Graphical models, including Graph Neural Networks (GNNs) and Probabilistic Graphical Models (PGMs), have demonstrated their exceptional capabilities across numerous fields. These models necessitate effective uncertainty quantification to ensure reliable decision-making amid the challenges posed by model training discrepancies and unpredictable testing scenarios. This survey examines recent works that address uncertainty quantification within the model architectures, training, and inference of GNNs and PGMs. We aim to provide an overview of the current landscape of uncertainty in graphical models by organizing the recent methods into uncertainty representation and handling. By summarizing state-of-the-art methods, this survey seeks to deepen the understanding of uncertainty quantification in graphical models, thereby increasing their effectiveness and safety in critical applications.

We consider a variant of online binary classification where a learner sequentially assigns labels ($0$ or $1$) to items with unknown true class. If, but only if, the learner chooses label $1$ they immediately observe the true label of the item. The learner faces a trade-off between short-term classification accuracy and long-term information gain. This problem has previously been studied under the name of the `apple tasting' problem. We revisit this problem as a partial monitoring problem with side information, and focus on the case where item features are linked to true classes via a logistic regression model. Our principal contribution is a study of the performance of Thompson Sampling (TS) for this problem. Using recently developed information-theoretic tools, we show that TS achieves a Bayesian regret bound of an improved order to previous approaches. Further, we experimentally verify that efficient approximations to TS and Information Directed Sampling via P\'{o}lya-Gamma augmentation have superior empirical performance to existing methods.

Formulating dynamical models for physical phenomena is essential for understanding the interplay between the different mechanisms and predicting the evolution of physical states. However, a dynamical model alone is often insufficient to address these fundamental tasks, as it suffers from model errors and uncertainties. One common remedy is to rely on data assimilation, where the state estimate is updated with observations of the true system. Ensemble filters sequentially assimilate observations by updating a set of samples over time. They operate in two steps: a forecast step that propagates each sample through the dynamical model and an analysis step that updates the samples with incoming observations. For accurate and robust predictions of dynamical systems, discrete solutions must preserve their critical invariants. While modern numerical solvers satisfy these invariants, existing invariant-preserving analysis steps are limited to Gaussian settings and are often not compatible with classical regularization techniques of ensemble filters, e.g., inflation and covariance tapering. The present work focuses on preserving linear invariants, such as mass, stoichiometric balance of chemical species, and electrical charges. Using tools from measure transport theory (Spantini et al., 2022, SIAM Review), we introduce a generic class of nonlinear ensemble filters that automatically preserve desired linear invariants in non-Gaussian filtering problems. By specializing this framework to the Gaussian setting, we recover a constrained formulation of the Kalman filter. Then, we show how to combine existing regularization techniques for the ensemble Kalman filter (Evensen, 1994, J. Geophys. Res.) with the preservation of the linear invariants. Finally, we assess the benefits of preserving linear invariants for the ensemble Kalman filter and nonlinear ensemble filters.

Automotive insurers increasingly have access to telematic information via black-box recorders installed in the insured vehicle, and wish to identify undesirable behaviour which may signify increased risk or uninsured activities. However, identification of such behaviour with machine learning is non-trivial, and results are far from perfect, requiring human investigation to verify suspected cases. An appropriately formed priority score, generated by automated analysis of GPS data, allows underwriters to make more efficient use of their time, improving detection of the behaviour under investigation. An example of such behaviour is the use of a privately insured vehicle for commercial purposes, such as delivering meals and parcels. We first make use of trip GPS and accelerometer data, augmented by geospatial information, to train an imperfect classifier for delivery driving on a per-trip basis. We make use of a mixture of Beta-Binomial distributions to model the propensity of a policyholder to undertake trips which result in a positive classification as being drawn from either a rare high-scoring or common low-scoring group, and learn the parameters of this model using MCMC. This model provides us with a posterior probability that any policyholder will be a regular generator of automated alerts given any number of trips and alerts. This posterior probability is converted to a priority score, which was used to select the most valuable candidates for manual investigation. Testing over a 1-year period ranked policyholders by likelihood of commercial driving activity on a weekly basis. The top 0.9% have been reviewed at least once by the underwriters at the time of writing, and of those 99.4% have been confirmed as correctly identified, showing the approach has achieved a significant improvement in efficiency of human resource allocation compared to manual searching.

We present Bayesian Diffusion Models (BDM), a prediction algorithm that performs effective Bayesian inference by tightly coupling the top-down (prior) information with the bottom-up (data-driven) procedure via joint diffusion processes. We show the effectiveness of BDM on the 3D shape reconstruction task. Compared to prototypical deep learning data-driven approaches trained on paired (supervised) data-labels (e.g. image-point clouds) datasets, our BDM brings in rich prior information from standalone labels (e.g. point clouds) to improve the bottom-up 3D reconstruction. As opposed to the standard Bayesian frameworks where explicit prior and likelihood are required for the inference, BDM performs seamless information fusion via coupled diffusion processes with learned gradient computation networks. The specialty of our BDM lies in its capability to engage the active and effective information exchange and fusion of the top-down and bottom-up processes where each itself is a diffusion process. We demonstrate state-of-the-art results on both synthetic and real-world benchmarks for 3D shape reconstruction.

We propose a constraint-based algorithm, which automatically determines causal relevance thresholds, to infer causal networks from data. We call these topological thresholds. We present two methods for determining the threshold: the first seeks a set of edges that leaves no disconnected nodes in the network; the second seeks a causal large connected component in the data. We tested these methods both for discrete synthetic and real data, and compared the results with those obtained for the PC algorithm, which we took as the benchmark. We show that this novel algorithm is generally faster and more accurate than the PC algorithm. The algorithm for determining the thresholds requires choosing a measure of causality. We tested our methods for Fisher Correlations, commonly used in PC algorithm (for instance in \cite{kalisch2005}), and further proposed a discrete and asymmetric measure of causality, that we called Net Influence, which provided very good results when inferring causal networks from discrete data. This metric allows for inferring directionality of the edges in the process of applying the thresholds, speeding up the inference of causal DAGs.

Simulation based inference (SBI) methods enable the estimation of posterior distributions when the likelihood function is intractable, but where model simulation is feasible. Popular neural approaches to SBI are the neural posterior estimator (NPE) and its sequential version (SNPE). These methods can outperform statistical SBI approaches such as approximate Bayesian computation (ABC), particularly for relatively small numbers of model simulations. However, we show in this paper that the NPE methods are not guaranteed to be highly accurate, even on problems with low dimension. In such settings the posterior cannot be accurately trained over the prior predictive space, and even the sequential extension remains sub-optimal. To overcome this, we propose preconditioned NPE (PNPE) and its sequential version (PSNPE), which uses a short run of ABC to effectively eliminate regions of parameter space that produce large discrepancy between simulations and data and allow the posterior emulator to be more accurately trained. We present comprehensive empirical evidence that this melding of neural and statistical SBI methods improves performance over a range of examples, including a motivating example involving a complex agent-based model applied to real tumour growth data.

Data assimilation is a core component of numerical weather prediction systems. The large quantity of data processed during assimilation requires the computation to be distributed across increasingly many compute nodes, yet existing approaches suffer from synchronisation overhead in this setting. In this paper, we exploit the formulation of data assimilation as a Bayesian inference problem and apply a message-passing algorithm to solve the spatial inference problem. Since message passing is inherently based on local computations, this approach lends itself to parallel and distributed computation. In combination with a GPU-accelerated implementation, we can scale the algorithm to very large grid sizes while retaining good accuracy and compute and memory requirements.

The practical utility of agent-based models in decision-making relies on their capacity to accurately replicate populations while seamlessly integrating real-world data streams. Yet, the incorporation of such data poses significant challenges due to privacy concerns. To address this issue, we introduce a paradigm for private agent-based modeling wherein the simulation, calibration, and analysis of agent-based models can be achieved without centralizing the agents attributes or interactions. The key insight is to leverage techniques from secure multi-party computation to design protocols for decentralized computation in agent-based models. This ensures the confidentiality of the simulated agents without compromising on simulation accuracy. We showcase our protocols on a case study with an epidemiological simulation comprising over 150,000 agents. We believe this is a critical step towards deploying agent-based models to real-world applications.

One of the main challenges in surrogate modeling is the limited availability of data due to resource constraints associated with computationally expensive simulations. Multi-fidelity methods provide a solution by chaining models in a hierarchy with increasing fidelity, associated with lower error, but increasing cost. In this paper, we compare different multi-fidelity methods employed in constructing Gaussian process surrogates for regression. Non-linear autoregressive methods in the existing literature are primarily confined to two-fidelity models, and we extend these methods to handle more than two levels of fidelity. Additionally, we propose enhancements for an existing method incorporating delay terms by introducing a structured kernel. We demonstrate the performance of these methods across various academic and real-world scenarios. Our findings reveal that multi-fidelity methods generally have a smaller prediction error for the same computational cost as compared to the single-fidelity method, although their effectiveness varies across different scenarios.