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Abstract Parallel Markov Chain Monte Carlo (pMCMC) algorithms generate clouds of proposals at each step to efficiently resolve a target probability distribution $$\mu $$. We build a rigorous foundational framework for pMCMC algorithms that situates these methods within a unified ‘extended phase space’ measure-theoretic formalism. Drawing on our recent work that provides a comprehensive theory for reversible single-proposal methods, we herein derive general criteria for multiproposal acceptance mechanisms that yield ergodic chains on general state spaces. Our formulation encompasses a variety of methodologies, including proposal cloud resampling and Hamiltonian methods, while providing a basis for the derivation of novel algorithms. In particular, we obtain a top-down picture for a class of methods arising from ‘conditionally independent’ proposal structures. As an immediate application of this formalism, we identify several new algorithms including a multiproposal version of the popular preconditioned Crank–Nicolson (pCN) sampler suitable for high- and infinite-dimensional target measures that are absolutely continuous with respect to a Gaussian base measure. To supplement the aforementioned theoretical results, we carry out a selection of numerical case studies that evaluate the efficacy of these novel algorithms. First, noting that the true potential of pMCMC algorithms arises from their natural parallelizability and the ease with which they map to modern high-performance computing architectures, we provide a limited parallelization study using TensorFlow and a graphics processing unit to scale pMCMC algorithms that leverage as many as 100k proposals at each step. Second, we use our multiproposal pCN algorithm (mpCN) to resolve a selection of problems in Bayesian statistical inversion for partial differential equations motivated by fluid measurement. These examples provide preliminary evidence of the efficacy of mpCN for high-dimensional target distributions featuring complex geometries and multimodal structures. 

Abstract We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct observations of the flow field as well as the measurement of concentrations of a solute passively advected by and diffusing within the flow. Adopting a statistical approach provides estimates of uncertainty in the shape due both to the non-invertibility of the forward map and to error in the measurements. When the shape represents a design problem of attempting to match desired target outcomes, this ‘uncertainty’ can be interpreted as identifying remaining degrees of freedom available to the designer. We demonstrate the viability of our framework on three concrete test problems. These problems illustrate the promise of our framework for applications while providing a collection of test cases for recently developed Markov chain Monte Carlo algorithms designed to resolve infinite-dimensional statistical quantities. 

ABSTRACT Modern engineered systems, and learning‐based systems, in particular, provide unprecedented complexity that requires advancement in our methods to achieve confidence in mission success through test and evaluation (T&E). We define learning‐based systems as engineered systems that incorporate a learning algorithm (artificial intelligence) component of the overall system. A part of the unparalleled complexity is the rate at which learning‐based systems change over traditional engineered systems. Where traditional systems are expected to steadily decline (change) in performance due to time (aging), learning‐based systems undergo a constant change which must be better understood to achieve high confidence in mission success. To this end, we propose pairing Bayesian methods with systems theory to quantify changes in operational conditions, changes in adversarial actions, resultant changes in the learning‐based system structure, and resultant confidence measures in mission success. We provide insights, in this article, into our overall goal and progress toward developing a framework for evaluation through an understanding of equivalence of testing. 

Training deep learning models requires having the right data for the problem and understanding both your data and the models’ performance on that data. Training deep learning models is difficult when data are limited, so in this paper, we seek to answer the following question: how can we train a deep learning model to increase its performance on a targeted area with limited data? We do this by applying rotation data augmentations to a simulated synthetic aperture radar (SAR) image dataset. We use the Uniform Manifold Approximation and Projection (UMAP) dimensionality reduction technique to understand the effects of augmentations on the data in latent space. Using this latent space representation, we can understand the data and choose specific training samples aimed at boosting model performance in targeted under-performing regions without the need to increase training set sizes. Results show that using latent space to choose training data significantly improves model performance in some cases; however, there are other cases where no improvements are made. We show that linking patterns in latent space is a possible predictor of model performance, but results require some experimentation and domain knowledge to determine the best options. 

Although the United States Safe Drinking Water Act (SDWA) theoretically ensures drinking water quality, recent studies have questioned the reliability and equity associated with community water system (CWS) service. This study aimed to identify SDWA violation differences (i.e., monitoring and reporting (MR) and health-based (HB)) between Virginia CWSs given associated service demographics, rurality, and system characteristics. A novel geospatial methodology delineated CWS service areas at the zip code scale to connect 2000 US Census demographics with 2006–2016 SDWA violations, with significant associations determined via negative binomial regression. The proportion of Black Americans within a service area was positively associated with the likelihood of HB violations. This effort supports the need for further investigation of racial and socioeconomic disparities in access to safe drinking water within the United States in particular and offers a geospatial strategy to explore demographics in other settings where data on infrastructure extents are limited. Further interdisciplinary efforts at multiple scales are necessary to identify the entwined causes for differential risks in adverse drinking water quality exposures and would be substantially strengthened by the mapping of official CWS service boundaries.