Sequential Monte Carlo methods can characterize arbitrary distributions using sequential importance sampling and resampling, but typically require very large sample sizes to mitigate weight collapse [Snyder et al., 2008, 2015]. By contrast, ensemble Kalman-type methods avoid the use of weights, but are based on affine prior-to-posterior updates that are consistent only if all distributions involved are Gaussian. In the context of smoothing, such methods include the ensemble Kalman smoother (EnKS) [Evensen and Van Leeuwen, 2000], which has inspired numerous algorithmic variations such as the ensemble smoother with multiple data assimilation [Emerick and Reynolds, 2013] and the iterative ensemble Kalman smoother (iEnKS) [Bocquet and Sakov, 2014, Evensen et al., 2019], as well as backwards smoothers such as the ensemble Rauch-Tung-Striebel smoother (EnRTSS) [Raanes, 2016]. These two classes of methods occupy opposite ends of a spectrum that ranges from an emphasis on statistical generality at one end to an emphasis on computational efficiency at the other. This trade-off complicates design decisions for smoothing problems that are at once non-Gaussian and computationally expensive.

Smoothers are algorithms for Bayesian time series re-analysis. Most operational smoothers rely either on affine Kalman-type transformations or on sequential importance sampling. These strategies occupy opposite ends of a spectrum that trades computational efficiency and scalability for statistical generality and consistency: non-Gaussianity renders affine Kalman updates inconsistent with the true Bayesian solution, while the ensemble size required for successful importance sampling can be prohibitive. This paper revisits the smoothing problem from the perspective of measure transport, which offers the prospect of consistent prior-to-posterior transformations for Bayesian inference. We leverage this capacity by proposing a general ensemble framework for transport-based smoothing. Within this framework, we derive a comprehensive set of smoothing recursions based on nonlinear transport maps and detail how they exploit the structure of state-space models in fully non-Gaussian settings. We also describe how many standard Kalman-type smoothing algorithms emerge as special cases of our framework. A companion paper (Ramgraber et al., 2023) explores the implementation of nonlinear ensemble transport smoothers in greater depth.

The concept of creating a virtual copy of a complete Cyber-Physical System opens up numerous possibilities, including real-time assessments of the physical environment and continuous learning from the system to provide reliable and precise information. This process, known as the twinning process or the development of a digital twin (DT), has been widely adopted across various industries. However, challenges arise when considering the computational demands of implementing AI models, such as those employed in digital twins, in real-time information exchange scenarios. This work proposes a digital twin framework for optimal and autonomous decision-making applied to a gas-lift process in the oil and gas industry, focusing on enhancing the robustness and adaptability of the DT. The framework combines Bayesian inference, Monte Carlo simulations, transfer learning, online learning, and novel strategies to confer cognition to the DT, including model hyperdimensional reduction and cognitive tack. Consequently, creating a framework for efficient, reliable, and trustworthy DT identification was possible. The proposed approach addresses the current gap in the literature regarding integrating various learning techniques and uncertainty management in digital twin strategies. This digital twin framework aims to provide a reliable and efficient system capable of adapting to changing environments and incorporating prediction uncertainty, thus enhancing the overall decision-making process in complex, real-world scenarios. Additionally, this work lays the foundation for further developments in digital twins for process systems engineering, potentially fostering new advancements and applications across various industrial sectors.

Resided at the intersection of multi-fidelity optimization (MFO) and Bayesian optimization (BO), MF BO has found a niche in solving expensive engineering design optimization problems, thanks to its advantages in incorporating physical and mathematical understandings of the problems, saving resources, addressing exploitation-exploration trade-off, considering uncertainty, and processing parallel computing. The increasing number of works dedicated to MF BO suggests the need for a comprehensive review of this advanced optimization technique. In this paper, we survey recent developments of two essential ingredients of MF BO: Gaussian process (GP) based MF surrogates and acquisition functions. We first categorize the existing MF modeling methods and MFO strategies to locate MF BO in a large family of surrogate-based optimization and MFO algorithms. We then exploit the common properties shared between the methods from each ingredient of MF BO to describe important GP-based MF surrogate models and review various acquisition functions. By doing so, we expect to provide a structured understanding of MF BO. Finally, we attempt to reveal important aspects that require further research for applications of MF BO in solving intricate yet important design optimization problems, including constrained optimization, high-dimensional optimization, optimization under uncertainty, and multi-objective optimization.

The posterior covariance matrix W defined by the log-likelihood of each observation plays important roles both in the sensitivity analysis and frequencist's evaluation of the Bayesian estimators. This study focused on the matrix W and its principal space; we term the latter as an essential subspace. First, it is shown that they appear in various statistical settings, such as the evaluation of the posterior sensitivity, assessment of the frequencist's uncertainty from posterior samples, and stochastic expansion of the loss; a key tool to treat frequencist's properties is the recently proposed Bayesian infinitesimal jackknife approximation (Giordano and Broderick (2023)). In the following part, we show that the matrix W can be interpreted as a reproducing kernel; it is named as W-kernel. Using the W-kernel, the essential subspace is expressed as a principal space given by the kernel PCA. A relation to the Fisher kernel and neural tangent kernel is established, which elucidates the connection to the classical asymptotic theory; it also leads to a sort of Bayesian-frequencist's duality. Finally, two applications, selection of a representative set of observations and dimensional reduction in the approximate bootstrap, are discussed. In the former, incomplete Cholesky decomposition is introduced as an efficient method to compute the essential subspace. In the latter, different implementations of the approximate bootstrap for posterior means are compared.

Bayesian Neural Networks (BNN) have emerged as a crucial approach for interpreting ML predictions. By sampling from the posterior distribution, data scientists may estimate the uncertainty of an inference. Unfortunately many inference samples are often needed, the overhead of which greatly hinder BNN's wide adoption. To mitigate this, previous work proposed propagating the first and second moments of the posterior directly through the network. However, on its own this method is even slower than sampling, so the propagated variance needs to be approximated such as assuming independence between neural nodes. The resulting trade-off between quality and inference time did not match even plain Monte Carlo sampling. Our contribution is a more principled variance propagation framework based on "spiked covariance matrices", which smoothly interpolates between quality and inference time. This is made possible by a new fast algorithm for updating a diagonal-plus-low-rank matrix approximation under various operations. We tested our algorithm against sampling based MC Dropout and Variational Inference on a number of downstream uncertainty themed tasks, such as calibration and out-of-distribution testing. We find that Favour is as fast as performing 2-3 inference samples, while matching the performance of 10-100 samples. In summary, this work enables the use of BNN in the realm of performance critical tasks where they have previously been out of reach.

Bayesian predictive synthesis (BPS) provides a method for combining multiple predictive distributions based on agent/expert opinion analysis theory and encompasses a range of existing density forecast pooling methods. The key ingredient in BPS is a ``synthesis'' function. This is typically specified parametrically as a dynamic linear regression. In this paper, we develop a nonparametric treatment of the synthesis function using regression trees. We show the advantages of our tree-based approach in two macroeconomic forecasting applications. The first uses density forecasts for GDP growth from the euro area's Survey of Professional Forecasters. The second combines density forecasts of US inflation produced by many regression models involving different predictors. Both applications demonstrate the benefits -- in terms of improved forecast accuracy and interpretability -- of modeling the synthesis function nonparametrically.

The availability of representative datasets is an essential prerequisite for many successful artificial intelligence and machine learning models. However, in real life applications these models often encounter scenarios that are inadequately represented in the data used for training. There are various reasons for the absence of sufficient data, ranging from time and cost constraints to ethical considerations. As a consequence, the reliable usage of these models, especially in safety-critical applications, is still a tremendous challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches. Knowledge augmented machine learning approaches offer the possibility of compensating for deficiencies, errors, or ambiguities in the data, thus increasing the generalization capability of the applied models. Even more, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-driven models with existing knowledge. The identified approaches are structured according to the categories knowledge integration, extraction and conformity. In particular, we address the application of the presented methods in the field of autonomous driving.

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance $\tau$. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of $\tilde{O}\left(\frac{H^7 A^2 d^4}{\tau^2 \epsilon^2}\right)$ to yield an $\epsilon$-optimal CVaR, where $H$ is the length of each episode, $A$ is the capacity of action space, and $d$ is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

We propose a method for the classification of objects that are structured as random trees. Our aim is to model a distribution over the node label assignments in settings where the tree data structure is associated with node attributes (typically high dimensional embeddings). The tree topology is not predetermined and none of the label assignments are present during inference. Other methods that produce a distribution over node label assignment in trees (or more generally in graphs) either assume conditional independence of the label assignment, operate on a fixed graph topology, or require part of the node labels to be observed. Our method defines a Markov Network with the corresponding topology of the random tree and an associated Gibbs distribution. We parameterize the Gibbs distribution with a Graph Neural Network that operates on the random tree and the node embeddings. This allows us to estimate the likelihood of node assignments for a given random tree and use MCMC to sample from the distribution of node assignments. We evaluate our method on the tasks of node classification in trees on the Stanford Sentiment Treebank dataset. Our method outperforms the baselines on this dataset, demonstrating its effectiveness for modeling joint distributions of node labels in random trees.