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1. Sequential Bayesian Registration for Functional Data

   https://doi.org/10.1007/s11222-025-10640-8  

   Kim, Yoonji; Chkrebtii, Oksana A; Kurtek, Sebastian A (August 2025, Statistics and Computing)

   Abstract In many modern applications, discretely-observed data may be naturally understood as a set of functions. Functional data often exhibit two confounded sources of variability: amplitude (y-axis) and phase (x-axis). The extraction of amplitude and phase, a process known as registration, is essential in exploring the underlying structure of functional data in a variety of areas, from environmental monitoring to medical imaging. Critically, such data are often gathered sequentially with new functional observations arriving over time. Despite this, existing registration procedures do not sequentially update inference based on the new data, requiring model refitting. To address these challenges, we introduce a Bayesian framework for sequential registration of functional data, which updates statistical inference as new sets of functions are assimilated. This Bayesian model-based sequential learning approach utilizes sequential Monte Carlo sampling to recursively update the alignment of observed functions while accounting for associated uncertainty. Distributed computing significantly reduces computational cost relative to refitting the model using an iterative method such as Markov chain Monte Carlo on the full data. Simulation studies and comparisons reveal that the proposed approach performs well even when the target posterior distribution has a challenging structure. We apply the proposed method to three real datasets: (1) functions of annual drought intensity near Kaweah River in California, (2) annual sea surface salinity functions near Null Island, and (3) a sequence of repeated patterns in electrocardiogram signals. 

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2. Bayesian Probabilistic Inference of Nonparametric Distance Distributions in DEER Spectroscopy

   https://doi.org/10.1021/acs.jpca.4c05056  

   Sweger, Sarah R; Cheung, Julian C; Zha, Lukas; Pribitzer, Stephan; Stoll, Stefan (October 2024, The Journal of Physical Chemistry A)

   Double electron−electron resonance (DEER) spectroscopy measures distance distributions between spin labels in proteins, yielding important structural and energetic information about conformational landscapes. Analysis of an experimental DEER signal in terms of a distance distribution is a nontrivial task due to the ill-posed nature of the underlying mathematical inversion problem. This work introduces a Bayesian probabilistic inference approach to analyze DEER data, assuming a nonparametric distance distribution with a Tikhonov smoothness prior. The method uses Markov Chain Monte Carlo sampling with a compositional Gibbs sampler to determine a posterior probability distribution over the entire parameter space, including the distance distribution, given an experimental data set. This posterior contains all of the information available from the data, including a full quantification of the uncertainty about the model parameters. The corresponding uncertainty about the distance distribution is visually captured via an ensemble of posterior predictive distributions. Several examples are presented to illustrate the method. Compared with bootstrapping, it performs faster and provides slightly larger uncertainty intervals. 

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3. Asymptotic Consistency of α-Rényi-Approximate Posteriors

   Jaiswal, Prateek; Rao, Vinayak; Honnappa, Harsha (May 2020, Journal of machine learning research)

   Nguyen, XuanLong (Ed.)

   We study the asymptotic consistency properties of α-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the α-Rényi divergence from the true posterior. Unique to our work is that we consider settings with α > 1, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a ‘good’ sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also extend our analysis to the setting where α equals one, corresponding to the minimizer of the reverse KL divergence, and to models with local latent variables. We also illustrate the existence of good sequence with a number of examples. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods. 

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4. Towards Robust and Discriminative Sequential Data Learning: When and How to Perform Adversarial Training?

   https://doi.org/10.1145/3292500.3330957  

   Jia, Xiaowei; Li, Sheng; Zhao, Handong; Kim, Sungchul; Kumar, Vipin (January 2019, Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (KDD '19))

   The last decade has witnessed a surge of interest in applying deep learning models for discovering sequential patterns from a large volume of data. Recent works show that deep learning models can be further improved by enforcing models to learn a smooth output distribution around each data point. This can be achieved by augmenting training data with slight perturbations that are designed to alter model outputs. Such adversarial training approaches have shown much success in improving the generalization performance of deep learning models on static data, e.g., transaction data or image data captured on a single snapshot. However, when applied to sequential data, the standard adversarial training approaches cannot fully capture the discriminative structure of a sequence. This is because real-world sequential data are often collected over a long period of time and may include much irrelevant information to the classification task. To this end, we develop a novel adversarial training approach for sequential data classification by investigating when and how to perturb a sequence for an effective data augmentation. Finally, we demonstrate the superiority of the proposed method over baselines in a diversity of real-world sequential datasets. 

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5. A Statistical Hypothesis Testing Strategy for Adaptively Blending Particle Filters and Ensemble Kalman Filters for Data Assimilation

   https://doi.org/10.1175/MWR-D-22-0108.1  

   Kurosawa, Kenta; Poterjoy, Jonathan (January 2023, Monthly Weather Review)

   Abstract Particle filters avoid parametric estimates for Bayesian posterior densities, which alleviates Gaussian assumptions in nonlinear regimes. These methods, however, are more sensitive to sampling errors than Gaussian-based techniques such as ensemble Kalman filters. A recent study by the authors introduced an iterative strategy for particle filters that match posterior moments—where iterations improve the filter’s ability to draw samples from non-Gaussian posterior densities. The iterations follow from a factorization of particle weights, providing a natural framework for combining particle filters with alternative filters to mitigate the impact of sampling errors. The current study introduces a novel approach to forming an adaptive hybrid data assimilation methodology, exploiting the theoretical strengths of nonparametric and parametric filters. At each data assimilation cycle, the iterative particle filter performs a sequence of updates while the prior sample distribution is non-Gaussian, then an ensemble Kalman filter provides the final adjustment when Gaussian distributions for marginal quantities are detected. The method employs the Shapiro–Wilk test to determine when to make the transition between filter algorithms, which has outstanding power for detecting departures from normality. Experiments using low-dimensional models demonstrate that the approach has a significant value, especially for nonhomogeneous observation networks and unknown model process errors. Moreover, hybrid factors are extended to consider marginals of more than one collocated variables using a test for multivariate normality. Findings from this study motivate the use of the proposed method for geophysical problems characterized by diverse observation networks and various dynamic instabilities, such as numerical weather prediction models. Significance Statement Data assimilation statistically processes observation errors and model forecast errors to provide optimal initial conditions for the forecast, playing a critical role in numerical weather forecasting. The ensemble Kalman filter, which has been widely adopted and developed in many operational centers, assumes Gaussianity of the prior distribution and solves a linear system of equations, leading to bias in strong nonlinear regimes. On the other hand, particle filters avoid many of those assumptions but are sensitive to sampling errors and are computationally expensive. We propose an adaptive hybrid strategy that combines their advantages and minimizes the disadvantages of the two methods. The hybrid particle filter–ensemble Kalman filter is achieved with the Shapiro–Wilk test to detect the Gaussianity of the ensemble members and determine the timing of the transition between these filter updates. Demonstrations in this study show that the proposed method is advantageous when observations are heterogeneous and when the model has an unknown bias. Furthermore, by extending the statistical hypothesis test to the test for multivariate normality, we consider marginals of more than one collocated variable. These results encourage further testing for real geophysical problems characterized by various dynamic instabilities, such as real numerical weather prediction models. 

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