Probabilistic Circuits

Ask about this topic

Overview

Probabilistic circuits represent distributions with structured sums and products, allowing exact computations such as likelihood evaluation and marginalization when their structure meets conditions like decomposability. Researchers learn these structures with methods such as sum-product networks, expectation-maximization, and reinforcement-learning-based search, and combine circuits with neural models to increase flexibility. Applications include tabular density estimation, time-series forecasting, out-of-distribution detection, causal reasoning, and decision support. A central tension is that constraints enabling tractable inference can limit how well circuits capture complex dependencies, while evaluation metrics and comparisons with neural and diffusion models remain important open issues.

Latest papers

CardsList

18 papers3 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Oct 21, 2026

Soft Ontology Priors over Latent Concept Posteriors for Weakly Supervised Neuro-Symbolic Learning

Neuro-symbolic (NeSy) predictors that supervise only the task label suffer from reasoning shortcuts: distinct concept mappings achieve the same maximum likelihood, leaving the recovered concepts unidentifiable and unsafe for downstream reuse. We attack this failure by injecting a soft ontology prior over the latent concepts into the training objective and compiling it together with the task constraint into a smooth, decomposable probabilistic circuit. Under independent Bernoulli perception, the normalisers of the resulting structured concept posterior reduce to weighted model counts that are linear in the compiled circuit size, because the neural leaf weights fold into the circuit before evaluation. The soft compilation strategy places expert axioms in the circuit's weights rather than its support (task-structural axioms guaranteed by the data-generating process stay hard), so noisy expert axioms only redistribute prior mass and cannot empty the support. We characterise how the prior reweights label-restricted shortcut mass through differences in ontology violation energy, show that soft priors incur bounded log-density perturbations under bounded axiom noise, and prove that the hard-support counterpart can be made degenerate by a single corrupted axiom. We validate these predictions on an audited six-concept view of the real-world BDD-OIA driving benchmark and on controlled NeSy tasks.

Tengwei Song, Robert Hoehndorf

CEMSE, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Neurosymbolic LearningWeakly Supervised LearningProbabilistic Circuits
AskPDF
Oct 16, 2026

Latent State Utilization in Probabilistic Circuits

Probabilistic circuits (PCs) are generative models that support efficient probabilistic inference. Adding latent states to a PC increases its representational capacity, but yields diminishing gains in likelihood. We use mutual information to examine how the dependence represented by a trained PC changes with latent size. Experiments with Hidden Chow-Liu Trees (HCLTs) on ImageNet32 show that learned dependence grows slowly and remains well below both the latent-state upper bound and estimates of dependence in the data. To understand how learning uses these states, we examine how strongly each sample favors particular latent states and how broadly states are used across the dataset. Tracking these measurements during HCLT training distinguishes rapid early posterior concentration from the more gradual stabilization of sample-dependent state preferences. Early measurements also guide initialization selection in HCLTs and hidden Markov models (HMMs), as well as tree selection in HCLTs. The resulting choices improve the ultimate performance across the tested latent sizes, providing a practical benefit from our study of latent-state utilization.

Zilei Shao, Anji Liu

University of California, Los Angeles · School of Computing, National University of Singapore

Probabilistic CircuitsLatent Variable Models
AskPDF
Sep 29, 2026

DisCoMBO: Steering Expert-in-the-Loop Black-Box Optimization via Distributional Conformance

Sequential Model-Based Optimization (SMBO) traditionally relies on Bayesian or ensembling surrogates for uncertainty quantification. While historically treated as fully data-driven, SMBO increasingly integrates external domain expertise to accelerate discovery. To overcome the opaque guidance and diminished integration fidelity of standard acquisition re-weighting, Probabilistic Circuits (PCs) have emerged as a generative surrogate alternative, enabling direct knowledge injection via conditional sampling. However, these generative routines lack the formal exploration-exploitation semantics required for rigorous optimization. We introduce the Distributional Conformance Score (DisCo), a novel metric that unifies the flexibility and efficiency of PCs with a formal uncertainty framework. DisCo provides a bounded, [0,1][0, 1]-normalized measure of model "surprise" that (1) recovers properties comparable to kernel-based uncertainty known from, e.g., Gaussian Processes, while maintaining linear-time inference, and (2) enables accurate assessment of conformance of external knowledge w.r.t. model evidence. We then present DisCoMBO, a framework leveraging these properties for robust, knowledge-aware optimization. We prove that DisCoMBO is a zero-regret algorithm and demonstrate its effectiveness across diverse benchmarks from AutoML, material optimization, and wind park optimization.

Jonas Seng, Bennet Wittelsbach, Kristian Kersting

Computer Science Department, TU Darmstadt · TU Darmstadt, Hessian.AI, DFKI

Black-Box OptimizationBayesian OptimizationSurrogate ModelingProbabilistic Circuits
AskPDF
Aug 31, 2026

On the Complexity of the Compatibility Problem for Succinctly Encoded Conditional Distributions

The motivation for this paper is the investigation of the trade-offs implicit in probabilistic models used in machine learning. Models are often used to make predictions in the form of conditional probabilities. However, a pair of conditional distributions p(x|y) and p(y|x) may not be compatible with any joint distribution p(x,y). Given two such conditionals, determining if there exists a compatible joint is known as the compatibility problem. For discrete random variables, when the conditionals are encoded as probability tables, the compatibility problem has a known solution, which is computationally tractable. In this paper, we formalise and study a succinct version of the problem, encoding conditional distributions as arithmetic circuits. This is applicable to practical applications of probabilistic modelling in high-dimensional settings, including neural network models. We show that, for succinct circuit representations of conditionals, the compatibility problem is intractable. In the case that all probabilities are non-zero, the problem is co-NP-complete. In the case that probabilities can be zero, we give examples to demonstrate that several notions of compatibility can be distinguished, and we prove that multiple versions of the problem are PSPACE-complete. Furthermore, we show that, assuming the polynomial hierarchy does not collapse, there exist compatible succinct conditionals whose joint cannot be expressed succinctly. Implications of these results for probabilistic modelling and machine learning are discussed.

Guy Emerson

Probabilistic Circuits
AskPDF
Aug 17, 2026

Probabilistic Circuits as Reasoning Machines in Artificial Intelligence (Part I)

This cumulative habilitation thesis studies probabilistic circuits (PCs) as a powerful and tractable framework for reasoning and learning under uncertainty in artificial intelligence (AI). It first advocates for probability as a core language for AI, emphasizing its connections to logic and information theory; the conceptual simplicity of probabilistic reasoning---based primarily on the sum and product rules; the parallels between probabilistic inference and human cognition; and the role of probability in optimal decision making. However, probability also faces significant computational challenges, as probabilistic inference is NP-hard in almost all probabilistic models. PCs address these challenges through structural constraints that ensure exact computation of a wide range of inference queries in polynomial time, such as marginals, conditionals, most probable explanations, expectations, and more advanced inference tasks. This thesis synthesizes a decade of research across foundations, algorithmic developments, and empirical validation of PCs. Key contributions highlighted in this work are foundational theory of PCs, Bayesian approaches for learning PCs, scalable implementations and integration with deep learning, hybrid models that combine PCs with intractable models, and connections with symbolic machine learning paradigms. This is the first part of my Habilitation Thesis. The second part is omitted, as it comprises the cumulative part of the thesis and has been published at various venues (see Chapter 5).

Robert Peharz

Institute of Machine Learning and Neural Computation Graz University of Technology

Probabilistic CircuitsProbabilistic Graphical Models
AskPDF
Aug 13, 2026

A Compositional Theory of Curvature in Probabilistic Circuits

Probabilistic Circuits (PCs) are generative models that support exact inference and, unlike deep neural networks, admit an exact and tractable measure of loss-surface curvature: the trace of the Hessian of the log-likelihood. Recent work regularizes this trace globally to bias learning toward flatter, better generalizing optima. We show that treating sharpness as a global regularizer can be misspecified for PCs, whose curvature is inherently compositional. We prove that each sum node's contribution to the Hessian trace factorizes exactly into its circuit flow, which measures how heavily the node is used, and a local sharpness term determined by its output distribution. This decomposition provides insights into why global sharpness regularization is depth biased and can lead to underfitting. Building on it, we introduce an adaptive sharpness aware regularizer that penalizes nodes based on intrinsic local curvature and preserves closed form EM updates. We also show that empirically, this targeted regularization recovers the generalization that global regularization sacrifices while retaining the robustness and benefits of sharpness aware learning.

Hrithik Suresh, Sahil Sidheekh, Shelar Parth Vijay +3

Mehta Family School of Data Science and Artificial Intelligence, Department of Data Science, Indian Institute of Technology Palakkad, Kerala, India · Erik Jonsson School of Engineering & Computer Science, The University of Texas at Dallas, Richardson, TX, USA · AT&T CDO

Sharpness-Aware MinimizationProbabilistic Circuits
AskPDF
Aug 10, 2026

A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection

Probabilistic Circuits (PCs) are tractable generative models whose internal nodes encode a hierarchy of probabilistic summaries over different variable scopes. Existing PC-based out-of-distribution (OOD) detection methods ignore this hierarchy, reducing the entire circuit to the scalar likelihood (or its uncertainty) computed at the root. We introduce Hierarchical Likelihood Vector (HLV), a representation whose entries are the likelihoods associated with selected PC nodes and define the Hierarchical Likelihood Distance (HLD), a PC-induced pseudo-metric that compares the probability distributions through the expectations of their HLVs. We show that HLD is an integral probability metric over a function class naturally induced by the PC and develop a principled goodness-of-fit hypothesis test for unsupervised OOD detection. Unlike existing approaches, the trained PC alone serves as the representation of the in-distribution: no held-out in-distribution data are required at deployment. We further show that the quantities required by the hypothesis test can be computed exactly, directly from the trained circuit, yielding an approximate analytic decision threshold. Experiments on tabular and MNIST datasets demonstrate that exploiting the hierarchical probabilistic summaries encoded through the PC improve OOD detection over root-likelihood, uncertainty-, typicality- and kernel-based baselines, while naturally localizing distribution shifts to the PC nodes responsible for the shift.

Bhumika K, Vidhya S, Narayanan C Krishnan

Mehta Family School of Data Science and Artificial Intelligence Indian Institute of Technology Palakkad Palakkad, Kerala, India

Probabilistic CircuitsOOD DetectionIntegral Probability Metrics
AskPDF
Aug 5, 2026

Diverse and Plausible Algorithmic Recourse via Tractable Recourse Distributions

Algorithmic recourse seeks to help individuals reverse unfavorable automated decisions by recommending actionable changes that achieve a desired outcome. As an individual usually has several distinct routes to a favorable decision, and different people can act on different ones, a recourse system should offer multiple realistic alternatives rather than one. Existing approaches formulate recourse as an optimization problem that constructs one or a small set of counterfactuals rather than modeling the underlying space of feasible solutions, and in practice each sacrifices diversity, plausibility, or feasibility to secure the others. We propose Tractable Recourse Distributions, a probabilistic framework that represents the space of feasible alternatives for a given factual instance as a probability distribution over favorable outcomes. For commonly used cost functions based on proximity and the number of feature changes, we show that this distribution admits an exact representation as a probabilistic circuit, obtained by exponentially tilting the circuit; each individual's distribution is therefore available in closed form, without retraining the model. Sampling from these distributions naturally produces diverse and plausible recourses, while the tilting parameters provide explicit control over their proximity and sparsity. Experiments on standard algorithmic recourse benchmark datasets demonstrate that the proposed framework attains diversity, plausibility, and feasibility simultaneously, while retaining sufficient probability mass over feasible counterfactuals for rejection sampling to be practical. A visual study on MNIST illustrates how the tilt strength trades proximity against validity.

Anagha Sabu, Hrithik Suresh, Narayanan C. Krishnan

Mehta Family School of Data Science and Artificial Intelligence Indian Institute of Technology Palakkad Palakkad, Kerala, India

Probabilistic CircuitsAlgorithmic Recourse
AskPDF
Jul 8, 2026

PeTeR: Post-Training Robustification of Probabilistic Circuits

Probabilistic circuits (PCs) can model complex joint distributions while supporting exact and efficient computation of many inference queries. However, standard likelihood-based PC learning is vulnerable to overfitting and fragile generalization when confronted with data noise, small sample sizes, or distribution shifts. This can be mitigated using distributionally-robust optimization which consider worst-case distributions within a Wasserstein ball of the empirical distribution, but current methods are limited to training a model from scratch in this framework. Instead, we propose PeTeR: a novel, data-free post-training framework designed to robustify pre-trained PCs against distribution shifts without retraining from scratch. Empirical evaluations across multiple density estimation benchmarks demonstrate that PeTeR effectively robustifies baseline models against both random and adversarial perturbations, achieving competitive or superior performance to data-dependent robust learning baselines.

Adrian Ciotinga, Yeming Dai, YooJung Choi

School of Computing and Augmented Intelligence, Arizona State University

Distribution Shift RobustnessProbabilistic CircuitsAdversarial Robustness
AskPDF
May 15, 2026

Scalable Uncertainty Reasoning in Knowledge Graphs

Knowledge Graphs are pivotal for semantic data integration. The real-world data they model is often inherently uncertain. Within knowledge graphs, uncertainty manifests in three distinct levels: imprecise attribute values, probabilistic triple existence, and incomplete schema knowledge. However, current Semantic Web standards lack native support for reasoning over such uncertainty, and naïve extensions often incur computational intractability. In this thesis, I aim to develop a modular framework that addresses each level through tailored techniques: (1) defining probabilistic literals and a corresponding query algebra for continuous attributes; (2) a compilation-based framework transforming SPARQL provenance into tractable probabilistic circuits for uncertain triples; and (3) topology-aware geometric embeddings for statistical schema reasoning. The central hypothesis is that specialized reasoning mechanisms, namely algebraic, logical, and geometric approaches, can reconcile semantic precision with computational tractability.

Jingcheng Wu

University of Stuttgart, Stuttgart, Germany

Probabilistic CircuitsUncertainty QuantificationKnowledge Graphs
AskPDF
May 13, 2026

The Expressivity Boundary of Probabilistic Circuits: A Comparison with Large Language Models

Probabilistic Circuits (PCs) are deep generative models that support exact and efficient probabilistic inference. Yet in autoregressive language modeling, PCs still lag behind Transformer-based large language models (LLMs), suggesting an important expressivity gap. In this work, we compare PCs and LLMs under a unified autoregressive formulation. First, an output bottleneck: PCs parameterize predictions as convex combinations in probability space, which struggles to represent the sharp distributions typical of language; adopting a logit-space parameterization substantially narrows this gap. Second, a context-encoding bottleneck: we prove that structured-decomposable PCs can match Transformer separation rank on vtree-aligned partitions, but show, both theoretically and empirically, that this capacity is limited to partitions aligned with the fixed routing structure, leading to severe degradation when the data exhibits heterogeneous dependency topologies. We further prove that decomposable PCs are strictly more expressive than structured-decomposable ones, though effectively optimizing them remains an open challenge.

Zhiyu Zhao, Xuejie Liu, Muhan Zhang +1

School of Computing, National University of Singapore · School of Intelligence Science and Technology, Peking University · Institute for Artificial Intelligence, Peking University

Probabilistic CircuitsAutoregressive Language Modeling
AskPDF
May 7, 2026

Hallucination as an Anomaly: Dynamic Intervention via Probabilistic Circuits

One of the most critical challenges in Large Language Models is their tendency to hallucinate, i.e., produce factually incorrect responses. Existing approaches show promising results in terms of hallucination correction, but still suffer from a main limitation: they apply corrections indiscriminately to every token, corrupting also the originally correct generations. To overcome this drawback, we propose PCNET, a Probabilistic Circuit trained as a tractable density estimator over the LLM residual stream. The method detects hallucinations as geometric anomalies on the factual manifold, which is done via exact Negative Log-Likelihood computation, hence without the need for sampling, external verifiers, or weight modifications, as in existing techniques. To demonstrate its effectiveness, we exploit PCNET as a dynamic gate that distinguishes hallucinated from factual hidden states at each decoding step. This triggers our second main contribution, PC-LDCD (Probabilistic Circuit Latent Density Contrastive Decoding), only when the latent geometry deviates from factual regions, while leaving correct generations untouched. Across four LLMs, ranging from 1B to 8B models, and four benchmarks covering conversational reasoning, knowledge-intensive QA, reading comprehension, and truthfulness, PCNET achieves near-perfect hallucination detection across CoQA, SQuAD v2.0, and TriviaQA, with AUROC reaching up to 99%. Moreover, PC-LDCD obtains the highest True+Info, MC2, and MC3 scores on TruthfulQA in three out of four models, in comparison with state-of-the-art baselines, while reducing the mean corruption rate to 53.7% and achieving a preservation rate of 79.3%. Our proposed method is publicly available on GitHub.

Erik Nielsen, Elia Cunegatti, Marcus Vukojevic +1

Department of Information Engineering and Computer Science University of Trento Trento, Italy

Probabilistic CircuitsLLM Hallucination MitigationHallucination in Language ModelsLLM ReliabilityLLM Hallucination DetectionContrastive Decoding
AskPDF
Apr 30, 2026

Probabilistic Circuits for Irregular Multivariate Time Series Forecasting

Joint probabilistic modeling is essential for forecasting irregular multivariate time series (IMTS) to accurately quantify uncertainty. Existing approaches often struggle to balance model expressivity with consistent marginalization, frequently leading to unreliable or contradictory forecasts. To address this, we propose CircuITS, a novel architecture for probabilistic IMTS forecasting based on sum-product networks (SPNs). Our model is flexible in capturing intricate dependencies between time series channels while structurally guaranteeing valid joint distributions. Experiments on five real-world datasets demonstrate that CircuITS achieves superior joint and marginal density estimation compared to state-of-the-art baselines.

Christian Klötergens, Lars Schmidt-Thieme, Vijaya Krishna Yalavarthi

ISMLL, University of Hildesheim, Germany · VWFS, Data Analytics Research Center

Irregular Time-Series ModelingMultivariate Time Series ForecastingProbabilistic CircuitsProbabilistic Time Series Forecasting
AskPDF
Mar 24, 2026

A Sobering Look at Tabular Data Generation via Probabilistic Circuits

Tabular data is more challenging to generate than text and images, due to its heterogeneous features and much lower sample sizes. On this task, diffusion-based models are the current state-of-the-art (SotA) model class, achieving almost perfect performance on commonly used benchmarks. In this paper, we question the perception of progress for tabular data generation. First, we highlight the limitations of current protocols to evaluate the fidelity of generated data, and advocate for alternative ones. Next, we revisit a simple baseline -- hierarchical mixture models in the form of deep probabilistic circuits (PCs) -- which delivers competitive or superior performance to SotA models for a fraction of the cost. PCs are the generative counterpart of decision forests, and as such can natively handle heterogeneous data as well as deliver tractable probabilistic generation and inference. Finally, in a rigorous empirical analysis we show that the apparent saturation of progress for SotA models is largely due to the use of inadequate metrics. As such, we highlight that there is still much to be done to generate realistic tabular data. Code available at https://github.com/april-tools/tabpc.

Davide Scassola, Dylan Ponsford, Adrián Javaloy +4

AILAB, University of Trieste, Trieste, Italy · Aindo SpA, AREA Science Park, Trieste, Italy · School of Informatics, University of Edinburgh, Edinburgh, UK

Synthetic Tabular Data GenerationProbabilistic CircuitsSynthetic Data EvaluationTabular Diffusion Models
AskPDF
Mar 20, 2026

SymCircuit: Bayesian Structure Inference for Tractable Probabilistic Circuits via Entropy-Regularized Reinforcement Learning

Probabilistic circuit (PC) structure learning is hampered by greedy algorithms that make irreversible, locally optimal decisions. We propose SymCircuit, which replaces greedy search with a learned generative policy trained via entropy-regularized reinforcement learning. Instantiating the RL-as-inference framework in the PC domain, we show the optimal policy is a tempered Bayesian posterior, recovering the exact posterior when the regularization temperature is set inversely proportional to the dataset size. The policy is implemented as SymFormer, a grammar-constrained autoregressive Transformer with tree-relative self-attention that guarantees valid circuits at every generation step. We introduce option-level REINFORCE, restricting gradient updates to structural decisions rather than all tokens, yielding an SNR (signal to noise ratio) improvement and >10 times sample efficiency gain on the NLTCS dataset. A three-layer uncertainty decomposition (structural via model averaging, parametric via the delta method, leaf via conjugate Dirichlet-Categorical propagation) is grounded in the multilinear polynomial structure of PC outputs. On NLTCS, SymCircuit closes 93% of the gap to LearnSPN; preliminary results on Plants (69 variables) suggest scalability.

Y. Sungtaek Ju

Department of Mechanical and Aerospace Engineering, University of California, Los Angeles

Probabilistic CircuitsUncertainty Quantification
AskPDF
Mar 12, 2026

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

Probabilistic circuits (PCs) enable exact and tractable inference but employ data independent mixture weights that limit their ability to capture local geometry of the data manifold. We propose Voronoi tessellations (VT) as a natural way to incorporate geometric structure directly into the sum nodes of a PC. However, naïvely introducing such structure breaks tractability. We formalize this incompatibility and develop two complementary solutions: (1) an approximate inference framework that provides guaranteed lower and upper bounds for inference, and (2) a structural condition for VT under which exact tractable inference is recovered. Finally, we introduce a differentiable relaxation for VT that enables gradient-based learning and empirically validate the resulting approach on standard density estimation tasks.

Sahil Sidheekh, Sriraam Natarajan

The University of Texas at Dallas, Richardson, TX, USA.

Probabilistic CircuitsDensity Estimation
AskPDF
Mar 2, 2026

Causal Neural Probabilistic Circuits

Concept Bottleneck Models (CBMs) enhance the interpretability of end-to-end neural networks by introducing a layer of concepts and predicting the class label from the concept predictions. A key property of CBMs is that they support interventions, i.e., domain experts can correct mispredicted concept values at test time to improve the final accuracy. However, typical CBMs apply interventions by overwriting only the corrected concept while leaving other concept predictions unchanged, which ignores causal dependencies among concepts. To address this, we propose the Causal Neural Probabilistic Circuit (CNPC), which combines a neural attribute predictor with a causal probabilistic circuit compiled from a causal graph. This circuit supports exact, tractable causal inference that inherently respects causal dependencies. Under interventions, CNPC models the class distribution based on a Product of Experts (PoE) that fuses the attribute predictor's predictive distribution with the interventional marginals computed by the circuit. We theoretically characterize the compositional interventional error of CNPC w.r.t. its modules and identify conditions under which CNPC closely matches the ground-truth interventional class distribution. Experiments on five benchmark datasets in both in-distribution and out-of-distribution settings show that, compared with five baseline models, CNPC achieves higher task accuracy across different numbers of intervened attributes.

Weixin Chen, Han Zhao

University of Illinois Urbana-Champaign

Concept Bottleneck ModelsCausal InterventionProbabilistic CircuitsCausal Inference
AskPDF
Feb 1, 2024

Building Expressive and Tractable Probabilistic Generative Models: A Review

We present a comprehensive survey of the advancements and techniques in the field of tractable probabilistic generative modeling, primarily focusing on Probabilistic Circuits (PCs). We provide a unified perspective on the inherent trade-offs between expressivity and tractability, highlighting the design principles and algorithmic extensions that have enabled building expressive and efficient PCs, and provide a taxonomy of the field. We also discuss recent efforts to build deep and hybrid PCs by fusing notions from deep neural models, and outline the challenges and open questions that can guide future research in this evolving field.

Sahil Sidheekh, Sriraam Natarajan

The University of Texas at Dallas

Probabilistic CircuitsGenerative Modeling
AskPDF