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Announcements
Homework 2 due this Wednesday (Nov 4) in class Project milestones due next Monday (Nov 9)
About half the work should be done 4 pages of writeup, NIPS format http://nips.cc/PaperInformation/StyleFiles
Probabilistic Graphical Models Lecture 10 Undirected Models - - PowerPoint PPT Presentation
Probabilistic Graphical Models Lecture 10 Undirected Models - - PowerPoint PPT Presentation
Probabilistic Graphical Models Lecture 10 Undirected Models CS/CNS/EE 155 Andreas Krause Announcements Homework 2 due this Wednesday (Nov 4) in class Project milestones due next Monday (Nov 9) About half the work should be done 4 pages
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Markov Networks
(a.k.a., Markov Random Field, Gibbs Distribution, …) A Markov Network consists of
An undirected graph, where each node represents a RV A collection of factors defined over cliques in the graph
Joint probability A distribution factorizes over undirected graph G if
X1 X2 X4 X5 X7 X8 X6 X3 X9
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Computing Joint Probabilities
Computing joint probabilities in BNs Computing joint probabilities in Markov Nets
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Local Markov Assumption for MN
The Markov Blanket MB(X)
- f a node X is the set of
neighbors of X Local Markov Assumption: X EverythingElse | MB(X) Iloc(G) = set of all local independences G is called an I-map of distribution P if Iloc(G) I(P) X1 X2 X4 X5 X7 X8 X6 X3 X9
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Factorization Theorem for Markov Nets “”
- Iloc(G) I(P)
G is an I-map of P (independence map) True distribution P can be represented exactly as a Markov net (G,P)
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Factorization Theorem for Markov Nets “” Hammersley-Clifford Theorem
- Iloc(G) I(P)
True distribution P can be represented exactly as G is an I-map of P (independence map) and P>0 i.e., P can be represented as a Markov net (G,P)
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Global independencies
A trail X—X1—…—Xm—Y is called active for evidence E, if none of X1,…,Xm E Variables X and Y are called separated by E if there is no active trail for E connecting X, Y Write sep(X,Y | E) I(G) = {X Y | E: sep(X,Y|E)} X1 X2 X4 X5 X7 X8 X6 X3 X9
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Soundness of separation
Know: For positive distributions P>0 Iloc(G) I(P) P factorizes over G Theorem: Soundness of separation For positive distributions P>0
Iloc(G) I(P) I(G) I(P)
Hence, separation captures only true independences How about I(G) = I(P)?
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Completeness of separation
Theorem: Completeness of separation
I(G) = I(P)
for “almost all” distributions P that factorize over G “almost all”: Except for of potential parameterizations
- f measure 0 (assuming no finite set have positive
measure)
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Minimal I-maps
For BNs: Minimal I-map not unique For MNs: For positive P, minimal I-map is unique!! E B A J M E B A J M J M A E B
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P-maps
Do P-maps always exist? For BNs: no How about Markov Nets?
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Exact inference in MNs
Variable elimination and junction tree inference work exactly the same way!
Need to construct junction trees by obtaining chordal graph through triangulation
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Pairwise MNs
A pairwise MN is a MN where all factors are defined
- ver single variables or pairs of variables
Can reduce any MN to pairwise MN! X1 X2 X4 X5 X3
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Logarithmic representation
Can represent any positive distribution in log domain
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Log-linear models
Feature functions φi(D) defined over cliques Log linear model over undirected graph G
Feature functions φ1(D1),…,φk(Dk) Domains Di can overlap Set of weights wi learnt from data
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Converting BNs to MNs
Theorem: Moralized Bayes net is minimal Markov I-map
C D I G S L J H
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Converting MNs to BNs
Theorem: Minimal Bayes I-map for MN must be chordal X1 X2 X7 X8 X6 X3 X9
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So far
Markov Network Representation
Local/Global Markov assumptions; Separation Soundness and completeness of separation
Markov Network Inference
Variable elimination and Junction Tree inference work exactly as in Bayes Nets
How about Learning Markov Nets?
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Parameter Learning for Bayes nets
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Algorithm for BN MLE
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MLE for Markov Nets
Log likelihood of the data
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Log-likelihood doesn’t decompose
Log likelihood l(D | θ) is concave function! Log Partition function log Z(θ) doesn’t decompose
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Derivative of log-likelihood
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Derivative of log-likelihood
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Computing the derivative
Derivative Computing P(ci | ) requires inference! Can optimize using conjugate gradient etc.
C
D
I
G S L J H
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Alternative approach: Iterative Proportional Fitting (IPF)
At optimum, it must hold that Solve fixed point equation Must recompute parameters every iteration
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Parameter learning for log-linear models
Feature functions (Ci) defined over cliques Log linear model over undirected graph G
Feature functions 1(C1),…,k(Ck) Domains Ci can overlap
Joint distribution How do we get weights wi?
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Derivative of Log-likelihood 1
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Derivative of Log-likelihood 2
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Optimizing parameters
Gradient of log-likelihood Thus, w is MLE
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Regularization of parameters
Put prior on parameters w
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Summary: Parameter learning in MN
MLE in BN is easy (score decomposes) MLE in MN requires inference (score doesn’t decompose) Can optimize using gradient ascent or IPF
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