters with ultra-high-dimensional likelihoods. We propose a shift in viewpoint from whole to local data perspectives that is rooted in distributed model building and integrated estimation and inference. The framework's backbone is a computationally and statistically efficient integration procedure that simultaneously incorporates dependence within and between spatial resolutions in a recursively partitioned spatial domain. Statistical and computational properties of our distributed approach are investigated theoretically and in simulations. The proposed approach is used to extract new insights on autism spectrum disorder from the Autism Brain Imaging Data Exchange., Comment: 21 pages, 4 figures, 5 tables

The recent explosion of genetic and high dimensional biobank and 'omic' data has provided researchers with the opportunity to investigate the shared genetic origin (pleiotropy) of hundreds to thousands of related phenotypes. However, existing methods for multi-phenotype genome-wide association studies (GWAS) do not model pleiotropy, are only applicable to a small number of phenotypes, or provide no way to perform inference. To add further complication, raw genetic and phenotype data are rarely observed, meaning analyses must be performed on GWAS summary statistics whose statistical properties in high dimensions are poorly understood. We therefore developed a novel model, theoretical framework, and set of methods to perform Bayesian inference in GWAS of high dimensional phenotypes using summary statistics that explicitly model pleiotropy, beget fast computation, and facilitate the use of biologically informed priors. We demonstrate the utility of our procedure by applying it to metabolite GWAS, where we develop new nonparametric priors for genetic effects on metabolite levels that use known metabolic pathway information and foster interpretable inference at the pathway level., Comment: 24 pages of main text, 7 figures, 1 table

Transfer learning uses a data model, trained to make predictions or inferences on data from one population, to make reliable predictions or inferences on data from another population. Most existing transfer learning approaches are based on fine-tuning pre-trained neural network models, and fail to provide crucial uncertainty quantification. We develop a statistical framework for model predictions based on transfer learning, called RECaST. The primary mechanism is a Cauchy random effect that recalibrates a source model to a target population; we mathematically and empirically demonstrate the validity of our RECaST approach for transfer learning between linear models, in the sense that prediction sets will achieve their nominal stated coverage, and we numerically illustrate the method's robustness to asymptotic approximations for nonlinear models. Whereas many existing techniques are built on particular source models, RECaST is agnostic to the choice of source model. For example, our RECaST transfer learning approach can be applied to a continuous or discrete data model with linear or logistic regression, deep neural network architectures, etc. Furthermore, RECaST provides uncertainty quantification for predictions, which is mostly absent in the literature. We examine our method's performance in a simulation study and in an application to real hospital data., Comment: 26 pages, 2 figures, 4 tables

Motivated by image-on-scalar regression with data aggregated across multiple sites, we consider a setting in which multiple independent studies each collect multiple dependent vector outcomes, with potential mean model parameter homogeneity between studies and outcome vectors. To determine the validity of jointly analyzing these data sources, we must learn which of these data sources share mean model parameters. We propose a new model fusion approach that delivers improved flexibility, statistical performance and computational speed over existing methods. Our proposed approach specifies a quadratic inference function within each data source and fuses mean model parameter vectors in their entirety based on a new formulation of a pairwise fusion penalty. We establish theoretical properties of our estimator and propose an asymptotically equivalent weighted oracle meta-estimator that is more computationally efficient. Simulations and application to the ABIDE neuroimaging consortium highlight the flexibility of the proposed approach. An R package is provided for ease of implementation., Comment: 28 pages, 4 figures

Modern longitudinal data, for example from wearable devices, measures biological signals on a fixed set of participants at a diverging number of time points. Traditional statistical methods are not equipped to handle the computational burden of repeatedly analyzing the cumulatively growing dataset each time new data is collected. We propose a new estimation and inference framework for dynamic updating of point estimates and their standard errors across serially collected dependent datasets. The key technique is a decomposition of the extended score function of the quadratic inference function constructed over the cumulative longitudinal data into a sum of summary statistics over data batches. We show how this sum can be recursively updated without the need to access the whole dataset, resulting in a computationally efficient streaming procedure with minimal loss of statistical efficiency. We prove consistency and asymptotic normality of our streaming estimator as the number of data batches diverges, even as the number of independent participants remains fixed. Simulations highlight the advantages of our approach over traditional statistical methods that assume independence between data batches. Finally, we investigate the relationship between physical activity and several diseases through the analysis of accelerometry data from the National Health and Nutrition Examination Survey., Comment: 18 pages, 2 figures. Biometrika (2023)

Estimation and inference with modern longitudinal data from wearable devices, which consist of biological signals at high-frequency time points, is burdened by massive computational costs. We propose a distributed estimation and inference procedure that efficiently estimates both functional and scalar parameters with intensively measured longitudinal outcomes. The procedure overcomes computational difficulties through a scalable divide-and-conquer algorithm that partitions the outcomes into smaller sets. We circumvent traditional basis selection problems by analyzing data using quadratic inference functions in smaller subsets such that the basis functions have a low dimension. To address the challenges of combining estimates from dependent subsets, we propose a statistically efficient one-step estimator derived from a constrained generalized method of moments objective function with a smoothing penalty. We show theoretically and numerically that the proposed estimator is as statistically efficient as non-distributed alternative approaches and more efficient computationally. We demonstrate the practicality of our approach with the analysis of accelerometer data from the National Health and Nutrition Examination Survey., Comment: 28 pages. 1 table. 6 figures

A fundamental aspect of statistics is the integration of data from different sources. Classically, Fisher and others were focused on how to integrate homogeneous (or only mildly heterogeneous) sets of data. More recently, as data are becoming more accessible, the question of if data sets from different sources should be integrated is becoming more relevant. The current literature treats this as a question with only two answers: integrate or don't. Here we take a different approach, motivated by information-sharing principles coming from the shrinkage estimation literature. In particular, we deviate from the do/don't perspective and propose a dial parameter that controls the extent to which two data sources are integrated. How far this dial parameter should be turned is shown to depend, for example, on the informativeness of the different data sources as measured by Fisher information. In the context of generalized linear models, this more nuanced data integration framework leads to relatively simple parameter estimates and valid tests/confidence intervals. Moreover, we demonstrate both theoretically and empirically that setting the dial parameter according to our recommendation leads to more efficient estimation compared to other binary data integration schemes., Comment: 46 pages, 10 figures, 15 tables

Extreme environmental events frequently exhibit spatial and temporal dependence. These data are often modeled using max stable processes (MSPs). MSPs are computationally prohibitive to fit for as few as a dozen observations, with supposed computationally-efficient approaches like the composite likelihood remaining computationally burdensome with a few hundred observations. In this paper, we propose a spatial partitioning approach based on local modeling of subsets of the spatial domain that delivers computationally and statistically efficient inference. Marginal and dependence parameters of the MSP are estimated locally on subsets of observations using censored pairwise composite likelihood, and combined using a modified generalized method of moments procedure. The proposed distributed approach is extended to estimate spatially varying coefficient models to deliver computationally efficient modeling of spatial variation in marginal parameters. We demonstrate consistency and asymptotic normality of estimators, and show empirically that our approach leads to a surprising reduction in bias of parameter estimates over a full data approach. We illustrate the flexibility and practicability of our approach through simulations and the analysis of streamflow data from the U.S. Geological Survey., Comment: 41 pages, 9 figures, 4 tables

Two dominant distributed computing strategies have emerged to overcome the computational bottleneck of supervised learning with big data: parallel data processing in the MapReduce paradigm and serial data processing in the online streaming paradigm. Despite the two strategies' common divide-and-combine approach, they differ in how they aggregate information, leading to different trade-offs between statistical and computational performance. In this paper, we propose a new hybrid paradigm, termed a Parallel-and-Stream Accelerator (PASA), that uses the strengths of both strategies for computationally fast and statistically efficient supervised learning. PASA's architecture nests online streaming processing into each distributed and parallelized data process in a MapReduce framework. PASA leverages the advantages and mitigates the disadvantages of both the MapReduce and online streaming approaches to deliver a more flexible paradigm satisfying practical computing needs. We study the analytic properties and computational complexity of PASA, and detail its implementation for two key statistical learning tasks. We illustrate its performance through simulations and a large-scale data example building a prediction model for online purchases from advertising data., Comment: 22 pages, 3 figures

We propose a distributed quadratic inference function framework to jointly estimate regression parameters from multiple potentially heterogeneous data sources with correlated vector outcomes. The primary goal of this joint integrative analysis is to estimate covariate effects on all outcomes through a marginal regression model in a statistically and computationally efficient way. We develop a data integration procedure for statistical estimation and inference of regression parameters that is implemented in a fully distributed and parallelized computational scheme. To overcome computational and modeling challenges arising from the high-dimensional likelihood of the correlated vector outcomes, we propose to analyze each data source using Qu, Lindsay and Li (2000)'s quadratic inference functions, and then to jointly reestimate parameters from each data source by accounting for correlation between data sources using a combined meta-estimator in a similar spirit to Hansen (1982)'s generalised method of moments. We show both theoretically and numerically that the proposed method yields efficiency improvements and is computationally fast. We illustrate the proposed methodology with the joint integrative analysis of the association between smoking and metabolites in a large multi-cohort study and provide an R package for ease of implementation., Comment: 26 pages, 4 figures, 4 tables

This paper presents a unified framework for supervised learning and inference procedures using the divide-and-conquer approach for high-dimensional correlated outcomes. We propose a general class of estimators that can be implemented in a fully distributed and parallelized computational scheme. Modelling, computational and theoretical challenges related to high-dimensional correlated outcomes are overcome by dividing data at both outcome and subject levels, estimating the parameter of interest from blocks of data using a broad class of supervised learning procedures, and combining block estimators in a closed-form meta-estimator asymptotically equivalent to estimates obtained by Hansen (1982)'s generalized method of moments (GMM) that does not require the entire data to be reloaded on a common server. We provide rigorous theoretical justifications for the use of distributed estimators with correlated outcomes by studying the asymptotic behaviour of the combined estimator with fixed and diverging number of data divisions. Simulations illustrate the finite sample performance of the proposed method, and we provide an R package for ease of implementation., Comment: 49 pages, 1 figure

This paper is motivated by a regression analysis of electroencephalography (EEG) neuroimaging data with high-dimensional correlated responses with multi-level nested correlations. We develop a divide-and-conquer procedure implemented in a fully distributed and parallelized computational scheme for statistical estimation and inference of regression parameters. Despite significant efforts in the literature, the computational bottleneck associated with high-dimensional likelihoods prevents the scalability of existing methods. The proposed method addresses this challenge by dividing responses into subvectors to be analyzed separately and in parallel on a distributed platform using pairwise composite likelihood. Theoretical challenges related to combining results from dependent data are overcome in a statistically efficient way using a meta-estimator derived from Hansen's generalized method of moments. We provide a rigorous theoretical framework for efficient estimation, inference, and goodness-of-fit tests. We develop an R package for ease of implementation. We illustrate our method's performance with simulations and the analysis of the EEG data, and find that iron deficiency is significantly associated with two auditory recognition memory related potentials in the left parietal-occipital region of the brain., Comment: 35 pages, 5 figures, 3 tables