Graph encoder embedding, a recent technique for graph data, offers speed and scalability in producing vertex-level representations from binary graphs. In this paper, we extend the applicability of this method to a general graph model, which includes weighted graphs, distance matrices, and kernel matrices. We prove that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis. Under certain condition, it achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis. These theoretical findings are validated through a series of experiments involving weighted graphs, as well as text and image data transformed into general graph representations using appropriate distance metrics.

In recent years, the inconsistency in Bayesian deep learning has garnered increasing attention. Tempered or generalized posterior distributions often offer a direct and effective solution to this issue. However, understanding the underlying causes and evaluating the effectiveness of generalized posteriors remain active areas of research. In this study, we introduce a unified theoretical framework to attribute Bayesian inconsistency to model misspecification and inadequate priors. We interpret the generalization of the posterior with a temperature factor as a correction for misspecified models through adjustments to the joint probability model, and the recalibration of priors by redistributing probability mass on models within the hypothesis space using data samples. Additionally, we highlight a distinctive feature of Laplace approximation, which ensures that the generalized normalizing constant can be treated as invariant, unlike the typical scenario in general Bayesian learning where this constant varies with model parameters post-generalization. Building on this insight, we propose the generalized Laplace approximation, which involves a simple adjustment to the computation of the Hessian matrix of the regularized loss function. This method offers a flexible and scalable framework for obtaining high-quality posterior distributions. We assess the performance and properties of the generalized Laplace approximation on state-of-the-art neural networks and real-world datasets.

PAC-Bayesian analysis is a frequentist framework for incorporating prior knowledge into learning. It was inspired by Bayesian learning, which allows sequential data processing and naturally turns posteriors from one processing step into priors for the next. However, despite two and a half decades of research, the ability to update priors sequentially without losing confidence information along the way remained elusive for PAC-Bayes. While PAC-Bayes allows construction of data-informed priors, the final confidence intervals depend only on the number of points that were not used for the construction of the prior, whereas confidence information in the prior, which is related to the number of points used to construct the prior, is lost. This limits the possibility and benefit of sequential prior updates, because the final bounds depend only on the size of the final batch. We present a novel and, in retrospect, surprisingly simple and powerful PAC-Bayesian procedure that allows sequential prior updates with no information loss. The procedure is based on a novel decomposition of the expected loss of randomized classifiers. The decomposition rewrites the loss of the posterior as an excess loss relative to a downscaled loss of the prior plus the downscaled loss of the prior, which is bounded recursively. As a side result, we also present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to discrete random variables, which we use for bounding the excess losses, and which can be of independent interest. In empirical evaluation the new procedure significantly outperforms state-of-the-art.

Directed acyclic graph (DAG) learning is a rapidly expanding field of research. Though the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. Our article addresses the difficult task of quantifying graph uncertainty by developing a variational Bayes inference framework based on novel distributions that have support directly on the space of DAGs. The distributions, which we use to form our prior and variational posterior, are induced by a projection operation, whereby an arbitrary continuous distribution is projected onto the space of sparse weighted acyclic adjacency matrices (matrix representations of DAGs) with probability mass on exact zeros. Though the projection constitutes a combinatorial optimization problem, it is solvable at scale via recently developed techniques that reformulate acyclicity as a continuous constraint. We empirically demonstrate that our method, ProDAG, can deliver accurate inference, and often outperforms existing state-of-the-art alternatives.

Label differential privacy (DP) is a framework that protects the privacy of labels in training datasets, while the feature vectors are public. Existing approaches protect the privacy of labels by flipping them randomly, and then train a model to make the output approximate the privatized label. However, as the number of classes $K$ increases, stronger randomization is needed, thus the performances of these methods become significantly worse. In this paper, we propose a vector approximation approach, which is easy to implement and introduces little additional computational overhead. Instead of flipping each label into a single scalar, our method converts each label into a random vector with $K$ components, whose expectations reflect class conditional probabilities. Intuitively, vector approximation retains more information than scalar labels. A brief theoretical analysis shows that the performance of our method only decays slightly with $K$. Finally, we conduct experiments on both synthesized and real datasets, which validate our theoretical analysis as well as the practical performance of our method.

Adversarial examples pose a security threat to many critical systems built on neural networks. Given that deterministic robustness often comes with significantly reduced accuracy, probabilistic robustness (i.e., the probability of having the same label with a vicinity is $\ge 1-\kappa$) has been proposed as a promising way of achieving robustness whilst maintaining accuracy. However, existing training methods for probabilistic robustness still experience non-trivial accuracy loss. It is unclear whether there is an upper bound on the accuracy when optimising towards probabilistic robustness, and whether there is a certain relationship between $\kappa$ and this bound. This work studies these problems from a Bayes error perspective. We find that while Bayes uncertainty does affect probabilistic robustness, its impact is smaller than that on deterministic robustness. This reduced Bayes uncertainty allows a higher upper bound on probabilistic robust accuracy than that on deterministic robust accuracy. Further, we prove that with optimal probabilistic robustness, each probabilistically robust input is also deterministically robust in a smaller vicinity. We also show that voting within the vicinity always improves probabilistic robust accuracy and the upper bound of probabilistic robust accuracy monotonically increases as $\kappa$ grows. Our empirical findings also align with our results.

Effective large-scale process optimization in manufacturing industries requires close cooperation between different human expert parties who encode their knowledge of related domains as Bayesian network models. For instance, Bayesian networks for domains such as lithography equipment, processes, and auxiliary tools must be conjointly used to effectively identify process optimizations in the semiconductor industry. However, business confidentiality across domains hinders such collaboration, and encourages alternatives to centralized inference. We propose CCBNet, the first Confidentiality-preserving Collaborative Bayesian Network inference framework. CCBNet leverages secret sharing to securely perform analysis on the combined knowledge of party models by joining two novel subprotocols: (i) CABN, which augments probability distributions for features across parties by modeling them into secret shares of their normalized combination; and (ii) SAVE, which aggregates party inference result shares through distributed variable elimination. We extensively evaluate CCBNet via 9 public Bayesian networks. Our results show that CCBNet achieves predictive quality that is similar to the ones of centralized methods while preserving model confidentiality. We further demonstrate that CCBNet scales to challenging manufacturing use cases that involve 16-128 parties in large networks of 223-1003 features, and decreases, on average, computational overhead by 23%, while communicating 71k values per request. Finally, we showcase possible attacks and mitigations for partially reconstructing party networks in the two subprotocols.

Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.

Bayesian networks are well-suited for clinical reasoning on tabular data, but are less compatible with natural language data, for which neural networks provide a successful framework. This paper compares and discusses strategies to augment Bayesian networks with neural text representations, both in a generative and discriminative manner. This is illustrated with simulation results for a primary care use case (diagnosis of pneumonia) and discussed in a broader clinical context.

In the context of reinforcement learning from human feedback (RLHF), the reward function is generally derived from maximum likelihood estimation of a random utility model based on pairwise comparisons made by humans. The problem of learning a reward function is one of preference aggregation that, we argue, largely falls within the scope of social choice theory. From this perspective, we can evaluate different aggregation methods via established axioms, examining whether these methods meet or fail well-known standards. We demonstrate that both the Bradley-Terry-Luce Model and its broad generalizations fail to meet basic axioms. In response, we develop novel rules for learning reward functions with strong axiomatic guarantees. A key innovation from the standpoint of social choice is that our problem has a linear structure, which greatly restricts the space of feasible rules and leads to a new paradigm that we call linear social choice.