[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82008-en":3,"doc-seo-82008-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82008,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","PETER: Post-Training Robustification of Probabilistic Circuits","Probabilistic circuits (PCs) model complex joint distributions and support exact, efficient inference, but likelihood-based learning can overfit and generalize poorly under data noise, small samples, or distribution shifts. Distributionally-robust optimization improves reliability via worst-case distributions in a Wasserstein ball, yet existing approaches typically retrain from scratch. PETER introduces a data-free post-training framework that robustifies pre-trained PCs against distribution shifts without full retraining, achieving competitive or superior results on multiple density estimation benchmarks against random and adversarial perturbations.","PETER: Post-Training Robustification of Probabilistic Circuits  \nAdrian Ciotinga 1 Yeming Dai 1 YooJung Choi 1  \n1 School of Computing and Augmented Intelligence, Arizona State University University  \narXiv :2607 .0767 1v2 [ cs .LG] 9 Jul 2026  \nAbstract  \nProbabilistic 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.  \n1 INTRODUCTION  \nDistributionally robust optimization (DRO) has emerged as a principled framework for learning estimators that remain reliable under data noise, limited sample sizes, and distribution shift, by hedging against an uncertainty set of plausible distributions rather than committing to the empirical distribution alone [Delage and Ye, 2010, Bertsimasand Nohadani, 2019] . A common approach is to define this uncertainty set as a Wasserstein ball around the empirical distribution, yielding a minimax formulation in which the estimator is optimized against the worst-case distribution, according to the appropriate performance measure, within some radius ϵ . Although widely studied in the context of  \ndeep generative models [Liu et al., 2023, Kuhn et al., 2019, Bauso et al., 2017], its application to tractable probabilistic models (TPMs) remains scarce; to the best of our knowledge, Peddi et al. [2022] is the sole work addressing this intersection. While effective, existing approaches to robust learning typically focus on training a model from scratch on a dataset, tying robustness to the availability of training data and requiring that any existing models be completely replaced rather than improved.  \nWe argue that TPMs, and probabilistic circuits (PCs) in particular, are uniquely positioned for an alternative, more flexible framework of post-training robustification. Rather than learning a new model from scratch or fine-tuning it with additional data as classic DRO settings, post-training robustification takes a pre-trained model and grants it the same distributionally-robust guarantees within an ϵ-Wasserstein ball without the use of any training data or complete retraining of the model. As we show in this paper, circuit structural properties enabling tractable inference allow us to not only compute the distance between two distributions, but also to explicitly represent and optimize against the worst-case adversarial distribution efficiently. Based on this insight, we propose PETER, a post-training robustification framework for PCs that hedges against worst-case perturbations within a Wasserstein ball, bounded using the recently introduced Circuit-Wasserstein distance [Ciotinga and Choi, 2025] . Our method reduces the optimization problem to an unconstrained problem and solves it via the gradient ascent-descent algorithm, leveraging tractable gradient computation for PCs. We empirically demonstrate the efficacy of our approach in comparison to an existing robust MLE baseline for PCs that requires a training dataset, as—to the best of our knowledge—no existing methods are fully data-free.  \n2 PRELIMINARIES  \nWe use capital letters (X) to denote random variables and lowercase l","cbCailRnzgV0S3dg","https://ap.wps.com/l/cbCailRnzgV0S3dg","pdf",262887,1,7,"English","en",105,"# Introduction\n# Preliminaries\n## Probabilistic Circuits\n## Robust Maximum-Likelihood Estimation","[{\"question\":\"What problem does PETER address in probabilistic circuit learning?\",\"answer\":\"Standard likelihood-based learning for probabilistic circuits can overfit and fail to generalize reliably under noise, limited samples, or distribution shifts.\"},{\"question\":\"How does PETER differ from existing distributionally-robust optimization methods?\",\"answer\":\"PETER is a data-free post-training approach that robustifies an already pre-trained probabilistic circuit without retraining from scratch.\"},{\"question\":\"What robustness set does PETER use to hedge against distribution shifts?\",\"answer\":\"PETER hedges against worst-case perturbations within an ε-Wasserstein ball, with the Wasserstein bound based on a circuit-oriented distance.\"}]",1784177544,18,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"peter-post-training-robustification-of-probabilistic-circuits","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/peter-post-training-robustification-of-probabilistic-circuits/82008/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does PETER address in probabilistic circuit learning?","Question",{"text":75,"@type":76},"Standard likelihood-based learning for probabilistic circuits can overfit and fail to generalize reliably under noise, limited samples, or distribution shifts.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does PETER differ from existing distributionally-robust optimization methods?",{"text":80,"@type":76},"PETER is a data-free post-training approach that robustifies an already pre-trained probabilistic circuit without retraining from scratch.",{"name":82,"@type":73,"acceptedAnswer":83},"What robustness set does PETER use to hedge against distribution shifts?",{"text":84,"@type":76},"PETER hedges against worst-case perturbations within an ε-Wasserstein ball, with the Wasserstein bound based on a circuit-oriented 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