第十讲 概率图模型导论 chapter 10 introduction to probabilistic graphical models
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第十讲 概率图模型导论 Chapter 10 Introduction to Probabilistic Gra
phical Models
Weike Pan, and Congfu Xu{panweike, xucongfu}@zju.edu.cn
Institute of Artificial Intelligence
College of Computer Science, Zhejiang University
October 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.cs.berkeley.edu/~jordan/graphical.html
Outline
PreparationsProbabilistic 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
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)
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
( | ) ( )( | )
( )i i
i
P O c P cP c O
P O
What we care is the conditional probability, and it’s is a ratio of two marginal probabilities.
A posteriori probability
Likelihood Priori probability
Class iNormalization factor
Observation
Problem description Observation Conclusion (classification or prediction)
Bayesian rule
Chain rules of probability theory
Conditional Independence
Outline
PreparationsProbabilistic 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)
Representation
Conditional Independence
Probability Distribution Queries
Implementation
Interpretation
Probability Distribution
Definition of Joint Probability Distribution
( , ) 1i
i
i ix
f x x ( , ) 0ii if x x
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 ancestors, conditional on its parents.
Interpret missing edges in terms of conditional independence
Conditional 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
( , , ) ( ) ( ) ( | , )
( , , ) ( ) ( )
( , )
p x y z p x p z p y x z
p x y zp x p z
p x z
( , ) ( ) ( )p x z p x p z
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
PreparationsProbabilistic Graphical Models (PGM)
Directed PGM Undirected 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 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 ( )
CX CxCx
CX
Probability Distribution(2)
Maximal cliques
Probability Distribution(3)
Joint probability distribution
Normalization factor
1( ) ( )
CX CC
p x xZ
( )CX C
x C
Z x
1( ) ( )
1 exp{ ( )}
1 exp{ ( )}
1 exp{ ( )}
CX CC
C CC
C CC
p x xZ
H xZ
H xZ
H xZ
( )
exp{ ( )}
CX Cx C
x
Z x
H x
Boltzman distribution
Conditional Independence
It’s a “reachability” problem in graph theory.
Representation
Outline
PreparationsProbabilistic 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 ).
Thanks!
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