{"type":"ScholarlyArticle","title":"A Complete Characterization of Projectivity for Statistical Relational Models","authors":[{"name":"Manfred Jaeger","slug":"manfred-jaeger","given":"Manfred","family":"Jaeger","affiliation":"Computer Science Department , Aalborg University , Aalborg , Denmark"},{"name":"Oliver Schulte","slug":"oliver-schulte","given":"Oliver","family":"Schulte","affiliation":"School of Computing Science , Simon Fraser University , Burnaby , Canada"}],"abstract":"A generative probabilistic model for relational data consists of a family of probability distributions for relational structures over domains of different sizes. In most existing statistical relational learning (SRL) frameworks, these models are not projective in the sense that the marginal of the distribution for size-n structures on induced substructures of size k<n is equal to the given distribution for size-k structures. Projectivity is very beneficial in that it directly enables lifted inference and statistically consistent learning from sub-sampled relational structures.   In earlier work some simple fragments of SRL languages have been identified that represent projective models. However, no complete characterization of, and representation framework for projective models has been given. In this paper we fill this gap: exploiting representation theorems for infinite exchangeable arrays we introduce a class of directed graphical latent variable models that precisely correspond to the class of projective relational models. As a by-product we also obtain a characterization for when a given distribution over size-k structures is the statistical frequency distribution of size-k substructures in much larger size-n structures. These results shed new light onto the old open problem of how to apply Halpern et al.'s ``random worlds approach'' for probabilistic inference to general relational signatures.","doi":"10.24963/ijcai.2020/591","url":"https://ijcai.org/proceedings/2020/591","pdf":"https://ijcai.org/proceedings/2020/0591.pdf","markdown":"https://ijcai.org/proceedings/2020/591.md","bibtex":"https://ijcai.org/proceedings/2020/bibtex/591","volume":{"slug":"2020","title":"Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2020","url":"https://ijcai.org/proceedings/2020/","year":2020,"location":"Yokohama","dates":"January 2021 (postponed from 11-17 July 2020)","editors":"Christian Bessiere","isbn":"978-0-9992411-6-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2020"},"section":"Uncertainty in AI","track":"Main track","number":591,"pages":{"first":4283,"last":4290},"keywords":["Machine Learning: Probabilistic Machine Learning","Machine Learning: Relational Learning","Uncertainty in AI: Statistical Relational AI"],"videos":[{"label":"Video","url":"https://videos.ijcai20.org/25453/file_25453.m3u8","length_sec":294,"provider":"hls"}],"references":[{"position":1,"raw":"[Aldous, 1981] David J Aldous. Representations for partially exchangeable arrays of random variables. Journal of Mul- tivariate Analysis, 11(4):581-598, 1981.","title":"Representations for partially exchangeable arrays of random variables","authors":"J David, Aldous","year":"1981","venue":"Journal of Multivariate Analysis","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bacchus et al., 1992] Fahiem Bacchus, Adam Grove, Joseph Y. Halpern, and Daphne Koller. From statistics to beliefs. In Proc. of National Conference on Artificial Intelligence (AAAI-92), 1992.","title":"From statistics to beliefs","authors":"Bacchus","year":"1992","venue":"Proc. of National Conference on Artificial Intelligence (AAAI-92)","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Bacchus, 1990] Fahiem Bacchus. Representing and Rea- soning with Probabilistic Knowledge: A Logical Approach to Probabilities. MIT Press, Cambridge, MA, USA, 1990.","title":"Fahiem Bacchus. Representing and Reasoning with Probabilistic Knowledge: A Logical Approach to Probabilities","authors":"Bacchus","year":"1990","venue":null,"doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Chatterjee and Diaconis, 2013] Sourav Chatterjee and Persi Diaconis. Estimating and understanding exponential ran- dom graph models. The Annals of Statistics, 41(5):2428- 2461, 2013.","title":"Estimating and understanding exponential random graph models","authors":"Diaconis Chatterjee, Sourav Chatterjee, Persi Diaconis","year":"2013","venue":"The Annals of Statistics","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Diaconis and Janson, 2007] Persi Diaconis and Svante Jan- son. Graph limits and exchangeable random graphs. arXiv preprint arXiv:0712.2749, 2007.","title":"Graph limits and exchangeable random graphs","authors":"Persi Diaconis, Svante Janson","year":"2007","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Fenstad, 1967] Jens Erik Fenstad. Representations of prob- abilities defined on first order languages. In J. N. Cross- ley, editor, Sets, Models and Recursion Theory, pages 156- 172. North Holland, Amsterdam, 1967.","title":"Representations of probabilities defined on first order languages","authors":"Jens Erik Fenstad, Fenstad","year":"1967","venue":"Sets, Models and Recursion Theory","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Halpern, 1990] Joseph Y. Halpern. An analysis of first-order logics of probability. Artificial Intelligence, 46(3):311- 350, 1990.","title":"An analysis of first-order logics of probability","authors":"Y Joseph, Halpern","year":"1990","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":"/proceedings/1989/220"},{"position":8,"raw":"[Halpern, 2017] Joseph Y Halpern. Reasoning about uncer- tainty. MIT press, 2017.","title":"Reasoning about uncertainty","authors":"Joseph Y Halpern","year":"2017","venue":null,"doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Hoff et al., 2002] Peter D Hoff, Adrian E Raftery, and Mark S Handcock. Latent space approaches to social net- work analysis. Journal of the American Statistical Associ- ation, 97(460):1090-1098, 2002.","title":"Latent space approaches to social network analysis","authors":"Adrian E Peter D Hoff, Mark S Raftery, Handcock","year":"2002","venue":"Journal of the American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Hoover, 1979] Douglas N. Hoover. Relations on probability spaces and arrays of random variables. HPreprint, Institute for Advanced Study, Princeton, NJ, 2, 1979.","title":"Relations on probability spaces and arrays of random variables","authors":"N Douglas, Hoover","year":"1979","venue":null,"doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Jaeger and Schulte, 2018] Manfred Jaeger and Oliver Schulte. Inference, learning, and population size: Pro- jectivity for srl models. IJCAI-StarAI Workshop on Statistical-Relational AI, July 2018.","title":"Inference, learning, and population size: Projectivity for srl models","authors":"Schulte Jaeger, Manfred Jaeger, Oliver Schulte","year":"2018","venue":"IJCAI-StarAI Workshop on Statistical-Relational AI","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Jain et al., 2010] Dominik Jain, Andreas Barthels, and Michael Beetz. Adaptive markov logic networks: Learn- ing statistical relational models with dynamic parameters. In ECAI, pages 937-942, 2010.","title":"Adaptive markov logic networks: Learning statistical relational models with dynamic parameters","authors":"Dominik Jain, Andreas Barthels, Michael Beetz","year":"2010","venue":"ECAI","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Kallenberg, 2006] Olav Kallenberg. Probabilistic symme- tries and invariance principles. Springer Science & Busi- ness Media, 2006.","title":"Probabilistic symmetries and invariance principles","authors":"Olav Kallenberg","year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Kemp et al., 2006] Charles Kemp, Joshua B Tenenbaum, Thomas L Griffiths, Takeshi Yamada, and Naonori Ueda. Learning systems of concepts with an infinite relational model. In AAAI, volume 3, 2006.","title":"Learning systems of concepts with an infinite relational model","authors":"Charles Kemp, Joshua B Tenenbaum, Thomas L Griffiths, Takeshi Yamada, Naonori Ueda","year":"2006","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Kern-Isberner and Thimm, 2010] Gabriele Kern-Isberner and Matthias Thimm. Novel semantical approaches to relational probabilistic conditionals. In Twelfth Inter- national Conference on the Principles of Knowledge Representation and Reasoning, 2010.","title":"Novel semantical approaches to relational probabilistic conditionals","authors":"Gabriele Kern-Isberner, Matthias Thimm","year":"2010","venue":"Twelfth International Conference on the Principles of Knowledge Representation and Reasoning","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Khot et al., 2013] Tushar Khot, Jude Shavlik, and Sriraam Natarajan. Boostr, 2013. URL = http://pages.cs.wisc.edu/ ∼ tushar/Boostr/.","title":"Boostr","authors":"Tushar Khot, Jude Shavlik, Sriraam Natarajan","year":"2013","venue":null,"doi":null,"url":"http://pages.cs.wisc.edu/∼tushar/Boostr/","internal_url":null},{"position":17,"raw":"[Kimmig et al., 2011] Angelika Kimmig, Bart Demoen, Luc De Raedt, Vitor Santos Costa, and Ricardo Rocha. On the implementation of the probabilistic logic programming language problog. Theory and Practice of Logic Program- ming, 11(2-3):235-262, 2011.","title":"On the implementation of the probabilistic logic programming language problog","authors":"Angelika Kimmig, Bart Demoen, Luc De Raedt, Santos Vitor, Ricardo Costa, Rocha","year":"2011","venue":"Theory and Practice of Logic Programming","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Kimmig et al., 2014] Angelika Kimmig, Lilyana Mi- halkova, and Lise Getoor. Lifted graphical models: a survey. Machine Learning, pages 1-45, 2014.","title":"Lifted graphical models: a survey","authors":"Angelika Kimmig, Lilyana Mihalkova, Lise Getoor","year":"2014","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kipf and Welling, 2016] Thomas N Kipf and Max Welling. Variational graph auto-encoders. arXiv preprint arXiv:1611.07308, 2016.","title":"Variational graph auto-encoders","authors":"N Thomas, Max Kipf, Welling","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kuzelka et al., 2018] Ondrej Kuzelka, Yuyi Wang, Jesse Davis, and Steven Schockaert. Relational marginal prob- lems: Theory and estimation. In Proceedings of the Thirty- Second AAAI Conference on Artificial Intelligence (AAAI- 18), pages 1-8. AAAI Press, 2018.","title":"Relational marginal problems: Theory and estimation","authors":"Kuzelka","year":"2018","venue":"Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18)","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Niepert and Van den Broeck, 2014] Mathias Niepert and Guy Van den Broeck. Tractability through exchangeabil- ity: A new perspective on efficient probabilistic inference. In Twenty-Eighth AAAI Conference on Artificial Intelli- gence, 2014.","title":"Tractability through exchangeability: A new perspective on efficient probabilistic inference","authors":"Van Niepert, Broeck Den, Mathias Niepert, Guy Van Den Broeck","year":"2014","venue":"Twenty-Eighth AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Orbanz and Roy, 2014] Peter Orbanz and Daniel M Roy. Bayesian models of graphs, arrays and other exchangeable random structures. IEEE transactions on pattern analysis and machine intelligence, 37(2):437-461, 2014.","title":"Peter Orbanz and Daniel M Roy. Bayesian models of graphs, arrays and other exchangeable random structures","authors":"Roy Orbanz","year":"2014","venue":"IEEE transactions on pattern analysis and machine intelligence","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Poole et al., 2014] David Poole, David Buchman, Seyed Mehran Kazemi, Kristian Kersting, and Sriraam Natarajan. Population size extrapolation in relational probabilistic modelling. In International Conference on Scalable Uncertainty Management, pages 292-305.","title":"Population size extrapolation in relational probabilistic modelling","authors":"David Poole, David Buchman, Mehran Seyed, Kristian Kazemi, Sriraam Kersting, Natarajan","year":null,"venue":"International Conference on Scalable Uncertainty Management","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Poole, 1997] David Poole. The independent choice logic for modelling multiple agents under uncertainty. Artificial Intelligence, 94(1-2):7-56, 1997.","title":"The independent choice logic for modelling multiple agents under uncertainty","authors":"David Poole","year":"1997","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Rinaldo et al., 2009] Alessandro Rinaldo, Stephen E. Fien- berg, and Yi Zhou. On the geometry of discrete exponen- tial families with application to exponential random graph models. Electron. J. Statist., 3:446-484, 2009.","title":"On the geometry of discrete exponential families with application to exponential random graph models","authors":"Alessandro Rinaldo, Stephen E Fienberg, Yi Zhou","year":"2009","venue":"Electron. J. Statist","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Sato, 1995] Taisuke Sato. A statistical learning method for logic programs with distribution semantics. In Proceed- ings of the 12th International Conference on Logic Pro- gramming (ICLP'95), pages 715-729, 1995.","title":"A statistical learning method for logic programs with distribution semantics","authors":"Taisuke Sato","year":"1995","venue":"Proceedings of the 12th International Conference on Logic Programming (ICLP'95)","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Schulte and Khosravi, 2012] Oliver Schulte and Hassan Khosravi. Learning graphical models for relational data via lattice search. Machine Learning, 88(3):331-368, 2012.","title":"Learning graphical models for relational data via lattice search","authors":"Khosravi Schulte, Oliver Schulte, Hassan Khosravi","year":"2012","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Shalizi and Rinaldo, 2013] Cosma Rohilla Shalizi and Alessandro Rinaldo. Consistency under sampling of exponential random graph models. Annals of statistics, 41(2):508, 2013.","title":"Consistency under sampling of exponential random graph models","authors":"Rohilla Cosma, Alessandro Shalizi, Rinaldo","year":"2013","venue":"Annals of statistics","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Van den Broeck, 2011] Guy Van den Broeck. On the com- pleteness of first-order knowledge compilation for lifted probabilistic inference. In Advances in Neural Informa- tion Processing Systems, pages 1386-1394, 2011.","title":"On the completeness of first-order knowledge compilation for lifted probabilistic inference","authors":"Broeck ; Van Den, Guy Van Den Broeck","year":"2011","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Wu et al., 2020] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A com- prehensive survey on graph neural networks. IEEE Trans- actions on Neural Networks and Learning Systems, 2020.","title":"A comprehensive survey on graph neural networks","authors":"Wu","year":"2020","venue":"IEEE Transactions on Neural Networks and Learning Systems","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Xu et al., 2006] Zhao Xu, Volker Tresp, Kai Yu, and Hans- Peter Kriegel. Learning infinite hidden relational models. Uncertainity in Artificial Intelligence (UAI2006), 2006.","title":"Learning infinite hidden relational models","authors":"Zhao Xu, Kai Volker Tresp, Hans-Peter Yu, Kriegel","year":"2006","venue":"Uncertainity in Artificial Intelligence (UAI2006)","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}