Integral Probability Metrics

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A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection
Aug 10, 2026Bhumika K, Vidhya S, Narayanan C KrishnanProbabilistic Circuits OOD Detection
Mehta Family School of Data Science and Artificial Intelligence Indian Institute of Technology Palakkad Palakkad, Kerala, India
Testing Distributions Against Bounded Distinguishers
Jul 17, 2026Mark Bun, Rathin Desai, Renato Ferreira PintoIntegral Probability Metrics
Boston University · Columbia University
A nonparametric two-sample test using a parametric integral probability metric
Jun 15, 2026Yuha Park, Yongdai KimTwo-Sample Testing Statistical Learning Theory
Department of Mathematics, University of Hamburg, Hamburg, Germany. · Department of Statistics, Seoul National University, Seoul, Republic of Korea.
Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space
May 17, 2026Kelvin Kan, Xingjian Li, Benjamin J. Zhang +3Integral Probability Metrics Diffusion Models
Department of Mathematics Oden Institute UCLA · Department of Mathematics University of Texas at Austin · Rutgers University +3
Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale
May 13, 2026Shashaank Aiyer, Yishay Mansour, Shay Moran +2Statistical Learning Theory Integral Probability Metrics
University of Maryland · Tel Aviv University and Google Research · Technion and Google Research +1
Beyond Expected Information Gain: Stable Bayesian Optimal Experimental Design with Integral Probability Metrics and Plug-and-Play Extensions
Apr 23, 2026Di Wu, Ling Liang, Haizhao YangBayesian Experimental Design Integral Probability Metrics
Department of Mathematics The University of Tennessee, Knoxville TN 37916, USA · Department of Mathematics and Department of Computer Science University of Maryland, College Park MD 20742, USA
Quantification of Credal Uncertainty: A Distance-Based Approach
Mar 28, 2026Xabier Gonzalez-Garcia, Siu Lun Chau, Julian Rodemann +6Uncertainty Quantification Aleatoric Uncertainty
Dept. of Statistics, CS & Mathematics, Public Univ. of Navarre · EPIC Lab, Nanyang Technological Univ. Singapore · Rational Intelligence Lab, CISPA Helmholtz Center for Information Security +7
Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance
Apr 15, 2025Peter Matthew Jacobs, Foad Namjoo, Jeff M. PhillipsTwo-Sample Testing Distribution Matching
University of Wisconsin–Madison, USA · University of Utah, USA

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

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 Circuits OOD Detection Integral Probability Metrics

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Probabilistic Circuits (PCs) are tractable generative models whose internal nodes encode a hierarchy of probabilistic sum- maries over different variable scopes. Existing PC-based out- of-distribution (OOD) detection methods ignore this hierar- chy, reducing the entire circuit to the scalar likelihood (or its uncertainty) computed at the root. We introduce Hierar- chical Likelihood Vector (HLV), a representation whose en- tries are the likelihoods associated with selected PC nodes and define the Hierarchical Likelihood Distance (HLD), a PC-induced pseudo-metric that compares the probability dis- tributions 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 detec- tion. 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 com- puted exactly, directly from the trained circuit, yielding an ap- proximate analytic decision threshold. Experiments on tabular and MNIST datasets demonstrate that exploiting the hierarchi- cal probabilistic summaries encoded through the PC improve OOD detection over root-likelihood, uncertainty-, typicality- and kernel-based baselines, while naturally localizing distri- bution shifts to the PC nodes responsible for the shift.

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