1、第十讲 概率图模型导论 Chapter 10 Introduction to Probabilistic Graphical Models,Weike Pan, and Congfu Xu panweike, Institute 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. Jo
2、rdan. http:/www.cs.berkeley.edu/jordan/graphical.html,Outline,Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM,Outline,Preparations PGM “is” a universal model Different thoughts of machine learning Different training approaches Different data types Bayesi
3、an Framework Chain rules of probability theory Conditional Independence Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM,Different thoughts of machine learning,Statistics (modeling uncertainty, detailed information) vs. Logics (modeling complexity, high level informat
4、ion)Unifying Logical and Statistical AI. Pedro 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)
5、vs. Discriminative Training: Maximum Margin (SVM)Speech: classical 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
6、 analysis)Undirected graph (Markov Random Fields, MRF) Modeling symmetric effects and dependencies: spatial dependence (e.g. image analysis),PGM “is” a universal model,To model both temporal and spatial data, by unifying Thoughts: Statistics + Logics Approaches: Maximum Likelihood Training + Discrim
7、inative Training Further more, the directed and undirected models together provide modeling 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
8、,Priori probability,Class i,Normalization factor,Observation,Problem descriptionObservation Conclusion (classification or prediction),Bayesian rule,Chain rules of probability theory,Conditional Independence,Outline,Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insight
9、s of PGM,PGM,Nodes represent random variables/states The missing arcs represent conditional independence assumptions The graph structure implies the decomposition,Directed PGM (BN),Representation,Conditional Independence,Probability Distribution,Queries,Implementation,Interpretation,Probability Dist
10、ribution,Definition of Joint Probability Distribution,Check:,Representation,Graphical models represent 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 ance
11、stors, conditional on its parents.,Interpret missing edges in terms of conditional independence,Conditional Independence (3 canonical graphs),Classical Markov chain “Past”, “present”, “future”,Common cause Y “explains” all the dependencies between X and Z,Marginal Independence,Common effectMultiple,
12、 competing explanation,Conditional Independence,Conditional Independence (check),One incoming arrow and one outgoing arrow,Two outgoing arrows,Two incoming arrows,Check through reachability,Bayes ball algorithm (rules),Outline,Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected
13、 PGM Insights of PGM,Undirected PGM (MRF),Representation,Conditional Independence,Probability Distribution,Queries,Implementation,Interpretation,Probability Distribution(1),Clique A clique of a graph is a fully-connected subset of nodes. Local functions should not be defined on domains of nodes that
14、 extend beyond the boundaries of cliques.Maximal cliques The maximal cliques of a graph are the cliques that cannot be 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
15、 possible dependencies.Potential function (local parameterization): potential function on the possible realizations of the maximal clique,Probability Distribution(2),Maximal cliques,Probability Distribution(3),Joint probability distributionNormalization factor,Boltzman distribution,Conditional Indep
16、endence,Its a “reachability” problem in graph theory.,Representation,Outline,Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM,Insights of PGM (Michael I. Jordan),Probabilistic Graphical Models are a marriage between probability theory and graph theory. A
17、graphical model can be thought of as a probabilistic database, a machine that can answer “queries” regarding the values of sets of random variables. We build up the database in pieces, using probability theory to ensure that the pieces have a consistent overall interpretation. Probability theory als
18、o 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) and inference (算法).,Thanks!,
