1. Happy Mittal Advisor : Parag Singla IIT Delhi Lifted Inference Rules With Constraints

2. Introduction Real World Problems : Relationship Between Entities Modeled by SQL, first-order logic etc Uncertainity in events Modeled by Markov networks, Bayesian networks etc Want to model both. Relational Models Probabilistic Models

3. Introduction : Markov Logic Networks Probability of a world x : Weight of formula iNo. of true groundings of formula i in x Intractable!!! InIIT(Ana) = True InIIT(Bob) = False GoodRsearch(Ana) = True …. Friends(Ana,Bob) = True World

4. Introduction : Markov Logic Networks Lifted Inference : Exploit symmetries (argue about group of objects together) Lifted Inference rules : Decomposer, Binomial [Gogate & Domingos 2011], Single Occurrence [Mittal et al, 2014] Probability of GoodResearch is same for every person. Symmetry breaks in case of evidence. Evidence imposes constraints on variables. How to do Lifted Inference in presence of constraints ? Focus of our work : Propose a new constraint language which supports efficient lifted inference. Modify Lifted Inference Rules to work with our proposed constraint language.

5. Outline Constraint Language Lifted Inference Rules Decomposer Binomial Experiments Conclusion and Future Work

6. Outline Constraint Language Lifted Inference Rules Decomposer Binomial Experiments Conclusion and Future Work

7. Constraint Language (SetInEq) Weighted Formulas with constraints. Three types of Atomic constraints : Subset Constraints Equality Constraints Inequality Constraints Constraint Tuple : Conjunction of atomic constraints Constraint Set : Disjunction of constraint tuples

8. Canonical Representation Require constrained theory be in canonical form to identify symmetries easily. Canonical form : Only one subset constraint per variable Identical supports for variables in eq/ineq constraints

9. Outline Constraint Language Lifted Inference Rules Decomposer Binomial Experiments Conclusion and Future Work

10. Lifted Inference Rules (Decomposer)

11. Condition for decomposer:  Variable should be in every predicate in a formula.

12. Decomposer with Constraints. Partitioning

13. Outline Constraint Language Lifted Inference Rules Decomposer Binomial Experiments Conclusion and Future Work

14. Lifted Inference Rules (Binomial)

15. Binomial Rule with Constraints Partitioning

16. Evidence Processing and Normal Form SetInEqNormal

17. Previous Approaches ApproachConstraint TypeConstraint Aggregation Tractable Solver Lifting Algorithm Lifted VE [Poole 2003] eq/ineq no subset intersection no union NoLifted VE CFOVE [Milch et al 2008] eq/ineq no subset intersection no union YesLifted VE GCFOVE [Taghipour et al 2012] Subset (tree based) No inequality Intersection Union YesLifted VE Approx LBP [Singla et al 2014] Subset (hypercube) No inequality Intersection Union NoLifted message passing Knowledge Compilation (KC) [Broeck et al 2011] eq/ineq subset Intersection no union NoFirst order knowledge compilation Lifted Inference from other side [Jha et al 2010] Normal forms (no constraints) NoneYesLifting rules : decomposer, binomial etc PTP [Gogate & Domingos 2011] eq/ineq no subset Intersection no union nolifted search & sampling: Decomposer, binomial Current Workeq/ineq subset Intersection Union Yeslifted search & sampling: Decomposer, binomial, single occurrence

18. Outline Constraint Language Lifted Inference Rules Decomposer Binomial Experiments Conclusion and Future Work

19. Experiments Algorithms Compared SetInEq : Our algorithm Normal : Normal form PTP : Available in alchemy 2 GCFOVE [Taghipour et al 2013]: Generalized counting for Lifted Variable Elimination

20. Experiments Exact Inference Performed exact inference on Friends & Smokers and Professors & students datasets Varying Domain size and evidence% PTP failed to scale to domain size of even 100. FS : size vs time (sec)FS : evidence vs time (sec)

21. Experiments Approximate Inference Performed approximate inference on WebKB and IMDB datasets GCFOVE doesn’t have approximate version. Approximate version of PTP is not fully implemented in alchemy-2. WebKB : size vs time (sec) IMDB : evidence vs time (sec)

22. Conclusion and Future work Proposed a new constraint language SetInEq. Subsumes existing formulations. Extended 3 key lifting rules : Decomposer Binomial Single Occurrence Experiments show efficacy of our approach. Future work : To do efficient learning in our constraint language.

23. THANK YOU