Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz's consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matern covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data.
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Standard neural networks struggle to generalize under distribution shifts in computer vision. Fortunately, combining multiple networks can consistently improve out-of-distribution generalization. In particular, weight averaging (WA) strategies were shown to perform best on the competitive DomainBed benchmark; they directly average the weights of multiple networks despite their nonlinearities. In this paper, we propose Diverse Weight Averaging (DiWA), a new WA strategy whose main motivation is to increase the functional diversity across averaged models. To this end, DiWA averages weights obtained from several independent training runs: indeed, models obtained from different runs are more diverse than those collected along a single run thanks to differences in hyperparameters and training procedures. We motivate the need for diversity by a new bias-variance-covariance-locality decomposition of the expected error, exploiting similarities between WA and standard functional ensembling. Moreover, this decomposition highlights that WA succeeds when the variance term dominates, which we show occurs when the marginal distribution changes at test time. Experimentally, DiWA consistently improves the state of the art on DomainBed without inference overhead.
Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.
Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.
Prior work has identified a resilient phenomenon that threatens the performance of human-AI decision-making teams: overreliance, when people agree with an AI, even when it is incorrect. Surprisingly, overreliance does not reduce when the AI produces explanations for its predictions, compared to only providing predictions. Some have argued that overreliance results from cognitive biases or uncalibrated trust, attributing overreliance to an inevitability of human cognition. By contrast, our paper argues that people strategically choose whether or not to engage with an AI explanation, demonstrating empirically that there are scenarios where AI explanations reduce overreliance. To achieve this, we formalize this strategic choice in a cost-benefit framework, where the costs and benefits of engaging with the task are weighed against the costs and benefits of relying on the AI. We manipulate the costs and benefits in a maze task, where participants collaborate with a simulated AI to find the exit of a maze. Through 5 studies (N = 731), we find that costs such as task difficulty (Study 1), explanation difficulty (Study 2, 3), and benefits such as monetary compensation (Study 4) affect overreliance. Finally, Study 5 adapts the Cognitive Effort Discounting paradigm to quantify the utility of different explanations, providing further support for our framework. Our results suggest that some of the null effects found in literature could be due in part to the explanation not sufficiently reducing the costs of verifying the AI's prediction.
Multi-channel imaging data is a prevalent data format in scientific fields such as astronomy and biology. The structured information and the high dimensionality of these 3-D tensor data makes the analysis an intriguing but challenging topic for statisticians and practitioners. The low-rank scalar-on-tensor regression model, in particular, has received widespread attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP) model with multi-linear kernel in Yu et al. (2018). In this paper, we extend the Tensor-GP model by integrating a dimensionality reduction technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor regression task with multi-channel imaging data. This is motivated by the solar flare forecasting problem with high dimensional multi-channel imaging data. We first estimate a latent, reduced-size tensor for each data tensor and then apply a multi-linear Tensor-GP on the latent tensor data for prediction. We introduce an anisotropic total-variation regularization when conducting the tensor contraction to obtain a sparse and smooth latent tensor. We then propose an alternating proximal gradient descent algorithm for estimation. We validate our approach via extensive simulation studies and applying it to the solar flare forecasting problem.
Motivated by environmental policy questions, we address the challenges of estimation, change point detection, and uncertainty quantification of a causal exposure-response function (CERF). Under a potential outcome framework, the CERF describes the relationship between a continuously varying exposure (or treatment) and its causal effect on an outcome. We propose a new Bayesian approach that relies on a Gaussian process (GP) model to estimate the CERF nonparametrically. To achieve the desired separation of design and analysis phases, we parametrize the covariance (kernel) function of the GP to mimic matching via a Generalized Propensity Score (GPS). The hyper-parameters as well as the form of the kernel function of the GP are chosen to optimize covariate balance. Our approach achieves automatic uncertainty evaluation of the CERF with high computational efficiency, and enables change point detection through inference on derivatives of the CERF. We provide theoretical results showing the correspondence between our Bayesian GP framework and traditional approaches in causal inference for estimating causal effects of a continuous exposure. We apply the methods to 520,711 ZIP-code-level observations to estimate the causal effect of long-term exposures to PM2.5, ozone, and NO2 on all-cause mortality among Medicare enrollees in the US. A computationally efficient implementation of the proposed GP models is provided in the GPCERF R package, which is available on CRAN.
Nowadays model uncertainty has become one of the most important problems in both academia and industry. In this paper, we mainly consider the scenario in which we have a common model set used for model averaging instead of selecting a single final model via a model selection procedure to account for this model's uncertainty to improve the reliability and accuracy of inferences. Here one main challenge is to learn the prior over the model set. To tackle this problem, we propose two data-based algorithms to get proper priors for model averaging. One is for meta-learner, the analysts should use historical similar tasks to extract the information about the prior. The other one is for base-learner, a subsampling method is used to deal with the data step by step. Theoretically, an upper bound of risk for our algorithm is presented to guarantee the performance of the worst situation. In practice, both methods perform well in simulations and real data studies, especially with poor-quality data.
For predictive evaluation based on quasi-posterior distributions, we develop a new information criterion, the posterior covariance information criterion (PCIC. PCIC generalises the widely applicable information criterion WAIC so as to effectively handle predictive scenarios where likelihoods for the estimation and the evaluation of the model may be different. A typical example of such scenarios is the weighted likelihood inference, including prediction under covariate shift and counterfactual prediction. The proposed criterion utilises a posterior covariance form and is computed by using only one Markov chain Monte Carlo run. Through numerical examples, we demonstrate how PCIC can apply in practice. Further, we show that PCIC is asymptotically unbiased to the quasi-Bayesian generalization error under mild conditions in weighted inference with both regular and singular statistical models.
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as the finite element method, the outputs of which consist of the solutions on a set of mesh nodes over the spatial domain. However, these simulations are often prohibitively costly to survey the input space. In this paper, we propose an efficient emulator that simultaneously predicts the outputs on a set of mesh nodes, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits a Gaussian process model in each. Most importantly, by revealing the underlying clustering structures, the proposed method can extract valuable flow physics present in the systems that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and provides valuable insights about the underlying physics of the systems. An R package for the proposed methodology is provided in an open repository.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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