Relational Dynamic Bayesian Networks: a report. Cristina Manfredotti


 Karen McDonald
 1 years ago
 Views:
Transcription
1 Relational Dynamic Bayesian Networks: a report Cristina Manfredotti Dipartimento di Informatica, Sistemistica e Comunicazione (D.I.S.Co.) Università degli Studi MilanoBicocca Bayesian Networks: C T T Encode the joint probability distribution of a set of variables, as a Direct Acyclic Graph A Direct Acyclic Graph which nodes are conditionally independent of its nondescendent given its parents B T T P(D C,B) D T P(E D) C E D D T A P(C) T.90 P( D) P(B A).05 B A P(A) 0.01 P(A,B,C,D,E,) = = P(A)P(B A)P(C)P(D C,B)P(E D)P( D) = P(Z i Pa(Z i )) Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 2
2 The alarm example(1) I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglary? Variables: BurglarEnter, EarthquakeAppens, AlarmRings, JohnCalls, MaryCalls Network topology reflects "causal" knowledge: A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call from Russel&Norvig Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 3 The alarm example(2) Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 4
3 Bayesian Networks Each node is a variable: Two different nodes in the network This is why we have such structure: Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 5 Bayesian Networks If we should have 4 neighbors? We have to construct a graph with 2 more knods. Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 6
4 A large BN Thanks to Mark Chavira Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 7 Relational Domain Objects: groups of attributes which belong together (tables of a database), c.f. a structure in a programming language e.g.: Object Relational Domain contains a set of objects with relations and/or predicates between them e.g.: Relation neighbor alarm burglar (honer of an house) neighbor s attributes: his capacity of hearing, his attention,... alarm s attributes: its volume, its sensibility,... e.g.: Predicate tocall (the honer of the house) tohear (the alarm) toring Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 8
5 The alarm Relational Domain: Burglary Alarm Volume Sensibility ToRing... Listening Neighbor DegOfDef NoiseAround Teleph... Calling Honer DegOfBelieve Teleph... Red words: predicates, that concern only the object itself Dashed arrows: relation between an object and an attribute of the object (or a predicate) Green arrows: dependence between two attributes Continouse arrows: relations between two objects Bold black words: objects names Black words: objects attributes (caracteristic of the variables, they make an instanciation of each object different by each other). Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 9 Relational Bayesian Networks difficult definition Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 10
6 Relational Bayesian Network Syntax RBN: a set of nodes, one per variable predicate/relation/attribute a directed, acyclic graph a conditional distribution for each node given its parents, this distribution must take into account the actual complexity of the nodes! Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 11 Alarm RBN: Earthquacke Neigh.DegOfDef Alarm.Volume Neigh.NoiseAround I relationated only that part of the graph, I could make the same for each knodes of the BN NeighborCalls Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 12
7 Conditional Probability Distribution/Table The CPTs will take into account the values of each attributes or each variable in the system (i.e. for each actor playing a role in the represented world) an object will be instantiated, the conditional probability of each variable will be the same but they will depend by the particular instantiation of their attributes. NOT ONLY BY THE ACT THAT THE ALARM HAS RANG E.g.: P(NeighCall Neigh.DegOfDef, Neigh.NoiseAround,Alarm.Vol) = =.90 if the Neighbor isn t def but he listen music (John case). =.70 if the Neighbor is def but his house is very quite (Mary case). Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 13 Relational Bayesian Networks Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 14
8 Dynamic Bayesian Networks: Extension of BN for modeling dynamic systems. State at time t represented by a set of random variables z t = (z 1,t,,z d,t ). The state at time t depends on the states at previous time steps. A 2TBN is a BN that contains variables from z t1 whose parents are variables from z t and/or z t1, and variables from z t without their parents. A 2TBN defines P(z t z t1 ) by means of a directed acyclic graph (DAG) as follows: P(z t z t1 ) = N i=1p(z i t Pa(z i t)) Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 15 Dynamic Bayesian Networks A Dynamic Bayesian Network (DBN) is defined to be a pair of Bayesian Networks (B 0, B ), where B 0 represents the initial distribution P(z 0 ), and B is a 2TBN, which defines the transition distribution P(z t+1 z t ). Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 16
9 Relational Dynamic Bayesian Nets: Once you defined a RBN and a DBN it is easy to define a RDBN... GUESS IT! Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 17 Particle ilters: Tecnique that implements a ricursive Bayesian Ilter through a Monte Carlo simulation. The key idea is to represent the posterior pdf with a set of random samples with associated weights and compute the estimation based on these samples and weights. As the number of samples becomes very large, this MC caratterization becomes an equivalent representation to the usual functional description of the posterior pdf, and the SIS algorithm filter approaches the optimal Bayesian estimate. Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 18
10 Particle iltering: steps ix the number of particles: M 1. Particle generation x ( m) k ~ p( xk xk 1) At time k arrives the observation/measure z k 2a. Weight computation w = p( z x *( m) ( m) k k k ) 2b. Weight normalization 3. Resampling w ( m) k = M w m= 1 *( m) k w *( m) k Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 19 Particle filtering operations Represents the required pdf by a set of samples with associated weights. Computs the estimate based in these samples and weights. Posterior pdf Sample space x ( m) k ~ p( xk xk 1 ) Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 20
11 Pros: Arbitrary pdf Most probable statespace NonGaussian noise More than one model Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 21 Cons: Computational complexity How to determine the number of particles Probable problems: density extraction, sampling variance Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 22
12 RaoBlackwellized P: Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 23 RaoBlackwellized P RaoBlackwellization: Some components of the model can have a liner dynamic and can be estimate by a traditional Kalman ilter. Kalman ilter is combine with P to reduce the number of particles to be used for a satisfying performance. Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 24
13 Domingos P Complex & Simple Predicates Abstractions: (set of pairs of objects which are related in some way) P smoothing on an Abstraction lattice Cristina Manfredotti D.I.S.Co. Università di Milano  Bicocca 25
Data Integration: A Theoretical Perspective
Data Integration: A Theoretical Perspective Maurizio Lenzerini Dipartimento di Informatica e Sistemistica Università di Roma La Sapienza Via Salaria 113, I 00198 Roma, Italy lenzerini@dis.uniroma1.it ABSTRACT
More informationA terminology model approach for defining and managing statistical metadata
A terminology model approach for defining and managing statistical metadata Comments to : R. Karge (49) 306576 2791 mail reinhard.karge@runsoftware.com Content 1 Introduction... 4 2 Knowledge presentation...
More informationQDquaderni. UPDRES User Profiling for a Dynamic REcommendation System E. Messina, D. Toscani, F. Archetti. university of milano bicocca
A01 084/01 university of milano bicocca QDquaderni department of informatics, systems and communication UPDRES User Profiling for a Dynamic REcommendation System E. Messina, D. Toscani, F. Archetti research
More informationLearning InstanceSpecific Predictive Models
Journal of Machine Learning Research 11 (2010) 33333369 Submitted 3/09; Revised 7/10; Published 12/10 Learning InstanceSpecific Predictive Models Shyam Visweswaran Gregory F. Cooper Department of Biomedical
More informationA Modeldriven Approach to Predictive Non Functional Analysis of Componentbased Systems
A Modeldriven Approach to Predictive Non Functional Analysis of Componentbased Systems Vincenzo Grassi Università di Roma Tor Vergata, Italy Raffaela Mirandola {vgrassi, mirandola}@info.uniroma2.it Abstract.
More informationVARIATIONAL ALGORITHMS FOR APPROXIMATE BAYESIAN INFERENCE
VARIATIONAL ALGORITHMS FOR APPROXIMATE BAYESIAN INFERENCE by Matthew J. Beal M.A., M.Sci., Physics, University of Cambridge, UK (1998) The Gatsby Computational Neuroscience Unit University College London
More informationProcess Query Systems
thesis_main 2005/11/21 13:09 page i #1 Process Query Systems Advanced Technologies for Process Detection and Tracking Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op
More informationMaximizing the Spread of Influence through a Social Network
Maximizing the Spread of Influence through a Social Network David Kempe Dept. of Computer Science Cornell University, Ithaca NY kempe@cs.cornell.edu Jon Kleinberg Dept. of Computer Science Cornell University,
More informationExpert in Disaster Recovery Scenarios. 1. Introduction. Michel Verheijen and Marcel E.M. Spruit
Expert in Disaster Recovery Scenarios Michel Verheijen and Marcel E.M. Spruit Many organizations rely heavily on the availability of their information systems. It is the responsibility of the disaster
More informationContextSpecific Independence in Bayesian Networks. P(z [ x) for all values x, y and z of variables X, Y and
ContextSpecific Independence in Bayesian Networks 115 ContextSpecific Independence in Bayesian Networks Craig Boutilier Dept. of Computer Science University of British Columbia Vancouver, BC V6T 1Z4
More informationCollective Learning in MultiAgent Systems Based on Cultural Algorithms
Collective Learning in MultiAgent Systems Based on Cultural Algorithms Abstract Juan Terán Universidad de Los Andes, CEMISID Mérida, Venezuela, 5101 carlostp@ula.ve José L. Aguilar Universidad de Los
More informationModeling Pixel Means and Covariances Using Factorized ThirdOrder Boltzmann Machines
Modeling Pixel Means and Covariances Using Factorized ThirdOrder Boltzmann Machines Marc Aurelio Ranzato Geoffrey E. Hinton Department of Computer Science  University of Toronto 10 King s College Road,
More informationMining Templates from Search Result Records of Search Engines
Mining Templates from Search Result Records of Search Engines Hongkun Zhao, Weiyi Meng State University of New York at Binghamton Binghamton, NY 13902, USA {hkzhao, meng}@cs.binghamton.edu Clement Yu University
More informationAnalysis of MapReduce Algorithms
Analysis of MapReduce Algorithms Harini Padmanaban Computer Science Department San Jose State University San Jose, CA 95192 4089241000 harini.gomadam@gmail.com ABSTRACT MapReduce is a programming model
More informationSelecting a Subset of Cases in SPSS: The Select Cases Command
Selecting a Subset of Cases in SPSS: The Select Cases Command When analyzing a data file in SPSS, all cases with valid values for the relevant variable(s) are used. If I opened the 1991 U.S. General Social
More informationAnalyzing the Security in the GSM Radio Network using Attack Jungles
Analyzing the Security in the GSM Radio Network using Attack Jungles Parosh Aziz Abdulla 1, Jonathan Cederberg 1, and Lisa Kaati 2 1 University of Uppsala, Sweden, email: {parosh, jonathan.cederberg}@it.uu.se
More informationEngineering Web Applications for Reuse
Engineering Web Applications for Reuse Daniel Schwabe *, Gustavo Rossi **, Luiselena Esmeraldo *, Fernando Lyardet** *Departamento de Informática, PUCRio, Brazil Email: {schwabe, luiselena} @inf.pucrio.br
More informationSupporting Keyword Search in Product Database: A Probabilistic Approach
Supporting Keyword Search in Product Database: A Probabilistic Approach Huizhong Duan 1, ChengXiang Zhai 2, Jinxing Cheng 3, Abhishek Gattani 4 University of Illinois at UrbanaChampaign 1,2 Walmart Labs
More informationProbabilistic Methods for Finding People
International Journal of Computer Vision 43(1), 45 68, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Probabilistic Methods for Finding People S. IOFFE AND D.A. FORSYTH Computer
More informationNo Free Lunch in Data Privacy
No Free Lunch in Data Privacy Daniel Kifer Penn State University dan+sigmod11@cse.psu.edu Ashwin Machanavajjhala Yahoo! Research mvnak@yahooinc.com ABSTRACT Differential privacy is a powerful tool for
More informationClean Answers over Dirty Databases: A Probabilistic Approach
Clean Answers over Dirty Databases: A Probabilistic Approach Periklis Andritsos University of Trento periklis@dit.unitn.it Ariel Fuxman University of Toronto afuxman@cs.toronto.edu Renée J. Miller University
More informationGaussian Process Dynamical Models
Gaussian Process Dynamical Models Jack M. Wang, David J. Fleet, Aaron Hertzmann Department of Computer Science University of Toronto, Toronto, ON M5S 3G4 {jmwang,hertzman}@dgp.toronto.edu, fleet@cs.toronto.edu
More informationProcessing Flows of Information: From Data Stream to Complex Event Processing
Processing Flows of Information: From Data Stream to Complex Event Processing GIANPAOLO CUGOLA and ALESSANDRO MARGARA Dip. di Elettronica e Informazione Politecnico di Milano, Italy A large number of distributed
More informationFactor Graphs and the SumProduct Algorithm
498 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 2, FEBRUARY 2001 Factor Graphs and the SumProduct Algorithm Frank R. Kschischang, Senior Member, IEEE, Brendan J. Frey, Member, IEEE, and HansAndrea
More informationGenerative or Discriminative? Getting the Best of Both Worlds
BAYESIAN STATISTICS 8, pp. 3 24. J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West (Eds.) c Oxford University Press, 2007 Generative or Discriminative?
More informationOn SetBased Multiobjective Optimization
1 On SetBased Multiobjective Optimization Eckart Zitzler, Lothar Thiele, and Johannes Bader Abstract Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can
More informationPerformance Matrix Exhibit 1
Background A White Paper Optimizing your Call Center through Simulation By Bill Hall, Call Center Services and Dr. Jon Anton, Purdue University The challenge for today's call centers is providing valueadded
More informationDirichlet Process Gaussian Mixture Models: Choice of the Base Distribution
Görür D, Rasmussen CE. Dirichlet process Gaussian mixture models: Choice of the base distribution. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 5(4): 615 66 July 010/DOI 10.1007/s1139001010511 Dirichlet
More informationUNIVERSITÀ DEGLI STUDI DI CATANIA facoltà di scienze matematiche, fisiche e naturali corso di laurea specialistica in fisica
UNIVERSITÀ DEGLI STUDI DI CATANIA facoltà di scienze matematiche, fisiche e naturali corso di laurea specialistica in fisica Alessio Vincenzo Cardillo Structural Properties of Planar Graphs of Urban Street
More informationModel Compression. Cristian Bucilă Computer Science Cornell University. Rich Caruana. Alexandru NiculescuMizil. cristi@cs.cornell.
Model Compression Cristian Bucilă Computer Science Cornell University cristi@cs.cornell.edu Rich Caruana Computer Science Cornell University caruana@cs.cornell.edu Alexandru NiculescuMizil Computer Science
More information