Distinguishing cause from effect using observations of a pair of random variables is a core problem in causal discovery. Most approaches proposed for this task, namely additive noise models (ANM), are only adequate for quantitative data. We propose a criterion to address the cause-effect problem with categorical variables (living in sets with no meaningful order), inspired by seeing a conditional probability mass function (pmf) as a discrete memoryless channel. We select as the most likely causal direction the one in which the conditional pmf is closer to a uniform channel (UC). The rationale is that, in a UC, as in an ANM, the conditional entropy (of the effect given the cause) is independent of the cause distribution, in agreement with the principle of independence of cause and mechanism. Our approach, which we call the uniform channel model (UCM), thus extends the ANM rationale to categorical variables. To assess how close a conditional pmf (estimated from data) is to a UC, we use statistical testing, supported by a closed-form estimate of a UC channel. On the theoretical front, we prove identifiability of the UCM and show its equivalence with a structural causal model with a low-cardinality exogenous variable. Finally, the proposed method compares favorably with recent state-of-the-art alternatives in experiments on synthetic, benchmark, and real data.

Neural-symbolic AI (NeSy) allows neural networks to exploit symbolic background knowledge in the form of logic. It has been shown to aid learning in the limited data regime and to facilitate inference on out-of-distribution data. Probabilistic NeSy focuses on integrating neural networks with both logic and probability theory, which additionally allows learning under uncertainty. A major limitation of current probabilistic NeSy systems, such as DeepProbLog, is their restriction to finite probability distributions, i.e., discrete random variables. In contrast, deep probabilistic programming (DPP) excels in modelling and optimising continuous probability distributions. Hence, we introduce DeepSeaProbLog, a neural probabilistic logic programming language that incorporates DPP techniques into NeSy. Doing so results in the support of inference and learning of both discrete and continuous probability distributions under logical constraints. Our main contributions are 1) the semantics of DeepSeaProbLog and its corresponding inference algorithm, 2) a proven asymptotically unbiased learning algorithm, and 3) a series of experiments that illustrate the versatility of our approach.

Variational inference is a technique for approximating intractable posterior distributions in order to quantify the uncertainty of machine learning. Although the unimodal Gaussian distribution is usually chosen as a parametric distribution, it hardly approximates the multimodality. In this paper, we employ the Gaussian mixture distribution as a parametric distribution. A main difficulty of variational inference with the Gaussian mixture is how to approximate the entropy of the Gaussian mixture. We approximate the entropy of the Gaussian mixture as the sum of the entropy of the unimodal Gaussian, which can be analytically calculated. In addition, we theoretically analyze the approximation error between the true entropy and approximated one in order to reveal when our approximation works well. Specifically, the approximation error is controlled by the ratios of the distances between the means to the sum of the variances of the Gaussian mixture. Furthermore, it converges to zero when the ratios go to infinity. This situation seems to be more likely to occur in higher dimensional parametric spaces because of the curse of dimensionality. Therefore, our result guarantees that our approximation works well, for example, in neural networks that assume a large number of weights.

The adaptive social learning paradigm helps model how networked agents are able to form opinions on a state of nature and track its drifts in a changing environment. In this framework, the agents repeatedly update their beliefs based on private observations and exchange the beliefs with their neighbors. In this work, it is shown how the sequence of publicly exchanged beliefs over time allows users to discover rich information about the underlying network topology and about the flow of information over the graph. In particular, it is shown that it is possible (i) to identify the influence of each individual agent to the objective of truth learning, (ii) to discover how well-informed each agent is, (iii) to quantify the pairwise influences between agents, and (iv) to learn the underlying network topology. The algorithm derived herein is also able to work under non-stationary environments where either the true state of nature or the graph topology are allowed to drift over time. We apply the proposed algorithm to different subnetworks of Twitter users, and identify the most influential and central agents by using their public tweets (posts).

The prospect of informed and optimal decision-making regarding the operation and maintenance (O&M) of structures provides impetus to the development of structural health monitoring (SHM) systems. A probabilistic risk-based framework for decision-making has already been proposed. However, in order to learn the statistical models necessary for decision-making, measured data from the structure of interest are required. Unfortunately, these data are seldom available across the range of environmental and operational conditions necessary to ensure good generalisation of the model. Recently, technologies have been developed that overcome this challenge, by extending SHM to populations of structures, such that valuable knowledge may be transferred between instances of structures that are sufficiently similar. This new approach is termed population-based structural heath monitoring (PBSHM). The current paper presents a formal representation of populations of structures, such that risk-based decision processes may be specified within them. The population-based representation is an extension to the hierarchical representation of a structure used within the probabilistic risk-based decision framework to define fault trees. The result is a series, consisting of systems of systems ranging from the individual component level up to an inventory of heterogeneous populations. The current paper considers an inventory of wind farms as a motivating example and highlights the inferences and decisions that can be made within the hierarchical representation.

Distributed Stein Variational Gradient Descent (DSVGD) is a non-parametric distributed learning framework for federated Bayesian learning, where multiple clients jointly train a machine learning model by communicating a number of non-random and interacting particles with the server. Since communication resources are limited, selecting the clients with most informative local learning updates can improve the model convergence and communication efficiency. In this paper, we propose two selection schemes for DSVGD based on Kernelized Stein Discrepancy (KSD) and Hilbert Inner Product (HIP). We derive the upper bound on the decrease of the global free energy per iteration for both schemes, which is then minimized to speed up the model convergence. We evaluate and compare our schemes with conventional schemes in terms of model accuracy, convergence speed, and stability using various learning tasks and datasets.

We propose methods for density estimation and data synthesis using a novel form of unsupervised random forests. Inspired by generative adversarial networks, we implement a recursive procedure in which trees gradually learn structural properties of the data through alternating rounds of generation and discrimination. The method is provably consistent under minimal assumptions. Unlike classic tree-based alternatives, our approach provides smooth (un)conditional densities and allows for fully synthetic data generation. We achieve comparable or superior performance to state-of-the-art probabilistic circuits and deep learning models on various tabular data benchmarks while executing about two orders of magnitude faster on average. An accompanying $\texttt{R}$ package, $\texttt{arf}$, is available on $\texttt{CRAN}$.

Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for highdimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI's key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O(100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.

A novel approach for vehicle tracking using a hybrid adaptive Kalman filter is proposed. The filter utilizes recurrent neural networks to learn the vehicle's geometrical and kinematic features, which are then used in a supervised learning model to determine the actual process noise covariance in the Kalman framework. This approach addresses the limitations of traditional linear Kalman filters, which can suffer from degraded performance due to uncertainty in the vehicle kinematic trajectory modeling. Our method is evaluated and compared to other adaptive filters using the Oxford RobotCar dataset, and has shown to be effective in accurately determining the process noise covariance in real-time scenarios. Overall, this approach can be implemented in other estimation problems to improve performance.

The Targeted Maximum Likelihood Estimation (TMLE) statistical data analysis framework integrates machine learning, statistical theory, and statistical inference to provide a least biased, efficient and robust strategy for estimation and inference of a variety of statistical and causal parameters. We describe and evaluate the epidemiological applications that have benefited from recent methodological developments. We conducted a systematic literature review in PubMed for articles that applied any form of TMLE in observational studies. We summarised the epidemiological discipline, geographical location, expertise of the authors, and TMLE methods over time. We used the Roadmap of Targeted Learning and Causal Inference to extract key methodological aspects of the publications. We showcase the contributions to the literature of these TMLE results. Of the 81 publications included, 25% originated from the University of California at Berkeley, where the framework was first developed by Professor Mark van der Laan. By the first half of 2022, 70% of the publications originated from outside the United States and explored up to 7 different epidemiological disciplines in 2021-22. Double-robustness, bias reduction and model misspecification were the main motivations that drew researchers towards the TMLE framework. Through time, a wide variety of methodological, tutorial and software-specific articles were cited, owing to the constant growth of methodological developments around TMLE. There is a clear dissemination trend of the TMLE framework to various epidemiological disciplines and to increasing numbers of geographical areas. The availability of R packages, publication of tutorial papers, and involvement of methodological experts in applied publications have contributed to an exponential increase in the number of studies that understood the benefits, and adoption, of TMLE.