D. Andrew Brown

Arvind K. Saibaba, Johnathan Bardsley, D. Andrew Brown et al.

Hierarchical models in Bayesian inverse problems are characterized by an assumed prior probability distribution for the unknown state and measurement error precision, and hyper-priors for the prior parameters. Combining these probability models using Bayes' law often yields a posterior distribution that cannot be sampled from directly, even for a linear model with Gaussian measurement error and Gaussian prior. Gibbs sampling can be used to sample from the posterior, but problems arise when the dimension of the state is large. This is because the Gaussian sample required for each iteration can be prohibitively expensive to compute, and because the statistical efficiency of the Markov chain degrades as the dimension of the state increases. The latter problem can be mitigated using marginalization-based techniques, but these can be computationally prohibitive as well. In this paper, we combine the low-rank techniques of Brown, Saibaba, and Vallelian (2018) with the marginalization approach of Rue and Held (2005). We consider two variants of this approach: delayed acceptance and pseudo-marginalization. We provide a detailed analysis of the acceptance rates and computational costs associated with our proposed algorithms, and compare their performances on two numerical test cases---image deblurring and inverse heat equation.

D. Andrew Brown, Arvind Saibaba, Sarah Vallélian

In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution. However, implementations of such algorithms can be computationally expensive. We present a computationally efficient scheme for sampling high-dimensional Gaussian distributions in ill-posed Bayesian linear inverse problems. Our approach uses Metropolis-Hastings independence sampling with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian. We show the dependence of the acceptance rate on the number of eigenvalues retained and discuss conditions under which the acceptance rate is high. We demonstrate our proposed sampler by using it with Metropolis-Hastings-within-Gibbs sampling in numerical experiments in image deblurring, computerized tomography, and NMR relaxometry.

D. Andrew Brown, Peter Kiessler, John Nicholson

Gaussian processes (GPs) are ubiquitous tools for modeling and predicting continuous processes in physical and engineering sciences. This is partly due to the fact that one may employ a Gaussian process as an interpolator while facilitating straightforward uncertainty quantification at other locations. In addition to training data, it is sometimes the case that available information is not in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, or underlying physics lead to known behavior on an entire uncountable subset of the domain of interest. While an approximation to such known information may be obtained via pseudo-training points in the known subset, such a procedure is ad hoc with little guidance on the number of points to use, nor the behavior as the number of pseudo-observations grows large. We propose and construct Gaussian processes that unify, via reproducing kernel Hilbert space, the typical finite training data case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections in Hilbert space. We show existence of the proposed process and establish that it is the limit of a conventional GP conditioned on an increasing number of training points. We illustrate the flexibility and advantages of our proposed approach via numerical experiments.

Jiajing Niu, Boyoung Hur, John Absher et al.

Graphical models, used to express conditional dependence between random variables observed at various nodes, are used extensively in many fields such as genetics, neuroscience, and social network analysis. While most current statistical methods for estimating graphical models focus on scalar data, there is interest in estimating analogous dependence structures when the data observed at each node are functional, such as signals or images. In this paper, we propose a fully Bayesian regularization scheme for estimating functional graphical models. We first consider a direct Bayesian analog of the functional graphical lasso proposed by Qiao et al. (2019). We then propose a regularization strategy via the graphical horseshoe. We compare these approaches via simulation study and apply our proposed functional graphical horseshoe to two motivating applications, electroencephalography data for comparing brain activation between an alcoholic group and controls, as well as changes in structural connectivity in the presence of traumatic brain injury (TBI). Our results yield insight into how the brain attempts to compensate for disconnected networks after injury.

Carl Ehrett, D. Andrew Brown, Evan Chodora et al.

Computer model calibration typically operates by choosing parameter values in a computer model so that the model output faithfully predicts reality. By using performance targets in place of observed data, we show that calibration techniques can be repurposed to wed engineering and material design, two processes that are traditionally carried out separately. This allows materials to be designed with specific engineering targets in mind while quantifying the associated sources of uncertainty. We demonstrate our proposed approach by "calibrating" material design settings to performance targets for a wind turbine blade.