Proceedings of Machine Learning Research

On Computing Total Variation Distance between Markov Logic Networks

Sat, 07 Nov 2026 00:00:00 +0000

The total variation (TV) distance is a fundamental metric for quantifying the difference between probability distributions. Bhattacharyya et al. (2024) showed that computing the TV distance between two directed graphical models is, in general, intractable. We extend this line of research to Markov Logic Networks (MLNs), a prominent class of statistical relational models that combine first-order logic with probabilistic graphical models. Specifically, we establish both tractability and hardness results regarding the computation of the TV distance between two MLNs. On the positive side, we reduce TV distance computation between MLNs to probabilistic inference, enabling the use of lifted inference techniques. Thereby, we identify classes of MLNs for which TV distance computation is tractable. On the negative side, we prove that, in general, the TV distance computation between two MLNs is intractable. In addition to these theoretical results, we empirically validate that soft rules with moderate weights can approximate hard rules, and we also present an application of the MLNs TV distance in benchmarking graph neural networks on synthetic data.

Federated Causal Discovery via Regression-Directed Cumulants

Sat, 07 Nov 2026 00:00:00 +0000

In this paper we study linear non-Gaussian acyclic models (LiNGAM) when used in federated environments. These causal models allow one to go beyond Markov equivalence. However, in many domains data are scarce, and increasing the sample size by centralising data from different clients is not advisable due to regulations such as the General Data Protection Regulation (GDPR). The federated environment offers an attractive option to balance privacy and causal discovery accuracy. Unfortunately, the standard centralised estimator in the LiNGAM setting, i.e., DirectLiNGAM, cannot be straightforwardly federated. Higher-order cumulant tensors offer a way around this obstacle: they depend only on the joint distribution of the variables involved and add exactly across independent sample groups, so a single communication round suffices in horizontal, vertical, and hybrid partitions. However, FedISHC, i.e., the current federated method along these lines, breaks down under near-symmetric noise. To overcome the above limitation, we introduce the FedRCD family of causal discovery algorithms, and investigate three variants that trade off communication rounds against algebraic noise; two of them are exact federated counterparts of the centralised high-order cumulant (HC) and HC-LiNGAM algorithms, and the single-round variants further effectively support exact unlearning at any granularity, from a single observation to a whole client. Numerical experiments show that at sample sizes typical of real deployments, the entire cumulant-based federated family does not actually rank variables by the population asymmetry that the scores encode at zero. It ranks them by a variance ladder induced by the DAG along its directed paths, the cumulant counterpart of varsortability. Marginal standardisation collapses every cumulant method to near-random ordering, while scale-invariant DirectLiNGAM, not federable under this protocol, is unaffected.

Predicting the Risk of Vehicle–Cyclist Crashes in Urban Intersections Using Bayesian Networks

Sat, 07 Nov 2026 00:00:00 +0000

The i4Driving European Union project aims to establish a credible and realistic human road safety baseline for virtual assessment of Connected, Cooperative, and Automated Mobility (CCAM) systems. As part of these efforts, multiple driving simulator experiments have been conducted to collect data to verify the developed human driver models. This paper focuses on the analysis of data from two urban road experiments in the same scenario involving a vehicle and a bicycle. The paper describes the development of a knowledge-driven Bayesian network (BN) to predict the risk of a right-turn maneuver involving a bicycle traveling straight through a four-way urban traffic intersection. The paper also reports on the results of an experimental analysis on data from the two urban road experiments, each involving several hundred driving sequences with a population of different human drivers. The predictive accuracy of the knowledge-driven model is high with a significant prediction horizon. That is, in most cases, the model can predict a high-risk scenario with high certainty in time to warn the driver, allowing a potential collision to be avoided.

Non-Symmetric Information Operators in GMRFs for PDE Inference

Sat, 07 Nov 2026 00:00:00 +0000

Probabilistic approaches to partial differential equations (PDEs) cast the solution as a Bayesian inference problem, in which the unknown field is inferred from a prior distribution and a likelihood derived from the governing equations. When the prior is Gaussian-Markov, the solution admits a Gaussian Markov Random Field (GMRF) representation with a sparse precision matrix, enabling efficient inference and marginal uncertainty estimates. In this framework, physical constraints enter through information operators that define the likelihood. Standard formulations rely on symmetric operators, which encode a non-directional flow of information. In advection-dominated regimes, this mismatch produces spurious oscillations and unreliable uncertainty estimates. In this work we introduce a non-symmetric information operator that embeds a physically motivated directional preference into the likelihood. Although the operator is non-symmetric, the resulting posterior precision matrix remains symmetric and positive definite, preserving full compatibility with standard sparse GMRF inference. We evaluate the method on the viscous Burgers’ equation across diffusion- to advection-dominated regimes. The non-symmetric operator suppresses spurious oscillations and matches finite-element accuracy. Its posterior uncertainty remains reliable in advection-dominated regimes where the symmetric operator becomes overconfident.

Forward sampling for State Variable Graphical Event Models with Markovian and piecewise constant dependencies

Sat, 07 Nov 2026 00:00:00 +0000

Graphical event models are dedicated to modelling continuous-time processes. Some of their extensions are used to represent event occurrences and state variable changes w.r.t time. But the existing models in the literature struggle to capture complex historical dependencies between variables, limiting their applications in real-world processes. To address this limitation, we define a new class of state variable graphical event models (SVGEMs) that deals with more complex piecewise constant historical dependencies. Sampling probabilistic graphical models is essential for approximate inference; previous studies have addressed this subject for several continuous-time models. However, sampling for graphical event models with various types of variables has not been explicitly described in the literature. To fill this gap, we also propose a forward sampling algorithm that unifies previous sampling algorithms into one single framework. A validation is performed with an experiment on randomly generated models, where we generate data using our sampling algorithm and show that the parameters of the models estimated from these data and priors converge to the ones of the underlying distribution. This work is the first one explicitly to describe a sampling algorithm for SVGEMs and pave the way for further research on the subject, including alternative SVGEM inference techniques such as importance sampling or Gibbs sampling.

Parameter Sensitivity Analysis in Mixtures of Polynomials

Sat, 07 Nov 2026 00:00:00 +0000

Parameter sensitivity analysis examines how a model is affected when some of its parameters are changed. In this study, we consider Mixtures of Polynomial (MoP) models to represent the probability distributions of continuous variables in Bayesian networks (BN). The objective of this study is to analyse the difference between a MoP density function and its perturbed versions, and how these perturbations propagate to other variables in the BN. A sensitivity measure, which is based on $L^1$ distance, is defined to compare the original distribution and the obtained distribution after perturbation. The theoretical results indicate that the sensitivity measure of a marginal distribution is at most the same as the sensitivity measure of the perturbed density. Overall, perturbations in local distributions of MoP Bayesian networks propagate to marginal distributions with decreasing intensity —meaning that their effect on marginal distributions of descendant variables becomes progressively weaker— which suggests that sensitivity analysis should primarily focus on local components.

Shapeshifter: A Dynamic Graph-Restructuring Algorithm for Improved Inference in Loopy-Structured PGMs

Sat, 07 Nov 2026 00:00:00 +0000

Inference in probabilistic graphical models (PGMs) is sensitive to the choice of graph structure used for message passing. While junction trees allow for exact inference, their use is practically infeasible due to exponential blow-up in cluster sizes. Loopy-structured PGMs, like factor graphs or cluster graphs, are computationally tractable alternatives, but the gap between the performance of such graphs and the ideal performance of junction trees still exists. We propose "shapeshifter", an algorithm that dynamically restructures a cluster graph during inference, by merging cluster pairs based on disagreement and redundancy criteria. We evaluate on two distinct domains: error correction and constraint satisfaction. The results show that shapeshifter offers a practical middle ground: improved inference when compared to existing PGM-based approaches—closing the performance gap by a substantial margin—without the prohibitive computational expense of junction trees.

Bayesian Network MPE via LLM-Compiled Semantic Maximizing Circuits

Sat, 07 Nov 2026 00:00:00 +0000

Recent research has shown that Large Language Models (LLMs) are able to construct the Directed Acyclic Graph (DAG) of a Bayesian Network (BN) at an accuracy level similar to the data-driven methods. A natural question that arises is whether LLMs can perform emph{logical inference} over these assembled graphical structures without having access to associated Conditional Probability Tables (CPTs). We tackle this question by focusing on Most Probable Explanation (MPE) inference, a task that closely aligns with LLM reasoning. We introduce a novel CPT-free emph{sequential semantic inference} method which is structured as a two-phase framework. An offline emph{semantic compilation} phase constructs a reusable emph{semantic maximizing circuit} by querying the LLM over assignment contexts induced by the DAG and a variable elimination ordering. An online emph{evaluation} phase applies evidence through circuit labels and resolves MPE queries by traversal, without making calls to the LLM at runtime. An empirical study shows that MPE accuracy surpasses 80% across six BNs and is even close to the levels reported in the literature for DAG learning, thereby demonstrating the potential of LLMs to be used as logical, parameter-free inference engines.

Partial Identification under Causal Orders by Linear Programming

Sat, 07 Nov 2026 00:00:00 +0000

Non-parametric (partial) identification of counterfactual queries typically relies on a fully specified causal graph. Motivated by settings with incomplete domain knowledge, we challenge this requirement by leveraging structural assumptions that are inherently implied by the query itself. We show that any counterfactual inquiry induces a, mostly partial, topological ordering over relevant variables, which, in turn, enables an explicit query parametrisation reducing the identification task to a linear program. This allows bounding arbitrary counterfactual and nested counterfactual queries. Our work can be viewed as a generalisation of the classical bounding framework of Tian and Pearl (2000), originally developed for probabilities of causation. We also prove the \emph{tightness} of our bounds by constructing structural causal models that attain the bounds whilst being compatible with both the observed data and the query-implied order. To assess both the generality and practical utility of the proposed bounding procedure, we revisit several case studies from the literature, demonstrating how the derived bounds can be used to yield informative insights even in the absence of an input causal graph.

BNqMark: Bayesian-Network Inference in Large Language Models

Sat, 07 Nov 2026 00:00:00 +0000

We present the first systematic evaluation of exact Bayesian-network inference in frontier reasoning large language models (LLMs). We introduce BNqMark, a benchmark-generation framework for exact inference in discrete Bayesian networks, and evaluate BNqMark-20, a benchmark instance comprising 78 procedurally generated binary networks and 434 conditional probability queries with exact ground truth. Across 9 frontier reasoning LLMs and 2 evaluation protocols, we find that current models can solve many exact inference queries on small and medium Bayesian networks, especially by externalizing inference as code. Several models correctly answered queries whose exact solution would require more than 13,000 arithmetic operations under variable elimination, rising to 68,000 in the code-generation protocol. Model capability degrades as inference complexity increases: graph and query complexity are both useful predictors of success after controlling for prompt length, network size, and model. These results position Bayesian networks as a controlled testbed for studying probabilistic reasoning in LLMs and motivate future work on richer and more challenging probabilistic inference settings.

Tensor Train Decomposition for BN2A Structure Learning

Sat, 07 Nov 2026 00:00:00 +0000

BN2A models are bipartite Bayesian networks with binary hidden and observed variables, characterized by Noisy-AND conditional probability tables with leak and penalty parameters. A fundamental challenge in BN2A modeling is identifying the bipartite graph structure, the Q-matrix in cognitive diagnostic models, from observed data alone. Existing tensor-based identification methods face the curse of dimensionality: the joint probability tensor has $2^J$ entries for $J$ observed variables, and constructive algorithms such as that of Gu (2026) require materializing it explicitly. We bridge BN2A structure learning and tensor network methods by showing that BN2A probability tensors have Tensor Train (TT) rank at most $2^K$, where $K$ is the number of hidden variables, reducing storage from $O(2^J)$ to $O(J \cdot 4^K)$, a bound that follows from a known equivalence between BN2A and Restricted Boltzmann Machines. Building on this representation, we show that Gu’s rank tests can be performed directly on TT cores via core contractions, without ever materializing the full tensor. Empirical validation on simulated data with $J = 15$ and $K = 3$ confirms the rank bound and achieves perfect structure learning; we further characterize the sample-size regime, showing that recovery is robust for $n$ as small as $30%$ of $2^J$. We also report two honest explorations of the boundaries of the approach: greedy mode ordering reduces the TT parameter count of the population tensor by $13%$ and, once an order-sensitive ambiguity in pure-item detection is resolved, composes fully with Gu’s rank tests, although the storage benefit vanishes on noisy empirical tensors; and Gu’s Step 2 itself degrades as $J$ grows with $K$ fixed, a limitation of the underlying algorithm rather than of the TT representation.

Partially Ordering Graphical Models with Latent Confounders by their Equality Constraints

Sat, 07 Nov 2026 00:00:00 +0000

A linear structural equation model can represent causal relations and latent confounders between its variables by means of directed and bidirected edges. The set of distributions in such a graphical model can be difficult to describe: conditional independence constraints do not suffice in this case, so more general equality constraints are needed. We present an algorithm that can check whether two linear structural equation models differ by an equality constraint. Using this algorithm, we empirically study the partially ordered set of these models, for different subclasses of mixed graphs. We show that for some of these classes, a natural generalization of Meek’s conjecture does not hold. We also investigate what set of constraints is needed to distinguish the models for various classes of graphs.

A Factor Graph Approach to Scalable Multi-Output Gaussian Process Regression

Sat, 07 Nov 2026 00:00:00 +0000

Multi-output Gaussian process regression scales cubically in the number of observations times outputs, and dense kernel-matrix methods need bespoke handling whenever different outputs are observed at different inputs. We express multi-output Gaussian process regression as a Forney-style factor graph in which a nearest-neighbor chain orders a fixed candidate set of $C$ inputs into a one-dimensional sequence. Along this chain, latent Matérn processes evolve through linear-Gaussian transition factors, while the linear model of coregionalization mixes $L$ latent processes into $D$ outputs through a deterministic mixing factor and per-output scalar observation factors. Posterior computation reduces to exact Gaussian message passing on the chain at cost $\mathcal{O}(C(DL^2 + L^3))$ after chain construction, and missing observations omit their local factor without any covariance-matrix restructuring. The formulation therefore scales in the number of data samples and in the rate of missing observations, while remaining best suited to candidate sets in low input dimension. We compare the factor-graph formulation against an exact kernel-matrix baseline, a sparse-variational inducing-point baseline, and a nearest-neighbor baseline on a synthetic input-dimension sweep and on electricity time series forecasting. At low input dimension the factor-graph posterior tracks the exact kernel-matrix posterior closely, and the gap grows gradually as input dimension increases while staying competitive with both approximate baselines. On the electricity time series our factor-graph formulation matches all three baselines in forecast accuracy while scaling linearly in the number of data points, where the exact kernel-matrix method becomes infeasible and the inducing-point baseline remains substantially slower.

Testable Implications of Causal Graphs in the Presence of Measurement Error

Sat, 07 Nov 2026 00:00:00 +0000

Measurement errors from sensors, instruments and human factors pose significant challenges for causal inference. Classical testable implications such as d-separations require fully observed variables and cannot be readily evaluated when variables are corrupted by measurement error. This paper shows that causal models nevertheless admit testable implications in such settings. We derive novel testable implications for causal graphs and present a general theorem stating sufficient conditions for their existence. Our results enable model falsification when traditional methods are inapplicable, enabling assessment of model compatibility with observed data when measurement error is present.

LOCO-FL: Scaling and Reference Choice for Bayesian Structural Analysis of Between-Center Heterogeneity in Federated Learning with Markov Boundaries

Sat, 07 Nov 2026 00:00:00 +0000

Multicenter quality-improvement programs still struggle to explain between-center heterogeneity using outcome measures alone. We investigate a Bayesian network method that focuses on Markov boundaries to characterize this heterogeneity at the structural level of risk models with multiple outcomes. Working with the full structure posterior rather than a point estimate allows the study of a phenomenon that point-identification learners overlook: under proper regularization, the resolvable Markov-boundary complexity grows with sample size. A naive comparison across centers, therefore, mixes sample-size induced model properties with genuine structural differences. We therefore study a global reference versus leave-one-center-out (LOCO) references together with sample-size correction, and show that both can be analyzed within a single reweighting of the per-family BD sufficient statistics, achieved exactly by adding the $\beta$-scaled LOCO counts as pseudocounts to the score. We investigate augmenting small centers to a larger, common effective size using the LOCO sample. We present two exact, privacy-preserving (PP) Bayesian federated learning schemes: a single global-reference scheme and a LOCO-reference scheme (LOCO-FL). We show that the LOCO-FL scheme supports per-center reference comparison and per-center scaling under the same institutional-privacy guarantees. Because the LOCO aggregate excludes a center’s own data, the same LOCO counts can also serve as an informative prior for that center’s local inference, without self-inclusion. We choose the scale of this prior empirically, as the value that most reduces the posterior spread, which reveals a trade-off between variance reduction and loss of the between-center signal. On synthetic data with known between-center differences, the method recovers these differences above the within-posterior spread. We report results for an eight-center Hungarian very preterm cohort (2{,}939 patients), measuring heterogeneity by posterior Jaccard distances between LOCO and global references. Results confirm the need to scale to a common effective sample size, the utility of a single global reference for coarse comparison, and that an empirically scaled LOCO prior reduces posterior spread.

Gradient-Based Fine-Tuning of Expert Bayesian Network Classifiers: Application in Forensic Porcine Wound Age Assessment

Sat, 07 Nov 2026 00:00:00 +0000

Bayesian network classifiers built from expert knowledge encode valuable domain understanding but may be improved when labelled data become available. We propose \textsc{EB-GFTBN}, a Gradient-based Fine-Tuning algorithm for Bayesian Networks with Expert-Bounded parameter updates. Building on the fine-tuning framework of El Hindi (2014) and the Brier-score variant of Mietkiewicz and Madsen (2022), we derive natural-gradient updates from the derivative of the Brier score and extend the algorithm to mixed discrete–continuous Bayesian networks. To remain faithful to the expert specification, each parameter is constrained to a relative tolerance around its expert elicited value. The methodology is evaluated in a case study regarding forensic age assessment of porcine wounds, where the parameters of a Bayesian network classifier were elicited from veterinary experts. Across $100$ cross-validation folds, \textsc{EB-GFTBN} improves both accuracy and Brier score over the expert baseline and the penalised EM alternative, while keeping every parameter inside the expert-allowed window.

Neuro-Causal Factor Analysis

Sat, 07 Nov 2026 00:00:00 +0000

Factor analysis (FA) is a statistical method for explaining how mutually dependent observed variables can be represented in terms of mutually independent latent factors, and it is widely used in the psychological, biological, and physical sciences. We revisit this classic method from the perspective of recent advances in causal structure learning and deep generative models, introducing a framework for Neuro-Causal Factor Analysis (NCFA). Our approach is fully nonparametric: it learns a directed graph between latent and observed variables, and then fits a deep generative model constrained to respect the Markov factorization of the graph. Empirically, on synthetic and real data, NCFA attains better reconstruction error compared to standard FA and better latent distribution recovery compared to a standard variational autoencoder, all with the advantages of sparser architecture, lower model complexity, and causal interpretability.

Semiparametric Inference for Half-Trek Estimators in Linear Structural Equation Models

Sat, 07 Nov 2026 00:00:00 +0000

Linear structural equation models on directed mixed graphs encode causal relationships among variables subject to latent confounding. The half-trek criterion (HTC) provides a graphical sufficient condition for the structural coefficients to be rationally identifiable from the observable covariance matrix, and yields a corresponding closed-form rational estimator. Despite this, the asymptotic distribution of the HTC estimator, and hence valid standard errors and confidence regions, have not been derived. We derive the semiparametric influence function of this estimator for every HTC-identified node, in arbitrary directed mixed graphs, including cyclic ones. The influence function combines the structural residual at the target node with the identification instruments, recursively corrected for uncertainty from earlier estimation stages. The HTC estimator is asymptotically normal with variance computable in closed form, yielding confidence regions, marginal intervals, and Wald tests for individual structural coefficients. Applied to the Fulton Fish Market dataset, our theory delivers a complete inferential summary for the causal effect of supply on demand.

Causal Modelling of Support Interventions for Student Competency Assessment

Sat, 07 Nov 2026 00:00:00 +0000

Accurate assessment of students’ competencies is essential for enabling educators to identify individual needs, design targeted interventions, and evaluate the effectiveness of educational strategies. Empirical assessment procedures are typically grounded in psychometric models, such as \emph{item response theory}, which relate students’ competence levels to their performance on assessment tasks. In this paper, we advocate adopting a structural causal modelling approach to educational assessment, moving beyond probabilistic belief updating toward a framework that explicitly supports interventional and counterfactual reasoning. We propose a corresponding protocol for its construction and analyse the practical relevance of forms of reasoning that remain inaccessible to standard associative models, including the explicit modelling of interventions such as hints and the related counterfactual scenario analysis. Although our protocol requires the structural equations to be elicited from experts, the necessary information is purely logical and does not rely on probabilistic, less tenable assumptions. We illustrate the approach using data from an assessment that employs complex tasks designed to measure compulsory school students’ algorithmic skills.

A Bayesian Network Approach to Earth Fault Localization in Medium Voltage Electricity Grids

Sat, 07 Nov 2026 00:00:00 +0000

Earth faults are the most commonly occurring fault type in Medium Voltage electricity distribution grids. Electricity distribution grids in Central Europe are compensated, meaning the network continues to operate after an earth fault. However, during the time-consuming search process via grid switching, secondary faults can occur, leading to outages. Digital twins, through their automated physics-informed topology and measurement-integration capabilities, can leverage generated training data and available measurements to pinpoint earth-fault locations without additional hardware to be deployed in the grid. This paper demonstrates how Bayesian networks can aid earth fault localization within a digital twin framework in electricity distribution grids. The proposed method uses stored voltage signatures from the digital twin to train a Bayesian network model. The results of an experimental analysis prove the efficacy of Bayesian networks when applied to observations of EMT voltages of the faulty phase. Ongoing work focuses on integrating this approach into a digital twin framework to actively support distribution system operators in locating earth faults.

Hierarchically Coupled Gaussian Bayesian Networks

Sat, 07 Nov 2026 00:00:00 +0000

Gaussian Bayesian networks are probabilistic graphical models for learning network structures from continuous data. Standard approaches typically assume that all observations arise from a homogeneous population. In many applications, however, the data originate from multiple related but heterogeneous groups that share a common network structure while having group-specific network parameters. Pooling all observations and inferring a single set of parameters ignores between-group heterogeneity and may lead to biased inference. Conversely, estimating group-specific parameters independently substantially increases model complexity and inferential uncertainty. We introduce a hierarchical prior that enables information sharing across groups. The proposed prior preserves group-specific parameters while introducing shrinkage by coupling the group-specific mean vectors and covariance structures through shared global hyperparameters. The corresponding coupling strengths are inferred from the data, allowing the degree of information sharing to adapt automatically to the observed heterogeneity across groups. We develop a Markov chain Monte Carlo (MCMC) algorithm for posterior inference and demonstrate empirically that the proposed approach improves both network structure learning and predictive performance.

On the Detection of Commutative Factors in Factor Graphs: Necessary and Sufficient Conditions

Sat, 07 Nov 2026 00:00:00 +0000

Exploiting the indistinguishability of objects in a probabilistic graphical model such as a factor graph is key to lifted probabilistic inference algorithms and allows for tractable probabilistic inference problems with respect to domain sizes. A central building block for the exploitation of indistinguishable objects in factor graphs is the identification of commutative factors, i.e., factors whose output values are invariant under permutations of input values assigned to a subset of their arguments. In this paper, we revisit the theoretical foundations underlying the state-of-the-art algorithm to detect commutative factors. Specifically, we show that in its current form, the state-of-the-art algorithm relies on a central theorem that is mistakenly regarded as a sufficient condition to identify commutative factors, while it actually only implies necessary condition. Consequently, the state of the art might, as we show in this paper, deliver incorrect results. To fix the flaws currently present in the state of the art, we prove a slightly modified version of the aforementioned theorem, which serves as a necessary condition to identify commutative factors. Moreover, we present a corrected version of the state-of-the-art algorithm, which keeps its efficiency while ensuring correctness and introduce a complementary algorithm with tighter worst-case bounds.

Bayesian estimation of causal effects in Bayesian networks with local structure

Sat, 07 Nov 2026 00:00:00 +0000

We consider the problem of estimating causal effects from observational categorical data when the data-generating process can be modeled by an unknown causal Bayesian network with local dependence structure in its conditional distributions. Capturing local structure in a categorical Bayesian network may significantly reduce the number of parameters and facilitate the discovery of weak dependencies and edge orientation, which is crucial for causal inference. We extend the Bayesian IDA (BIDA) framework to Bayesian networks with local structure, focusing on a specific class of local independencies called context-specific independencies (CSI). To this end, we build upon previous work that combines Markov chain Monte Carlo methods over DAGs with deterministic greedy optimisation of CSI-trees to learn the full network structure, and propose a local approach for efficient estimation of pairwise causal effects in a given structure, where we apply the same greedy optimisation procedure to infer the conditional distributions involved in the backdoor formula. We evaluate the proposed method in a simulation study in which we simulate data from synthetic Bayesian networks with local structure, as well as a set of benchmark networks.

LLM-Augmented Causal Discovery: Probabilistic Fusion of Edge Existence and Orientation

Sat, 07 Nov 2026 00:00:00 +0000

Bayesian network structure learning (BNSL) from observational data struggles with orientation identifiability, while large language models (LLMs) offer broad but often unreliable causal knowledge. We propose combining these complementary sources through a novel representation, termed Probabilistic Dependency Graphs (PDGs). In a PDG, each edge is associated with a distribution over directed, undirected, and absent states, enabling fusion via weighted averaging. We evaluate this approach on 26 benchmark networks, combining ensembles of three BNSL algorithms (FGES, Tabu, PC) with three LLMs (Gemini, Claude, GPT) across multiple prompts and random seeds. A simple 50/50 fusion improves F1 over the better of either source alone in 22 of 26 networks, with a statistically significant mean improvement of 0.056 (p<0.001). Analysis reveals that the two sources play complementary roles: BNSL contributes a high-recall edge skeleton (80% vs 60% for LLM), while LLM contributes accurate edge orientation (96% vs 77% for BNSL). Our results show that representing both sources as probabilistic uncertainty over edge existence and orientation is a practical and effective way to improve causal graph accuracy.

Differentiable Structural EM for Explicit-Latent Linear–Gaussian DAGs

Sat, 07 Nov 2026 00:00:00 +0000

We bridge two largely separate lines of work on directed graphical models: Classical Structural EM, which handles hidden variables and is commonly instantiated through search over discrete graph structures; and Continuous DAG learning methods, which avoid combinatorial search through smooth optimization, but are usually designed for fully observed data. We combine these ideas for linear–Gaussian models with explicit hidden variables. Our proposed method treats the graph weights as parameters of a generative Gaussian model over both observed and hidden variables. This makes the E-step exact: instead of filling in hidden variables with point estimates, it computes the needed posterior first and second moments. The M-step then uses smooth sparsity and acyclicity penalties to guide the graph search, but accepts an update only when it does not decrease the unpenalized expected complete-data objective. This preserves the usual monotone improvement of EM while still benefiting from continuous DAG optimization. Experiments show that both ingredients are necessary: exact second-moment inference in the E-step and the unpenalized acceptance check in the M-step. On hidden-confounding synthetic models and a linear–Gaussian version of Friedman’s hidden-variable benchmark, a spectral warm start helps the explicit-latent model recover hidden structure, while observed-only and random-initialized alternatives perform substantially worse.

What to Check Next? Bayesian Networks and Rollout-Based Lookahead for Quicker Inspections

Sat, 07 Nov 2026 00:00:00 +0000

Regulatory inspections, such as labour inspections, take time and money. Time and money are always limited. Inspectors often work with checklists. It could save a lot of time if they could focus first on checklist items that are likely to be non-compliant with the law. We implement a method that can guide an inspector in the field by suggesting which item on the checklist they should inspect next, based on their answers to previous checklist items. To do this, we train a Bayesian Network model on filled checklist data from the Norwegian Labour Inspection Authority. We test a myopic policy that always picks the item that is most likely to be non-compliant against a policy that looks one step ahead. The lookahead policy computes for each candidate item the expected value ($\gamma$-discounted returns) of inspecting it under a fixed policy that no longer updates probabilities, also known as a rollout. The expected value under the fixed policy is cheaply computed in closed form. We find that this lookahead policy increases performance greatly over the myopic policy, halving the catching times of non-compliances from roughly $5$ to $2.7$ checks (out of a maximum of $14$). The Bayesian network is however slightly outperformed across the board by a Binary-Relevance Naive Bayes model intended as a baseline.

A Comparative Review of Probabilistic Inference for Timescale Graphical Event Models

Sat, 07 Nov 2026 00:00:00 +0000

Timescale Graphical Event Models (TGEMs) provide a continuous-time probabilistic framework for representing dependencies among multivariate point processes using a directed graph, where each node corresponds to an event label and each edge specifies how occurrences of a parent event label influence the conditional intensity of a child event label over future time intervals, called timescales. Although TGEMs have been mainly investigated from a modeling and learning perspective, the practical behavior of sampling-based inference methods in these models remains insufficiently characterized. In this paper, we empirically compare three sampling strategies for probabilistic inference in TGEMs: forward sampling, Poisson Superposition Importance Sampling (PSIS), and auxiliary Gibbs sampling. We evaluate these methods on episodic sequence queries with incomplete and imprecise evidence. We examine the consistency of probability estimates, convergence behavior, and sensitivity to the rarity of the evidence. Our results provide a comparative experimental framework for assessing the strengths and limitations of each strategy and offer practical guidance for method selection in TGEM-based inference.

Using P&IDs and Process Data To Construct Bayesian Networks For Anomaly Detection

Sat, 07 Nov 2026 00:00:00 +0000

Industrial processes are often safety-critical and consist of multiple interacting components. The complexity of such systems, combined with the scarcity of faulty-operation data, makes the construction of reliable anomaly detection models challenging. To address this issue, we investigate how qualitative information in technical process documentation - specifically Piping and Instrumentation Diagrams (P&IDs) - can guide the construction of Bayesian networks for anomaly detection. P&IDs describe the instrumentation and piping of industrial systems, including the placement of sensors and actuators and their interconnections. We show how this structural information can be exploited to decompose an industrial process into smaller subsystems for which local models can be specified or learned independently. For anomaly detection, we employ a conflict-based measure that quantifies inconsistencies in observed process data. We adapt existing threshold calibration and smoothing concepts to the conflict-measure setting, introducing three sequential filters that balance false-alarm frequency against detection latency for true anomalies. The proposed methodology has been integrated into PredictIT, a predictive maintenance platform for industrial processes and systems. Experiments on a simulated formaldehyde production plant demonstrate that P&ID-informed models outperform corresponding full-system models, yielding lower false positive rates and reduced detection latency.

How Does Bayesian Causal Discovery Fail: Characterising Structural Consequences in Linear Gaussian Networks under Latent Confounding

Sat, 07 Nov 2026 00:00:00 +0000

Bayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size - more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.

Spectral Sparsification of Laplacian-Constrained Gaussian and Hüsler–Reiss Graphical Models

Sat, 07 Nov 2026 00:00:00 +0000

Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the Hüsler–Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well: we run simulations for Spectral-LCGGM on Erdős–Rényi and stochastic block model graphs, and we showcase Spectral-HR on two real extremal data sets.

Local Robustness Quantification for Naive Bayes Classifiers and Generative Forests: a General Approach

Sat, 07 Nov 2026 00:00:00 +0000

We provide methods for calculating the robustness of the predictions of two types of generative classifiers whose underlying distribution is a Probabilistic Graphical Model (PGM): naive Bayes classifiers and generative forests (a probabilistic extension of random forests). Following the paradigm of robustness quantification, we define the robustness of a prediction as the extent to which the distribution of the classifier can be perturbed without changing this prediction. We consider perturbations obtained by varying the local models of the PGMs within general neighborhoods and focus in particular on epsilon-contamination, total variation distance and chi-squared divergence balls. We test our methods on benchmark datasets, demonstrate that the robustness value of a prediction serves as an indicator for its trustworthiness and compare our approach with other such indicators.

Scope-Restricted Backtracking Counterfactuals

Sat, 07 Nov 2026 00:00:00 +0000

Interventional counterfactuals evaluate statements such as "had $A$ been $a$" by replacing the assignment for $A$ while keeping the unit’s exogenous background fixed. Backtracking counterfactuals reverse this logic: they preserve the structural assignments and instead vary background conditions across worlds. This semantics is natural for diagnostic explanation, but full backtracking may be too permissive because every exogenous coordinate can vary. We introduce scope-restricted backtracking counterfactuals, in which the modeler specifies which coordinates may vary while the remainder are shared across worlds. The scope determines the admissible carriers of the contrast, while a cross-world kernel governs the plausibility of their alternative values. We present model-based inference for general scope-restricted backtracking counterfactuals, derive sharp marginal bounds for a specified explanatory query, and establish frontier and compatibility conditions under which this explanatory query admits a modified-value policy representation. A linear-Gaussian example distinguishes local-noise, ancestor, irrelevant-noise, and infeasible explanations across scopes, while an application to the UCI Adult benchmark illustrates kernel specification, sensitivity, and how much of the explanation is carried by the cross-world coupling itself.

I-FLOP: Fast Learning of Order and Parents from Interventional Data

Sat, 07 Nov 2026 00:00:00 +0000

We extend the FLOP (fast learning of order and parents) algorithm recently developed by Wienöbst et al. (2026) from observational to interventional data. In particular, we use the interventional BIC score of Hauser and Bühlmann (2012), adapting it to be used with the iterative Cholesky-based score updates that are partly responsible for FLOP’s speed. We show that, in the sample limit, I-FLOP recovers a DAG in the same interventional Markov equivalence class as the data-generating DAG. We compare I-FLOP to existing causal structure learning algorithms on simulated and real interventional data, where it performs favorably in terms of both performance and run time.

Algorithmics for Graphical Separation Criteria

Sat, 07 Nov 2026 00:00:00 +0000

Statistical graphical models are representations of the dependences between the components of a random vector. They encode these dependences via graphical separation criteria. We consider the algorithmic problem of checking separation statements for the various graph classes and separation criteria that have been developed by the community. We establish that all criteria we study can be checked by modular variations of reachability. For all separation criteria but $\sigma$-separation, this yields a linear-time algorithm. This is effectively the optimal attainable time since loading the graph itself is linear-time in the number of edges in the graph. Using the CIfly framework for deploying modular variations of reachability, we also provide R and Python code implementing our algorithms. We evaluate empirical runtime and scaling behavior in a simulation study.

Learning Causal Structure of Time Series using Best Order Score Search

Sat, 07 Nov 2026 00:00:00 +0000

Causal structure learning from observational data is central to many scientific and policy domains, but the time series setting common to many disciplines poses several challenges due to temporal dependence. In this paper we focus on score-based causal discovery for multivariate time series and introduce TS-BOSS, a time series extension of the recently proposed Best Order Score Search (BOSS) (Andrews et al., 2023). TS-BOSS performs a permutation-based search over dynamic Bayesian network structures while leveraging grow–shrink trees to cache intermediate score computations, preserving the scalability and strong empirical performance of BOSS in the static setting. We provide theoretical guarantees establishing the soundness of TS-BOSS under suitable assumptions, and we present an intermediate result that extends classical subgraph minimality results for permutation-based methods to the dynamic (time series) setting. This result paves the way for time series extensions of sparsest permutation-based causal discovery algorithms under suitable stationarity assumptions. We present experimental results on synthetic data that highlight the high accuracy of TS-BOSS.

Parameterising Gaussian Graphical Models

Sat, 07 Nov 2026 00:00:00 +0000

Gaussian graphical models (GGMs) describe the dependence structure among jointly Gaussian random variables. However, the most common parameterisation of GGMs, the precision matrix, describes both the dependence and scale of the variables. This has been shown to lead to model selection methods that depend on the scale of the variables, despite graphical models being scale invariant. Even after standardising data to have unit sample variances, entries of the precision matrix can be on different scales leading to poor model selection. This paper decomposes the precision matrix into marginal variances and interpretable scale-invariant parameters - the partial correlations and variance inflation factors. This decomposition gives new insights into the precision matrix entries and the dynamics of model selection methods such as penalised likelihoods. In particular, it explains the observed phenomenon that penalties on the precision matrix perform poorly at selecting hub variables and motivates the necessity of data standardisation. It also shows why methods based on partial correlations have better hub detection properties. The effect of penalisation of different quantities on the estimation of marginal variances is then investigated and an interesting simplification of the log-likelihood is shown when using maximum likelihood estimation of the marginal variances.

Preface: The 13th International Conference on Probabilistic Graphical Models (PGM 2026)

Sat, 07 Nov 2026 00:00:00 +0000

This volume contains the papers presented at the 13th International Conference on Probabilistic Graphical Models (PGM 2026), held in València, Spain, from 9 to 11 September 2026 at the Institut Valencià d’Art Modern (IVAM). PGM is a biennial conference that brings together researchers working on all aspects of graphical models for probabilistic reasoning, learning and decision making, from theory and algorithms to applications.

Accounting for Data Uncertainty in Counterfactual Reasoning

Sat, 07 Nov 2026 00:00:00 +0000

Structural causal models with discrete variables are a class of probabilistic graphical models used for causal and counterfactual reasoning. In the presence of observational (i.e., non-experimental) data, some queries in such models are non-identifiable and therefore cannot be computed exactly. Instead, several methods derive bounds on non-identifiable queries by solving linear programming problems. However, these methods usually assume a precise specification of the data distribution. In this work, we consider a more realistic scenario in which uncertainty in the data is represented through an interval-valued specification of the data distribution. We formulate the corresponding linear programming problem under this setting, which covers both identifiable and non-identifiable counterfactual queries under data uncertainty. We show how a recently proposed divide-and-conquer algorithm for bounding counterfactual probabilities (Bj{ø}ru et al., 2025) can be adapted to such imprecise specifications of the data distribution without a substantial increase in computational complexity.

Selective Indicator Elimination in Arithmetic Circuits for Bipartite Noisy-OR Bayesian Networks

Sat, 07 Nov 2026 00:00:00 +0000

Bipartite noisy-OR Bayesian networks (BN2Os) have been compiled into arithmetic circuits (ACs) for exact inference, where evaluation traditionally consists of assigning indicator values at the circuit leaves and propagating values upward to the root. We introduce selective indicator elimination (SIE), a transformation for compiled BN2O ACs that removes indicator-controlled single-variable subcircuits and replaces them with cached values. The resulting method has two phases: an offline transformation phase that prunes such subcircuits and precomputes their replacement values, and an online evaluation phase that parses a query, assigns the appropriate cached values, and propagates over the transformed AC. Unlike prior AC compression methods that simplify static numeric computation by compressing arithmetic subcircuits consisting of multiplication and addition nodes, our approach is the first to eliminate indicator leaves together with their associated indicator-controlled computation while preserving support for general exact inference queries. We formally prove that SIE preserves exact inference correctness. Statistically significant improvements in both circuit size and exact inference runtime are empirically observed for $58%$ of the tested synthetic BN2Os.

Safe Inverse Marginalization in Bayesian Networks

Sat, 07 Nov 2026 00:00:00 +0000

Safe inverse marginalization (SIM) was recently proposed for augmenting a Bayesian network with new discrete variables such that their marginalization exactly recovers the original network, allowing re-certification to focus exclusively on the new variables. Graph-safe marginalization (GSM) was introduced to describe when marginalizing the new variable yields a distribution that factorizes according to the original DAG, with constraint-optimization methods proposed for constructing new conditional probability tables (CPTs). In this paper, we first show that arc-reversal without fill-in (AR-NFI) strictly subsumes GSM as a sufficient graphical condition for SIM. We then propose constructing new CPTs via unity-potential expansion (UPE) rather than optimization. AR-NFI combined with UPE provides a concrete structural route to SIM. More fundamentally, UPE recovers the original Bayesian network exactly by reconstructing its factorization, from which the original DAG is uniquely identified. Hence, UPE alone guarantees SIM, establishing that neither AR-NFI nor GSM is necessary. Taken together, these results show that SIM is fundamentally a parametric property, with structural conditions characterizing only when recovery can be realized via structure-preserving transformations.

Fast Constraint-Based Structure Learning for Continuous-Time Bayesian Networks

Sat, 07 Nov 2026 00:00:00 +0000

Continuous-time Bayesian networks (CTBNs) are a class of probabilistic graphical models representing complex processes evolving over time. Applying Bayesian networks’ state-space factorisation to the time-space mathematical structure of continuous-time Markov processes (CTMPs) significantly reduces the number of parameters required to model them. However, learning the structure of a CTBN that correctly factorises the corresponding CTMP is difficult, as the set of candidate graphs grows exponentially with the size of the CTMP random vector. In this work, we present HITON-CTPC, a novel constraint-based algorithm to navigate the search space of CTBNs more efficiently. Specifically, we incorporate the key steps of the HITON-PC BN structure-learning algorithm into the CTPC CTBN algorithm, thereby enabling us to add heuristics for determining the order of conditional independence tests. We assessed the performance of the proposed solution through an extensive set of synthetic experiments spanning a wide range of state-space and time-space configurations. We observed that HITON-CTPC achieves competitive structural accuracy while significantly reducing execution time. Furthermore, we evaluated the new HITON-CTPC algorithm using the Beijing Multi-Site Air Quality data set (Chen, 2017) to assess its ability to reconstruct significant relationships in a real-world scenario.

Counterfactual Fairness under Hidden Confounding

Sat, 07 Nov 2026 00:00:00 +0000

Designing algorithms that are non-discriminatory has attracted major attention in recent years. It has been shown that a causal perspective is necessary to mitigate unfair discrimination. However, little attention has been paid to how algorithms can make fair decisions under hidden confounding. In this paper, we study counterfactual fairness in the presence of hidden confounding and make two contributions. First, we show that under commonly adopted assumptions, counterfactual fairness is invariant across all causal models consistent with the observed data and causal ordering of the variables. In particular, even in the presence of hidden confounding and non-identifiability of the underlying causal model, all compatible models yield the same counterfactual fairness conclusions. Second, we show how instrumental variables, a tool commonly used in causal inference to address hidden confounding, can help to design fair predictors when the protected attribute is unobserved. Together, our results extend the scope of causal fairness methods to more realistic settings where both confounders and sensitive attributes may be partially or entirely unobserved.

Table-based Probability Trees: a New Hybrid Potential Representation for Probabilistic Inference in Graphical Models

Sat, 07 Nov 2026 00:00:00 +0000

Probabilistic inference in graphical models, such as Bayesian and Markov networks, has traditionally relied either on tabular representations of conditional probability distributions (CPDs) or on probability trees. Tables offer straightforward marginalization and combination at the cost of exponential memory in the number of variables, while pure probability trees exploit zero entries and repeated subdistributions to achieve compact representations, but their arithmetic is subject to structural overhead when non-zero subtrees dominate. This paper introduces Table-based Probability Tree (TBPT) potentials as a hybrid potential representation that unifies both paradigms within a single, flexible data structure: a rooted ordered tree in which internal nodes are discrete random-variable selectors as usual, and terminal nodes are not only scalar values, but also table nodes storing multidimensional arrays covering the variables below a given branching point. After formalizing the operations to be applied in these potentials, empirical evaluation on standard UAI competition benchmarks is conducted by using the TBPT structure running exact inference via variable elimination (VE). This data structure is useful in cases in which potential have a high zero percentage. We propose to compute the total set of zeros in advance with a previous Boolean propagation.

Augmenting Bayesian Networks with Categorical Principal Components

Sat, 07 Nov 2026 00:00:00 +0000

We propose a method for learning interpretable Bayesian networks (BNs) over high-dimensional categorical data by augmenting the original variable set with discrete components derived from expert-defined groups of variables. Because the components are computed deterministically and entered as fully observed columns, the augmented dataset defines a BN in a curved exponential family, making standard BIC asymptotically valid for joint model scoring. To enable a fair comparison of models with different numbers of component nodes — which induce different model classes over the original variables — we introduce a modified BIC that evaluates marginal likelihood over the original variables while retaining the full parameter penalty. We show that this criterion is conservative relative to the true Geiger-Heckerman effective dimension and establish when the marginal and joint criteria agree asymptotically. We compare PCA-derived components (applied to ordinally coded variables) with Multiple Correspondence Analysis (MCA), the discrete analog of PCA, and show theoretically and empirically that MCA is the more natural choice. Experiments on the Czech Attitude Barometer dataset demonstrate that MCA models achieve the best marginal BIC as well as the best held-out marginal log-likelihood across all augmented and non-augmented models, yielding a sparse, stable, and sociologically interpretable network.

An AutoML-Powered Architecture for Causal Discovery

Sat, 07 Nov 2026 00:00:00 +0000

Causal discovery is fundamental across a wide range of scientific domains. While numerous algorithms have been proposed in the literature, they often rely on methods that suffer from limitations such as strong parametric assumptions or restrictions on variable types, making them ill-suited for application to complex, real-world datasets. Recent advances in Automated Machine Learning (AutoML) leverage and tune flexible model architectures to process complex non-linear relationships and mixed data types, while automating data preprocessing for real-world applications. In this work, we design an end-to-end AutoML-based architecture for Causal Discovery (MLCD) that fully leverages these capabilities. The proposed framework consists of three main components: (i) selecting the optimal configuration for each node—including data transformations, feature sets, and predictive models and hyperparameters; (ii) constructing the graph skeleton by taking the union of the selected feature sets; and (iii) applying conditional independence tests that employ the selected optimal configurations to compute the p-values. To account for the presence of latent common causes in real-world scenarios, we formulate MLCD for causally insufficient systems, though it can easily adapt to causally sufficient environments as well. We evaluate our method on both synthetic and real-world datasets. Notably, MLCD processes raw data directly, bypassing the preprocessing pipelines required by most causal discovery methods.

Finite Dimensional Approximations of Incentive Functions for Causal Incentive Design Models

Sat, 07 Nov 2026 00:00:00 +0000

Single-stage principal-agent problems with bilateral private information can be represented as additive-noise Causal Incentive Design Models (CIDMs), where an incentive policy $\gamma$ acts as a functional intervention on a function-valued variable $\Gamma$. This research extends the Causal Incentive Design framework to address the computational challenge of densely sampling CIDMs when the intervention space $\Gamma$ is a Reproducing Kernel Hilbert Space (RKHS). This challenge arises because RKHS feature maps are typically infinite-dimensional, while CIDM simulation requires that each incentive policy be instantiated as an explicit callable function and evaluated within a structural equation involving the follower’s best-response optimization. To overcome this, quasi-Monte Carlo (QMC) and MC Random Fourier Features (RFF) frameworks are used to construct finite-dimensional approximations of RKHS-valued incentive functions and to densely sample CIDMs. The developed samplers are evaluated on five distinct CIDM instantiations. The experiments compare MC-RFF and QMC-RFF in terms of the approximation error of the RKHS kernel and the subsequent estimation of the causal policy-value target $V(\gamma)$ using Functional Kernel Ridge Regression and a Functional Multi-Layer Perceptron, benchmarked against an analytical Gaussian-smoothed baseline.

Optimized Priority Scheduling for Faster Scalable Belief Propagation

Sat, 07 Nov 2026 00:00:00 +0000

Belief propagation (BP) is a marginalization method using local message passing on probabilistic graphical models, seen in disciplines like Bayesian inference, statistical physics, and wireless communications. With the proliferation of manycore processors with 100s of cores, BP parallelization solves larger marginalization problems in reasonable time. On one hand, the classic synchronous algorithm easily maps to multithreaded execution with abundant parallelism per synchronization barrier. On the other hand, asynchronous algorithms converge on more graphs and reduce total work and sequential time to convergence, but are more difficult to parallelize due to priority-ordered scheduling. Recently proposed relaxed ordered parallel algorithms are plagued by the overheads of a decade-old scheduler. This paper unifies recent innovations in concurrent priority schedulers with relaxed parallel BP, using the Stealing MultiQueue (SMQ) with lazy priority updates. Prior work proposes different relaxed BP algorithms but fails to identify one BP algorithm that outperforms alternatives across inputs. We are the first to empirically show relaxed residual BP (RRBP) consistently outperforming all other baseline BP algorithms: at 384 threads our SMQ-based RRBP outperforms synchronous and other asynchronous BP algorithms by geomean 2.66$\times$ and $>$1.6$\times$, respectively. Our code is available at github.com/mcj-group/rbp.