{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T19:18:13Z","timestamp":1792783093481,"version":"4.5.0"},"reference-count":109,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T00:00:00Z","timestamp":1787961600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T00:00:00Z","timestamp":1787961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Amazon-Illinois Center on AI for Interactive Conversational Experiences"},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS 2311085"],"award-info":[{"award-number":["ECCS 2311085"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000104","name":"National Aeronautics and Space Administration","doi-asserted-by":"publisher","award":["80NSSC22M0070"],"award-info":[{"award-number":["80NSSC22M0070"]}],"id":[{"id":"10.13039\/100000104","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Google Research Scholar Award","id":[{"id":"https:\/\/ror.org\/00njsd438","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>End-to-end deep neural networks have achieved remarkable success across various domains but are often criticized for their lack of interpretability. While post hoc explanation methods attempt to address this issue, they often fail to accurately represent these black-box models, resulting in misleading or incomplete explanations. To overcome these challenges, we propose an inherently transparent model architecture called Neural Probabilistic Circuits (NPCs), which enable compositional and interpretable predictions through logical reasoning. In particular, an NPC\u00a0consists of two modules: an attribute recognition model, which predicts probabilities for various attributes, and a task predictor built on a probabilistic circuit, which enables logical reasoning over recognized attributes to make class predictions. To train NPCs, we introduce a three-stage training algorithm comprising attribute recognition, circuit construction, and joint optimization. Moreover, we theoretically demonstrate that an NPC\u2019s error is upper-bounded by a linear combination of the errors from its modules. To further demonstrate the interpretability of NPC, we provide both the most probable explanations and the counterfactual explanations. Empirical results on four benchmark datasets show that NPCs strike a balance between interpretability and performance, achieving results competitive even with those of end-to-end black-box models while providing enhanced interpretability.<\/jats:p>","DOI":"10.1007\/s10994-026-07118-7","type":"journal-article","created":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T11:47:06Z","timestamp":1788004026000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neural Probabilistic Circuits: Enabling Compositional and Interpretable Predictions Through Logical Reasoning"],"prefix":"10.1007","volume":"115","author":[{"given":"Weixin","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simon","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huajie","family":"Shao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lui","family":"Sha","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Han","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,29]]},"reference":[{"key":"7118_CR1","unstructured":"Adel, T., Balduzzi, D., & Ghodsi, A. (2015). Learning the structure of sum\u2013product networks via an SVD-based algorithm. In UAI, 2015."},{"key":"7118_CR2","doi-asserted-by":"crossref","unstructured":"Ahmed, K., Chang, K., & Broeck, G. V. (2023). A pseudo-semantic loss for autoregressive models with logical constraints. In NeurIPS, 2023.","DOI":"10.52202\/075280-0806"},{"key":"7118_CR3","doi-asserted-by":"crossref","unstructured":"Ahmed, K., Teso, S., Chang, K., Broeck, G. V., & Vergari, A. (2022). Semantic probabilistic layers for neuro-symbolic learning. In NeurIPS, 2022.","DOI":"10.52202\/068431-2171"},{"key":"7118_CR4","unstructured":"Alvarez-Melis, D., & Jaakkola, T. S. (2018). On the robustness of interpretability methods. arXiv preprint. arXiv:1806.08049"},{"key":"7118_CR5","doi-asserted-by":"publisher","first-page":"103649","DOI":"10.1016\/j.artint.2021.103649","volume":"303","author":"S Badreddine","year":"2022","unstructured":"Badreddine, S., Garcez, A. S., Serafini, L., & Spranger, M. (2022). Logic tensor networks. Artificial Intelligence, 303, 103649.","journal-title":"Artificial Intelligence"},{"key":"7118_CR6","doi-asserted-by":"crossref","unstructured":"Barbiero, P., Ciravegna, G., Giannini, F., Li\u00f3, P., Gori, M., & Melacci, S. (2022). Entropy-based logic explanations of neural networks. In AAAI, 2022.","DOI":"10.3233\/FAIA230157"},{"key":"7118_CR7","unstructured":"Barbiero, P., Ciravegna, G., Giannini, F., Zarlenga, M. E., Magister, L. C., Tonda, A., Lio, P., Precioso, F., Jamnik, M., & Marra, G. (2023). Interpretable neural-symbolic concept reasoning. In ICML, 2023."},{"key":"7118_CR8","doi-asserted-by":"crossref","unstructured":"Bortolotti, S., Marconato, E., Carraro, T., Morettin, P., Krieken, E., Vergari, A., Teso, S., & Passerini, A. (2024). A neuro-symbolic benchmark suite for concept quality and reasoning shortcuts. In NeurIPS track on datasets and benchmarks, 2024.","DOI":"10.52202\/079017-3679"},{"key":"7118_CR9","unstructured":"Butz, C. J., Oliveira, J. S., Santos, A. E., Teixeira, A. L., Poupart, P., & Kalra, A. (2018). An empirical study of methods for SPN learning and inference. In PGM, 2018."},{"key":"7118_CR10","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1023\/A:1007379606734","volume":"28","author":"R Caruana","year":"1997","unstructured":"Caruana, R. (1997). Multitask learning. Machine Learning, 28, 41\u201375.","journal-title":"Machine Learning"},{"key":"7118_CR11","unstructured":"Chavira, M., & Darwiche, A. (2005). Compiling Bayesian networks with local structure. In IJCAI, 2005."},{"key":"7118_CR12","doi-asserted-by":"crossref","unstructured":"Choi, A., Kisa, D., & Darwiche, A. (2013). Compiling probabilistic graphical models using sentential decision diagrams. In ECSQARU, 2013.","DOI":"10.1007\/978-3-642-39091-3_11"},{"key":"7118_CR13","unstructured":"Choi, Y., Vergari, A., & Broeck, G. (2020). Probabilistic circuits: A unifying framework for tractable probabilistic models (p. 6). UCLA. http:\/\/starai.cs.ucla.edu\/papers\/ProbCirc20.pdf"},{"key":"7118_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2022.103822","author":"G Ciravegna","year":"2023","unstructured":"Ciravegna, G., Barbiero, P., Giannini, F., Gori, M., Li\u00f2, P., Maggini, M., & Melacci, S. (2023). Logic explained networks. Artificial Intelligence. https:\/\/doi.org\/10.1016\/j.artint.2022.103822","journal-title":"Artificial Intelligence"},{"key":"7118_CR15","unstructured":"Darwiche, A. (2002). A compiler for deterministic, decomposable negation normal form. In AAAI, 2002."},{"key":"7118_CR16","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1613\/jair.989","volume":"17","author":"A Darwiche","year":"2002","unstructured":"Darwiche, A., & Marquis, P. (2002). A knowledge compilation map. Journal of Artificial Intelligence Research, 17, 229\u2013264.","journal-title":"Journal of Artificial Intelligence Research"},{"key":"7118_CR17","doi-asserted-by":"crossref","unstructured":"Debot, D., Barbiero, P., Giannini, F., Ciravegna, G., Diligenti, M., & Marra, G. (2024). Interpretable concept-based memory reasoning. In NeurIPS, 2024.","DOI":"10.52202\/079017-0608"},{"issue":"7","key":"7118_CR18","first-page":"1829","volume":"119","author":"A Deeks","year":"2019","unstructured":"Deeks, A. (2019). The judicial demand for explainable artificial intelligence. Columbia Law Review, 119(7), 1829\u20131850.","journal-title":"Columbia Law Review"},{"key":"7118_CR19","unstructured":"Dennis, A., & Ventura, D. (2012). Learning the architecture of sum\u2013product networks using clustering on variables. In NeurIPS, 2012."},{"key":"7118_CR20","doi-asserted-by":"crossref","unstructured":"Devlin, J., Chang, M., Lee, K., & Toutanova, K. (2019). BERT: Pre-training of deep bidirectional transformers for language understanding. In NAACL-HLT, 2019.","DOI":"10.18653\/v1\/N19-1423"},{"key":"7118_CR21","unstructured":"Dominici, G., Barbiero, P., Zarlenga, M. E., Termine, A., Gjoreski, M., Marra, G., & Langheinrich, M. (2025). Causal concept graph models: Beyond causal opacity in deep learning. In ICLR, 2025."},{"key":"7118_CR22","unstructured":"Gens, R., & Domingos, P. M. (2012). Discriminative learning of sum\u2013product networks. In NeurIPS, 2012."},{"key":"7118_CR23","unstructured":"Gens, R., & Domingos, P. M. (2013). Learning the structure of sum\u2013product networks. In ICML, 2013."},{"key":"7118_CR24","doi-asserted-by":"publisher","first-page":"3482","DOI":"10.1007\/s10489-024-05300-9","volume":"54","author":"E Gr\u00e9goire","year":"2024","unstructured":"Gr\u00e9goire, E., Chaudhary, M. H., & Verboven, S. (2024). Sample-level weighting for multi-task learning with auxiliary tasks. Applied Intelligence, 54, 3482\u20133501.","journal-title":"Applied Intelligence"},{"key":"7118_CR25","unstructured":"G\u00fcrel, N. M., Qi, X., Rimanic, L., Zhang, C., & Li, B. (2021). Knowledge enhanced machine learning pipeline against diverse adversarial attacks. In ICML, 2021."},{"key":"7118_CR26","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In CVPR.","DOI":"10.1109\/CVPR.2016.90"},{"issue":"6","key":"7118_CR27","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1109\/MSP.2012.2205597","volume":"29","author":"G Hinton","year":"2012","unstructured":"Hinton, G., Deng, L., Yu, D., Dahl, G. E., Mohamed, A.-R., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T. N., et al. (2012). Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Processing Magazine, 29(6), 82\u201397.","journal-title":"IEEE Signal Processing Magazine"},{"key":"7118_CR28","unstructured":"Hou, J., Liu, S., Bie, Y., Wang, H., Tan, A., Luo, L., & Chen, H. (2024). Self-explainable AI for medical image analysis: A survey and new outlooks. arXiv preprint. arXiv:2410.02331"},{"key":"7118_CR29","unstructured":"Kang, M., G\u00fcrel, N. M., Li, L., & Li, B. (2024). COLEP: Certifiably robust learning-reasoning conformal prediction via probabilistic circuits. In ICLR, 2024."},{"key":"7118_CR30","unstructured":"Kazhdan, D., Dimanov, B., Jamnik, M., Li\u00f2, P., & Weller, A. (2020). Now you see me (CME): Concept-based model extraction. In CIKM (workshops), 2020."},{"key":"7118_CR31","unstructured":"Kendall, A., Gal, Y., & Cipolla, R. (2018). Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In CVPR, 2018."},{"key":"7118_CR32","unstructured":"Kim, E., Jung, D., Park, S., Kim, S., & Yoon, S. (2023). Probabilistic concept bottleneck models. In ICML, 2023."},{"key":"7118_CR33","unstructured":"Koh, P. W., Nguyen, T., Tang, Y. S., Mussmann, S., Pierson, E., Kim, B., & Liang, P. (2020). Concept bottleneck models. In ICML, 2020."},{"key":"7118_CR34","unstructured":"Koller, D., & Friedman, N. (2009). Probabilistic graphical models: Principles and techniques. MIT Press."},{"key":"7118_CR35","unstructured":"Krakovna, V., & Looks, M. (2016). A minimalistic approach to sum\u2013product network learning for real applications. arXiv preprint. arXiv:1602.04259"},{"key":"7118_CR36","unstructured":"Krieken, E., Minervini, P., Ponti, E. M., & Vergari, A. (2024). On the independence assumption in neurosymbolic learning. In ICML, 2024."},{"key":"7118_CR37","unstructured":"Krieken, E., Minervini, P., Ponti, E., & Vergari, A. (2025a). Neurosymbolic diffusion models. In NeurIPS, 2025."},{"key":"7118_CR38","unstructured":"Krieken, E., Minervini, P., Ponti, E., & Vergari, A. (2025b). Neurosymbolic reasoning shortcuts under the independence assumption. arXiv preprint. arXiv:2507.11357"},{"key":"7118_CR39","doi-asserted-by":"crossref","unstructured":"Krieken, E., Thanapalasingam, T., Tomczak, J. M., Harmelen, F., & Teije, A. (2023). A-NeSI: A scalable approximate method for probabilistic neurosymbolic inference. In NeurIPS, 2023.","DOI":"10.52202\/075280-1069"},{"key":"7118_CR40","unstructured":"Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). ImageNet classification with deep convolutional neural networks. In NeurIPS, 2012."},{"key":"7118_CR41","unstructured":"Laugel, T., Lesot, M.-J., Marsala, C., & Detyniecki, M. (2019). Issues with post hoc counterfactual explanations: A discussion. arXiv preprint. arXiv:1906.04774"},{"issue":"11","key":"7118_CR42","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","volume":"86","author":"Y LeCun","year":"1998","unstructured":"LeCun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998). Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11), 2278\u20132324.","journal-title":"Proceedings of the IEEE"},{"key":"7118_CR43","unstructured":"Lee, S.-W., Watkins, C., & Zhang, B.-T. (2014). Non-parametric Bayesian sum\u2013product networks. In ICML workshop on learning tractable probabilistic models, 2014."},{"key":"7118_CR44","doi-asserted-by":"crossref","unstructured":"Li, Z., Huang, J., & Naik, M. (2023). Scallop: A language for neurosymbolic programming. In Proceedings of the ACM on programming languages, 2023.","DOI":"10.1145\/3591280"},{"issue":"4","key":"7118_CR47","doi-asserted-by":"publisher","first-page":"2365","DOI":"10.1109\/JBHI.2023.3336721","volume":"19","author":"W Liu","year":"2023","unstructured":"Liu, W., Zhao, F., Shankar, A., Maple, C., Peter, J. D., Kim, B.-G., Slowik, A., Parameshachari, B., & Lv, J. (2023). Explainable AI for medical image analysis in medical cyber\u2013physical systems: Enhancing transparency and trustworthiness of IoMT. IEEE Journal of Biomedical and Health Informatics, 19(4), 2365\u20132376.","journal-title":"IEEE Journal of Biomedical and Health Informatics"},{"key":"7118_CR46","doi-asserted-by":"publisher","DOI":"10.1016\/j.jvcir.2019.02.012","author":"Y Liu","year":"2019","unstructured":"Liu, Y., & Luo, T. (2019). The optimization of sum\u2013product network structure learning. Journal of Visual Communication and Image Representation. https:\/\/doi.org\/10.1016\/j.jvcir.2019.02.012","journal-title":"Journal of Visual Communication and Image Representation"},{"key":"7118_CR45","doi-asserted-by":"crossref","unstructured":"Liu, Z., Luo, P., Wang, X., & Tang, X. (2015). Deep learning face attributes in the wild. In CVPR, 2015.","DOI":"10.1109\/ICCV.2015.425"},{"key":"7118_CR48","doi-asserted-by":"crossref","unstructured":"Long, J., Shelhamer, E., & Darrell, T. (2015). Fully convolutional networks for semantic segmentation. In CVPR, 2015.","DOI":"10.1109\/CVPR.2015.7298965"},{"key":"7118_CR49","unstructured":"Lundberg, S. M., & Lee, S. (2017). A unified approach to interpreting model predictions. In NeurIPS, 2017."},{"key":"7118_CR50","unstructured":"Mahinpei, A., Clark, J., Lage, I., Doshi-Velez, F., & Pan, W. (2021). Promises and pitfalls of black-box concept learning models. arXiv preprint. arXiv:2106.13314"},{"key":"7118_CR51","unstructured":"Manhaeve, R., Dumancic, S., Kimmig, A., Demeester, T., & Raedt, L. D. (2018). DeepProbLog: Neural probabilistic logic programming. In NeurIPS, 2018."},{"key":"7118_CR52","doi-asserted-by":"crossref","unstructured":"Marconato, E., Bortolotti, S., Krieken, E., Morettin, P., Umili, E., Vergari, A., Tsamoura, E., Passerini, A., & Teso, S. (2025). Symbol grounding in neuro-symbolic AI: A gentle introduction to reasoning shortcuts. arXiv preprint. arXiv:2510.14538","DOI":"10.1613\/jair.1.22389"},{"key":"7118_CR53","unstructured":"Marconato, E., Bortolotti, S., Krieken, E., Vergari, A., Passerini, A., & Teso, S. (2024). BEARS make neuro-symbolic models aware of their reasoning shortcuts. In UAI, 2024."},{"key":"7118_CR54","doi-asserted-by":"crossref","unstructured":"Marconato, E., Passerini, A., & Teso, S. (2022). GlanceNets: Interpretable, leak-proof concept-based models. In NeurIPS, 2022.","DOI":"10.52202\/068431-1542"},{"key":"7118_CR55","doi-asserted-by":"crossref","unstructured":"Marconato, E., Teso, S., Vergari, A., & Passerini, A. (2023). Not all neuro-symbolic concepts are created equal: Analysis and mitigation of reasoning shortcuts. In NeurIPS, 2023.","DOI":"10.52202\/075280-3170"},{"key":"7118_CR56","doi-asserted-by":"crossref","unstructured":"Mauro, N. D., Esposito, F., Ventola, F. G., & Vergari, A. (2017). Alternative variable splitting methods to learn sum\u2013product networks. In AI*IA, 2017.","DOI":"10.1007\/978-3-319-70169-1_25"},{"key":"7118_CR57","doi-asserted-by":"crossref","unstructured":"Molina, A., Vergari, A., Mauro, N. D., Natarajan, S., Esposito, F., & Kersting, K. (2018). Mixed sum\u2013product networks: A deep architecture for hybrid domains. In AAAI, 2018.","DOI":"10.1609\/aaai.v32i1.11731"},{"key":"7118_CR58","unstructured":"Oikarinen, T. P., Das, S., Nguyen, L. M., & Weng, T. (2023). Label-free concept bottleneck models. In ICLR, 2023."},{"key":"7118_CR59","unstructured":"Peharz, R. (2015). Foundations of sum\u2013product networks for probabilistic modeling. PhD Thesis, Medical University of Graz."},{"key":"7118_CR60","unstructured":"Peharz, R., Gens, R., & Domingos, P. (2014). Learning selective sum\u2013product networks. In ICML, 2014."},{"key":"7118_CR61","unstructured":"Peharz, R., Lang, S., Vergari, A., Stelzner, K., Molina, A., Trapp, M., Broeck, G. V., Kersting, K., & Ghahramani, Z. (2020). Einsum networks: Fast and scalable learning of tractable probabilistic circuits. In ICML, 2020."},{"key":"7118_CR62","unstructured":"Peharz, R., Tschiatschek, S., Pernkopf, F., & Domingos, P. M. (2015). On theoretical properties of sum\u2013product networks. In AISTATS, 2015."},{"key":"7118_CR63","unstructured":"Peharz, R., Vergari, A., Stelzner, K., Molina, A., Trapp, M., Shao, X., Kersting, K., & Ghahramani, Z. (2019). Random sum\u2013product networks: A simple and effective approach to probabilistic deep learning. In UAI, 2019."},{"key":"7118_CR64","unstructured":"Pipatsrisawat, K., & Darwiche, A. (2008). New compilation languages based on structured decomposability. In AAAI, 2008."},{"key":"7118_CR65","doi-asserted-by":"crossref","unstructured":"Poon, H., & Domingos, P. M. (2011). Sum\u2013product networks: A new deep architecture. In UAI, 2011.","DOI":"10.1109\/ICCVW.2011.6130310"},{"key":"7118_CR66","unstructured":"Rahman, T., & Gogate, V. (2016). Merging strategies for sum\u2013product networks: From trees to graphs. In UAI, 2016."},{"key":"7118_CR67","unstructured":"Rashwan, A., Poupart, P., & Chen, Z. (2018). Discriminative training of sum\u2013product networks by extended Baum\u2013Welch. In PGM, 2018."},{"key":"7118_CR68","unstructured":"Rashwan, A., Zhao, H., & Poupart, P. (2016). Online and distributed Bayesian moment matching for parameter learning in sum\u2013product networks. In AISTATS, 2016."},{"key":"7118_CR69","doi-asserted-by":"crossref","unstructured":"Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). \u201cWhy should I trust you?\u201d: Explaining the predictions of any classifier. In KDD, 2016.","DOI":"10.18653\/v1\/N16-3020"},{"issue":"1","key":"7118_CR70","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s44206-023-00081-z","volume":"3","author":"KM Richmond","year":"2024","unstructured":"Richmond, K. M., Muddamsetty, S. M., Gammeltoft-Hansen, T., Olsen, H. P., & Moeslund, T. B. (2024). Explainable AI and law: An evidential survey. Digital Society, 3(1), 1.","journal-title":"Digital Society"},{"key":"7118_CR71","doi-asserted-by":"publisher","first-page":"3667","DOI":"10.1007\/s10489-024-05321-4","volume":"54","author":"DM Rodr\u00edguez","year":"2024","unstructured":"Rodr\u00edguez, D. M., Cu\u00e9llar, M. P., & Morales, D. P. (2024). Concept logic trees: Enabling user interaction for transparent image classification and human-in-the-loop learning. Applied Intelligence, 54, 3667\u20133679.","journal-title":"Applied Intelligence"},{"key":"7118_CR72","unstructured":"Rooshenas, A., & Lowd, D. (2014). Learning sum\u2013product networks with direct and indirect variable interactions. In ICML, 2014."},{"key":"7118_CR73","unstructured":"Ruder, S. (2017). An overview of multi-task learning in deep neural networks. arXiv preprint. arXiv:1706.05098"},{"key":"7118_CR74","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1038\/s42256-019-0048-x","volume":"1","author":"C Rudin","year":"2019","unstructured":"Rudin, C. (2019). Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence, 1, 206\u2013215.","journal-title":"Nature Machine Intelligence"},{"issue":"7","key":"7118_CR75","first-page":"3821","volume":"44","author":"R S\u00e1nchez-Cauce","year":"2021","unstructured":"S\u00e1nchez-Cauce, R., Par\u00eds, I., & D\u00edez, F. J. (2021). Sum\u2013product networks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(7), 3821\u20133839.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"7118_CR76","first-page":"3821","volume":"44","author":"R S\u00e1nchez-Cauce","year":"2022","unstructured":"S\u00e1nchez-Cauce, R., Par\u00eds, I., & D\u00edez, F. J. (2022). Sum\u2013product networks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44, 3821\u20133839.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"7118_CR77","doi-asserted-by":"crossref","unstructured":"Sarkar, A., Vijaykeerthy, D., Sarkar, A., Balasubramanian, V. N. (2022). A framework for learning ante-hoc explainable models via concepts. In CVPR, 2022.","DOI":"10.1109\/CVPR52688.2022.01004"},{"key":"7118_CR78","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2022.3167702","author":"Y Sawada","year":"2022","unstructured":"Sawada, Y., & Nakamura, K. (2022). Concept bottleneck model with additional unsupervised concepts. IEEE Access. https:\/\/doi.org\/10.1109\/ACCESS.2022.3167702","journal-title":"IEEE Access"},{"key":"7118_CR79","doi-asserted-by":"crossref","unstructured":"Selvaraju, R. R., Cogswell, M., Das, A., Vedantam, R., Parikh, D., & Batra, D. (2017). Grad-CAM: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.","DOI":"10.1109\/ICCV.2017.74"},{"key":"7118_CR80","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/j.ijar.2021.10.011","volume":"140","author":"X Shao","year":"2022","unstructured":"Shao, X., Molina, A., Vergari, A., Stelzner, K., Peharz, R., Liebig, T., & Kersting, K. (2022). Conditional sum\u2013product networks: Modular probabilistic circuits via gate functions. International Journal of Approximate Reasoning, 140, 298\u2013313.","journal-title":"International Journal of Approximate Reasoning"},{"key":"7118_CR81","unstructured":"Shih, A., Sadigh, D., & Ermon, S. (2021). HyperSPNs: Compact and expressive probabilistic circuits. In NeurIPS, 2021."},{"key":"7118_CR82","doi-asserted-by":"crossref","unstructured":"Slack, D., Hilgard, S., Jia, E., Singh, S., & Lakkaraju, H. (2020). Fooling LIME and SHAP: Adversarial attacks on post hoc explanation methods. In AIES, 2020.","DOI":"10.1145\/3375627.3375830"},{"key":"7118_CR83","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2012.02.016","author":"J Stallkamp","year":"2012","unstructured":"Stallkamp, J., Schlipsing, M., Salmen, J., & Igel, C. (2012). Man vs. computer: Benchmarking machine learning algorithms for traffic sign recognition. Neural Networks. https:\/\/doi.org\/10.1016\/j.neunet.2012.02.016","journal-title":"Neural Networks"},{"key":"7118_CR84","unstructured":"Sutskever, I., Vinyals, O., & Le, Q. V. (2014). Sequence to sequence learning with neural networks. In NeurIPS, 2014."},{"key":"7118_CR85","unstructured":"Trapp, M., Peharz, R., Ge, H., Pernkopf, F., & Ghahramani, Z. (2019). Bayesian learning of sum\u2013product networks. In NeurIPS, 2019."},{"key":"7118_CR86","unstructured":"Trapp, M., Peharz, R., Skowron, M., Madl, T., Pernkopf, F., & Trappl, R. (2016). Structure inference in sum\u2013product networks using infinite sum\u2013product trees. In NeurIPS workshop on practical Bayesian nonparametrics, 2016."},{"key":"7118_CR87","doi-asserted-by":"crossref","unstructured":"Vandenhirtz, M., Laguna, S., Marcinkevics, R., & Vogt, J. E. (2024). Stochastic concept bottleneck models. In NeurIPS, 2024.","DOI":"10.52202\/079017-1641"},{"key":"7118_CR88","unstructured":"Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., & Polosukhin, I. (2017). Attention is all you need. In NeurIPS, 2017."},{"key":"7118_CR89","unstructured":"Ventola, F., Stelzner, K., Molina, A., & Kersting, K. (2020). Residual sum\u2013product networks. In International conference on probabilistic graphical models, 2020. PMLR."},{"key":"7118_CR90","doi-asserted-by":"crossref","unstructured":"Vergari, A., Mauro, N. D., & Esposito, F. (2015). Simplifying, regularizing and strengthening sum\u2013product network structure learning. In ECML PKDD, 2015.","DOI":"10.1007\/978-3-319-23525-7_21"},{"key":"7118_CR91","doi-asserted-by":"crossref","unstructured":"Verreet, V., De Smet, L., De Raedt, L., & Sansone, E. (2024). Explain, agree, learn: Scaling learning for neural probabilistic logic. arXiv preprint. arXiv:2408.08133","DOI":"10.3233\/FAIA240634"},{"key":"7118_CR92","doi-asserted-by":"publisher","DOI":"10.48550\/arXiv.1711.00399","author":"S Wachter","year":"2017","unstructured":"Wachter, S., Mittelstadt, B., & Russell, C. (2017). Counterfactual explanations without opening the black box: Automated decisions and the GDPR. Harvard Journal of Law and Technology. https:\/\/doi.org\/10.48550\/arXiv.1711.00399","journal-title":"Harvard Journal of Law and Technology"},{"key":"7118_CR94","doi-asserted-by":"crossref","unstructured":"Wang, T., & Chen, S.-C. (2020). Multi-label multi-task learning with dynamic task weight balancing. In IEEE international conference on information reuse and integration for data science (IRI), 2020.","DOI":"10.1109\/IRI49571.2020.00042"},{"key":"7118_CR93","unstructured":"Wang, W., & Carreira-Perpin\u00e1n, M. A. (2013). Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application. arXiv preprint. arXiv:1309.1541"},{"key":"7118_CR96","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2023.05.010","author":"R Xia","year":"2023","unstructured":"Xia, R., Zhang, Y., Liu, X., & Yang, B. (2023). A survey of sum\u2013product networks structural learning. Neural Networks. https:\/\/doi.org\/10.1016\/j.neunet.2023.05.010","journal-title":"Neural Networks"},{"key":"7118_CR95","doi-asserted-by":"publisher","first-page":"2251","DOI":"10.1109\/TPAMI.2018.2857768","volume":"41","author":"Y Xian","year":"2018","unstructured":"Xian, Y., Lampert, C. H., Schiele, B., & Akata, Z. (2018). Zero-shot learning\u2014A comprehensive evaluation of the good, the bad and the ugly. IEEE Transactions on Pattern Analysis and Machine Intelligence, 41, 2251\u20132265.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"7118_CR97","unstructured":"Xu, J., Zhang, Z., Friedman, T., Liang, Y., & Broeck, G. V. (2018). A semantic loss function for deep learning with symbolic knowledge. In ICML, 2018."},{"key":"7118_CR98","doi-asserted-by":"crossref","unstructured":"Yang, Z., Zhao, Z., Wang, B., Zhang, J., Li, L., Pei, H., Karlas, B., Liu, J., Guo, H., Zhang, C., & Li, B. (2022). Improving certified robustness via statistical learning with logical reasoning. In NeurIPS, 2022.","DOI":"10.52202\/068431-2526"},{"key":"7118_CR99","unstructured":"Yeh, C., Kim, B., Arik, S. \u00d6., Li, C., Pfister, T., & Ravikumar, P. (2020). On completeness-aware concept-based explanations in deep neural networks. In NeurIPS, 2020."},{"key":"7118_CR100","unstructured":"Yu, Z., Dhami, D. S., & Kersting, K. (2021). Sum\u2013product-attention networks: Leveraging self-attention in probabilistic circuits. arXiv preprint. arXiv:2109.06587"},{"key":"7118_CR101","unstructured":"Y\u00fcksekg\u00f6n\u00fcl, M., Wang, M., & Zou, J. (2023). Post hoc concept bottleneck models. In ICLR, 2023."},{"key":"7118_CR102","doi-asserted-by":"crossref","unstructured":"Zarlenga, M. E., Barbiero, P., Ciravegna, G., Marra, G., Giannini, F., Diligenti, M., Shams, Z., Precioso, F., Melacci, S., Weller, A., Li\u00f3, P., & Jamnik, M. (2022). Concept embedding models: Beyond the accuracy\u2013explainability trade-off. In NeurIPS, 2022.","DOI":"10.52202\/068431-1555"},{"key":"7118_CR103","doi-asserted-by":"crossref","unstructured":"Zarlenga, M. E., Collins, K., Dvijotham, K., Weller, A., Shams, Z., & Jamnik, M. (2023). Learning to receive help: Intervention-aware concept embedding models. In NeurIPS, 2023.","DOI":"10.52202\/075280-1648"},{"key":"7118_CR104","doi-asserted-by":"crossref","unstructured":"Zhang, J., Li, L., Zhang, C., & Li, B. (2023). CARE: Certifiably robust learning with reasoning via variational inference. In SaTML, 2023.","DOI":"10.1109\/SaTML54575.2023.00043"},{"issue":"12","key":"7118_CR105","doi-asserted-by":"publisher","first-page":"5586","DOI":"10.1109\/TKDE.2021.3070203","volume":"34","author":"Y Zhang","year":"2021","unstructured":"Zhang, Y., & Yang, Q. (2021). A survey on multi-task learning. IEEE Transactions on Knowledge and Data Engineering, 34(12), 5586\u20135609.","journal-title":"IEEE Transactions on Knowledge and Data Engineering"},{"key":"7118_CR106","unstructured":"Zhao, H., Adel, T., Gordon, G. J., & Amos, B. (2016). Collapsed variational inference for sum\u2013product networks. In ICML, 2016."},{"key":"7118_CR107","unstructured":"Zhao, H., Melibari, M., & Poupart, P. (2015a). On the relationship between sum\u2013product networks and Bayesian networks. In ICML, 2015."},{"key":"7118_CR108","unstructured":"Zhao, H., Melibari, M., & Poupart, P. (2015b). On the relationship between sum\u2013product networks and Bayesian networks. In International conference on machine learning, 2015 (pp. 116\u2013124). PMLR."},{"key":"7118_CR109","unstructured":"Zhao, H., Poupart, P., & Gordon, G. J. (2016). A unified approach for learning the parameters of sum\u2013product networks. In NeurIPS, 2016."}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-026-07118-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-026-07118-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-026-07118-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T06:24:51Z","timestamp":1790749491000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-026-07118-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,29]]},"references-count":109,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["7118"],"URL":"https:\/\/doi.org\/10.1007\/s10994-026-07118-7","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,29]]},"assertion":[{"value":"12 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"207"}}