Relational Dynamic Bayesian Networks: a report. Cristina Manfredotti

Transcription

1 Relational Dynamic Bayesian Networks: a report Cristina Manfredotti Dipartimento di Informatica, Sistemistica e Comunicazione (D.I.S.Co.) Università degli Studi Milano-Bicocca [email protected] 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 non-descendent 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 t-1 whose parents are variables from z t and/or z t-1, and variables from z t without their parents. A 2TBN defines P(z t z t-1 ) by means of a directed acyclic graph (DAG) as follows: P(z t z t-1 ) = 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 state-space Non-Gaussian 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 Rao-Blackwellized P: Cristina Manfredotti D.I.S.Co. Università di Milano - Bicocca 23 Rao-Blackwellized P Rao-Blackwellization: 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