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Overview Introduction Mixed graphical models SampleSearch Sampling with Searching Exploiting structure in samplinig AND OR Importance sampling Overview Introduction Mixed graphical models SampleSearch Sampling with Searching Exploiting structure in samplinig AND OR Importance sampling Bayesian Networks Representation Pearl 1988 P S Smoking P C S P B S Bronchitis lung Cancer CPTs P Xi pa Xi n P X P Xi pa Xi i 1 P X C S X ray P D C B Dyspnoea P S C B X D P S P C S P B S P X C S P D C B Belief Updating P lung cancer yes smoking no dyspnoea yes Mixed Networks Mixing Belief and Constraints Belief or Bayesian Networks Constraint Networks A A F B C P D B C B C E B D F B D 0 D 1 0 0 0 1 0 1 1 9 1 0 3 7 1 1 1 0 Variables A B C D E F Domains DA DB DC DD DE DF 0 1 CPTS P A P B A P C A P D B C P E A B P F A C R3 BCD E D B 0 C 0 D 0 is not allowed B 1 C 1 D 1 is not allowed R Variables A B C D E F Domains DA DB DC DD DE DF 0 1 Constraints R1 ABC R2 ACF R3 BCD R4 A E Expresses the set of solutions sol R Constraints could be specified externally or may occur as zeros in the Belief network Mixed networks Distribution and Queries The distribution represented by a mixed network 1 T B R P x if x sol R PT x M B 0 otherwise Queries Weighted Counting Equivalent to P e partition function solution counting M PB x x sol R Marginal distribution PT X i Applications Determinism More Ubiquitous than you may think Transportation Planning Liao et al 2004 Gogate et al 2005 Predicting and Inferring Car Travel Activity of individuals Genetic Linkage Analysis Fischelson and Geiger 2002 associate functionality of genes to their location on chromosomes Functional Software Verification Bergeron 2000 Generating random test programs to check validity of hardware First Order Probabilistic models Domingos et al 2006 Milch et al 2005 Citation matching L11m L11f L12m X11 S13m X12 L13m Model for locus 1 L21m S23m X13 L22m X21 Model for locus 2 S13f L13f L21f Functions on Orange nodes are deterministic L12f X22



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