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1. VARIANCE-BASED SENSITIVITY OF BAYESIAN INVERSE PROBLEMS TO THE PRIOR DISTRIBUTION

   https://doi.org/10.1615/Int.J.UncertaintyQuantification.2024051475  

   Darges, John E; Alexanderian, Alen; Gremaud, Pierre A (January 2025, International Journal for Uncertainty Quantification)

   The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reason-able may lead to significantly different conclusions. We develop a computational approach to understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of Bayesian inverse problems with respect to the prior hyperparameters. This, however, is a challenging problem-a naive double loop sampling approach would require running a prohibitive number of Markov chain Monte Carlo (MCMC) sampling procedures. The present work takes a foundational step in making such a sensitivity analysis practical by combining efficient surrogate models and a tailored importance sampling approach. In particular, we can perform accurate GSA of posterior statistics of quantities of interest with respect to prior hyperparameters without the need to repeat MCMC runs. We demonstrate the effectiveness of the approach on a simple Bayesian linear inverse problem and a nonlinear inverse problem governed by an epidemiological model. 

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2. Adaptive importance sampling for extreme quantile estimation with stochastic black box computer models

   https://doi.org/10.1002/nav.21938  

   Pan, Qiyun; Byon, Eunshin; Ko, Young Myoung; Lam, Henry (August 2020, Naval Research Logistics (NRL))

   Abstract Quantile is an important quantity in reliability analysis, as it is related to the resistance level for defining failure events. This study develops a computationally efficient sampling method for estimating extreme quantiles using stochastic black box computer models. Importance sampling has been widely employed as a powerful variance reduction technique to reduce estimation uncertainty and improve computational efficiency in many reliability studies. However, when applied to quantile estimation, importance sampling faces challenges, because a good choice of the importance sampling density relies on information about the unknown quantile. We propose an adaptive method that refines the importance sampling density parameter toward the unknown target quantile value along the iterations. The proposed adaptive scheme allows us to use the simulation outcomes obtained in previous iterations for steering the simulation process to focus on important input areas. We prove some convergence properties of the proposed method and show that our approach can achieve variance reduction over crude Monte Carlo sampling. We demonstrate its estimation efficiency through numerical examples and wind turbine case study. 

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3. Learning structured neural dynamics from single trial population recording

   Nassar, J.; Linderman, S.; Zhao, Y.; Bugallo, M.; Park, I. M. (December 2018, Conference record - Asilomar Conference on Signals, Systems, & Computers)

   To understand the complex nonlinear dynamics of neural circuits, we fit a structured state-space model called tree-structured recurrent switching linear dynamical system (TrSLDS) to noisy high-dimensional neural time series. TrSLDS is a multi-scale hierarchical generative model for the state-space dynamics where each node of the latent tree captures locally linear dynamics. TrSLDS can be learned efficiently and in a fully Bayesian manner using Gibbs sampling. We showcase TrSLDS' potential of inferring low-dimensional interpretable dynamical systems on a variety of examples. 

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4. Approximate Confidence Distribution Computing

   https://doi.org/10.51387/23-NEJSDS38  

   Thornton, Suzanne; Li, Wentao; Xie, Minge (January 2023, The New England Journal of Statistics in Data Science)

   Approximate confidence distribution computing (ACDC) offers a new take on the rapidly developing field of likelihood-free inference from within a frequentist framework. The appeal of this computational method for statistical inference hinges upon the concept of a confidence distribution, a special type of estimator which is defined with respect to the repeated sampling principle. An ACDC method provides frequentist validation for computational inference in problems with unknown or intractable likelihoods. The main theoretical contribution of this work is the identification of a matching condition necessary for frequentist validity of inference from this method. In addition to providing an example of how a modern understanding of confidence distribution theory can be used to connect Bayesian and frequentist inferential paradigms, we present a case to expand the current scope of so-called approximate Bayesian inference to include non-Bayesian inference by targeting a confidence distribution rather than a posterior. The main practical contribution of this work is the development of a data-driven approach to drive ACDC in both Bayesian or frequentist contexts. The ACDC algorithm is data-driven by the selection of a data-dependent proposal function, the structure of which is quite general and adaptable to many settings. We explore three numerical examples that both verify the theoretical arguments in the development of ACDC and suggest instances in which ACDC outperform approximate Bayesian computing methods computationally. 

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5. Conditional score-based diffusion models for Bayesian inference in infinite dimensions

   Baldassari, L; Siahkoohi, A; Garnier, J; Solna, K; de_Hoop, V M (December 2023, NeurIPS)

   Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions—the conditional denoising estimator—can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference. 

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