第十讲 概率图模型导论 Chapter 10 Introduction to Probabilistic Graphical Models 浙江大学计算机学院《人工智能引论》课件 第十讲 概率图模型导论 Chapter 10 Introduction to Probabilistic Graphical Models Weike Pan, and Congfu Xu {panweike, xucongfu}@zju.edu.cn Institute of Artificial Intelligence College of Computer Science, Zhejiang University October 12, 2006

References An Introduction to Probabilistic Graphical Models. Michael I. Jordan. http://www.cs.berkeley.edu/~jordan/graphical.html

Outline Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM

Outline Preparations Probabilistic Graphical Models (PGM) PGM “is” a universal model Different thoughts of machine learning Different training approaches Different data types Bayesian 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 information) 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) Machine Learning: supervised, unsupervised, reinforcement learning, multi-instance learning Thoughts: Statistics, Logics Trends: unifying the two thoughts together

Different training approaches Maximum Likelihood Training: MAP (Maximum a Posteriori) 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 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 + Discriminative Training Further more, the directed and undirected models together provide modeling power beyond that which could be provided by either alone.

Bayesian Framework Problem description Bayesian rule Observation  Conclusion (classification or prediction) Bayesian rule Likelihood Priori probability Observation A posteriori probability Class i Normalization factor What we care is the conditional probability, and it’s is a ratio of two marginal probabilities.

Chain rules of probability theory

Conditional Independence

Outline Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM

PGM Nodes represent random variables/states The missing arcs represent conditional independence assumptions The graph structure implies the decomposition

Directed PGM (BN) Probability Distribution Queries Representation Implementation Conditional Independence Interpretation

Probability Distribution 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) Interpret missing edges in terms of conditional independence Assert the conditional independence of a node from its ancestors, conditional on its parents.

Conditional Independence (3 canonical graphs) Marginal Independence Classical Markov chain “Past”, “present”, “future” Common cause Y “explains” all the dependencies between X and Z Common effect Multiple, competing explanation

Conditional Independence (check) Bayes ball algorithm (rules) One incoming arrow and one outgoing arrow Two outgoing arrows Two incoming arrows Check through reachability

Outline Preparations Probabilistic Graphical Models (PGM) Directed PGM Undirected PGM Insights of PGM

Undirected PGM (MRF) Probability Distribution Queries Representation Implementation Conditional Independence 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 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 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 distribution Normalization factor Boltzman distribution

Conditional Independence It’s 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 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 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) and inference (算法).

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