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1、第十讲 概率图模型导论 Chapter 10 Introduction to Probabilistic Graphical Models,Weike Pan,and Congfu XuInstitute of Artificial Intelligence College of Computer Science,Zhejiang UniversityOctober 12,2006,浙江大学计算机学院人工智能引论课件,References,An Introduction to Probabilistic Graphical Models.Michael I.Jordan.http:/www.c
2、s.berkeley.edu/jordan/graphical.html,Outline,PreparationsProbabilistic Graphical Models(PGM)Directed PGMUndirected PGMInsights of PGM,Outline,PreparationsPGM“is”a universal modelDifferent thoughts of machine learningDifferent training approachesDifferent data typesBayesian FrameworkChain rules of pr
3、obability theoryConditional IndependenceProbabilistic Graphical Models(PGM)Directed PGMUndirected PGMInsights of PGM,Different thoughts of machine learning,Statistics(modeling uncertainty,detailed information)vs.Logics(modeling complexity,high level information)Unifying Logical and Statistical AI.Pe
4、dro Domingos,University of Washington.AAAI 2006.Speech:Statistical information(Acoustic model+Language model+Affect model)+High level information(Expert/Logics),Different training approaches,Maximum Likelihood Training:MAP(Maximum a Posteriori)vs.Discriminative Training:Maximum Margin(SVM)Speech:cla
5、ssical combination Maximum Likelihood+Discriminative Training,Different data types,Directed acyclic graph(Bayesian Networks,BN)Modeling asymmetric effects and dependencies:causal/temporal dependence(e.g.speech analysis,DNA sequence analysis)Undirected graph(Markov Random Fields,MRF)Modeling symmetri
6、c effects and dependencies:spatial dependence(e.g.image analysis),PGM“is”a universal model,To model both temporal and spatial data,by unifyingThoughts:Statistics+LogicsApproaches:Maximum Likelihood Training+Discriminative Training Further more,the directed and undirected models together provide mode
7、ling power beyond that which could be provided by either alone.,Bayesian Framework,What we care is the conditional probability,and its is a ratio of two marginal probabilities.,A posteriori probability,Likelihood,Priori probability,Class i,Normalization factor,Observation,Problem description Observa
8、tion Conclusion(classification or prediction),Bayesian rule,Chain rules of probability theory,Conditional Independence,Outline,PreparationsProbabilistic Graphical Models(PGM)Directed PGMUndirected PGMInsights of PGM,PGM,Nodes represent random variables/statesThe missing arcs represent conditional in
9、dependence assumptions The graph structure implies the decomposition,Directed PGM(BN),Representation,Conditional Independence,Probability Distribution,Queries,Implementation,Interpretation,Probability Distribution,Definition of Joint Probability Distribution,Check:,Representation,Graphical models re
10、present joint probability distributions more economically,using a set of“local”relationships among variables.,Conditional Independence(basic),Assert the conditional independence of a node from its ancestors,conditional on its parents.,Interpret missing edges in terms of conditional independence,Cond
11、itional Independence(3 canonical graphs),Classical Markov chain“Past”,“present”,“future”,Common causeY“explains”all the dependencies between X and Z,Marginal Independence,Common effect Multiple,competing explanation,Conditional Independence,Conditional Independence(check),One incoming arrow and one
12、outgoing arrow,Two outgoing arrows,Two incoming arrows,Check through reachability,Bayes ball algorithm(rules),Outline,PreparationsProbabilistic Graphical Models(PGM)Directed PGMUndirected PGMInsights of PGM,Undirected PGM(MRF),Representation,Conditional Independence,Probability Distribution,Queries,
13、Implementation,Interpretation,Probability Distribution(1),CliqueA clique of a graph is a fully-connected subset of nodes.Local functions should not be defined on domains of nodes that extend beyond the boundaries of cliques.Maximal cliquesThe maximal cliques of a graph are the cliques that cannot be
14、 extended to include additional nodes without losing the probability of being fully connected.We restrict ourselves to maximal cliques without loss of generality,as it captures all possible dependencies.Potential function(local parameterization):potential function on the possible realizations of the
15、 maximal clique,Probability Distribution(2),Maximal cliques,Probability Distribution(3),Joint probability distribution Normalization factor,Boltzman distribution,Conditional Independence,Its a“reachability”problem in graph theory.,Representation,Outline,PreparationsProbabilistic Graphical Models(PGM
16、)Directed PGMUndirected PGMInsights of PGM,Insights of PGM(Michael I.Jordan),Probabilistic Graphical Models are a marriage between probability theory and graph theory.A graphical model can be thought of as a probabilistic database,a machine that can answer“queries”regarding the values of sets of ran
17、dom variables.We build up the database in pieces,using probability theory to ensure that the pieces have a consistent overall interpretation.Probability theory also justifies the inferential machinery that allows the pieces to be put together“on the fly”to answer the queries.In principle,all“queries”of a probabilistic database can be answered if we have in hand the joint probability distribution.,Insights of PGM(data structure&algorithm),A graphical model is a natural/perfect tool for representation(数据结构)andinference(算法).,Thanks!,
