Lecture 33: Inference in Graphical Models and Intro to Structure Learning, Slides of Artificial Intelligence

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Inference in Graphical Models

and Intro to Structure Learning

Lecture 33 of 41

Lecture Outline

• More Bayesian Belief Networks (BBNs)
  • Inference: applying CPTs
  • Learning: CPTs from data, elicitation
  • In-class exercises
    • Hugin , BKD demos
    • CPT elicitation, application
• Learning BBN Structure
  • K2 algorithm
  • Other probabilistic scores and search algorithms
• Causal Discovery: Learning Causality from Observations
• Incomplete Data: Learning and Inference (Expectation-Maximization)
• Next Week: BBNs Concluded; Review for Midterm (11 October 2001)
• After Midterm: EM Algorithm, Unsupervised Learning, Clustering

Bayesian Networks:

Quick Review

X 1

X 2

X 3

X 4

Season: Spring Summer Fall Winter

Sprinkler: On, Off

Rain: None, Drizzle, Steady, Downpour

Ground-Moisture: Wet, Dry X 5 Ground-Slipperiness: Slippery, Not-Slippery

P ( Summer , Off , Drizzle , Wet , Not-Slippery ) = P ( S ) · P ( O | S ) · P ( D | S ) · P ( W | O , D ) · P ( N | W )

• Recall: Conditional Independence (CI) Assumptions
• Bayesian Network: Digraph Model
  • Vertices (nodes): denote events (each a random variable)
  • Edges (arcs, links): denote conditional dependencies
• Chain Rule for (Exact) Inference in BBNs
  • Arbitrary Bayesian networks: NP -complete
  • Polytrees: linear time
• Example (“Sprinkler” BBN)
• MAP, ML Estimation over BBNs



n i

P X 1 ,X 2 , ,Xn P Xi|parents Xi 1

hML  argmaxh  H P  D |h 

Learning Distributions in BBNs:

Quick Review

• Learning Distributions
  • Shortcomings of Naïve Bayes
  • Making judicious CI assumptions
  • Scaling up to BBNs: need to learn a CPT for all parent sets
  • Goal: generalization
    • Given D (e.g., {1011, 1001, 0100})
    • Would like to know P ( schema ): e.g., P (11**)  P ( x 1 = 1, x 2 = 1)
• Variants
  • Known or unknown structure
  • Training examples may have missing values
• Gradient Learning Algorithm
  • Weight update rule
  • Learns CPTs given data points D



x D ijk

h ij ik ijk ijk w

P y ,u |x w w r

• Constraint-Based
  • Perform tests of conditional independence
  • Search for network consistent with observed dependencies (or lack thereof)
  • Intuitive; closely follows definition of BBNs
  • Separates construction from form of CI tests
  • Sensitive to errors in individual tests
• Score-Based
  • Define scoring function ( aka score) that evaluates how well (in)dependencies in a structure match observations
  • Search for structure that maximizes score
  • Statistically and information theoretically motivated
  • Can make compromises
• Common Properties
  • Soundness: with sufficient data and computation, both learn correct structure
  • Both learn structure from observations and can incorporate knowledge

Learning Structure:

Constraints Versus Scores

Learning Structure:

Maximum Weight Spanning Tree (Chow-Liu)

• Algorithm Learn-Tree-Structure-I ( D )
  • Estimate P ( x ) and P ( x , y ) for all single RVs, pairs; I( X ; Y ) = D( P ( X , Y ) || P ( X ) · P ( Y ))
  • Build complete undirected graph: variables as vertices, I( X ; Y ) as edge weights
  • T  Build-MWST ( V  V , Weights ) // Chow-Liu algorithm: weight function  I
  • Set directional flow on T and place the CPTs on its edges (gradient learning)
  • RETURN: tree-structured BBN with CPT values
• Algorithm Build-MWST-Kruskal ( E  V  V , Weights : E  R +)
  • H  Build-Heap ( E , Weights ) // aka priority queue (| E |)
  • E’  Ø; Forest  {{ v } | v  V } // E’ : set; Forest : union-find (| V |)
  • WHILE Forest.Size > 1 DO (| E |)
    • e  H. Delete-Max () // e  new edge from H (lg | E |)
    • IF (( TS  Forest.Find ( e. Start ))  ( TE  Forest.Find ( e. End ))) THEN (lg*^ | E |) E’.Union ( e ) // append edge e ; E’. Size ++ (1) Forest.Union ( TS , TE ) // Forest.Size-- (1)
  • RETURN E’ (1)
• Running Time: (| E | lg | E |) = (| V |^2 lg | V |^2 ) = (| V |^2 lg | V |) = ( n^2 lg n )

• General-Case BBN Structure Learning: Use Inference to Compute Scores
• Recall: Bayesian Inference aka Bayesian Reasoning
  • Assumption: h  H are mutually exclusive and exhaustive
  • Optimal strategy: combine predictions of hypotheses in proportion to likelihood
    • Compute conditional probability of hypothesis h given observed data D
    • i.e., compute expectation over unknown h for unseen cases
    • Let h  structure, parameters   CPTs

Scores for Learning Structure:

The Role of Inference

 ^     ^ ^ ^ ^ 

 ^    







 

h H

m

m 1 2 n

P x |D,h P h| D

P x |D P x ,x , ,x |x ,x , ,x 1

1 1 2 m 

P  h  P  D |h,Θ  P  Θ |h  dΘ

P h|D P D|h P h

Posterior Score Marginal Likelihood

Prior over Structures Likelihood

Prior over Parameters

• Likelihood L ( : D )
  • Definition: L ( : D )  P ( D | ) =  x  D P ( x | )
  • General BBN (i.i.d data x ): L ( : D )   x  D  i P ( xi | Parents ( xi ) ~ ) =  i L ( i : D )
    • NB:  specifies CPTs for Parents ( xi )
    • Likelihood decomposes according to the structure of the BBN
• Estimating Prior over Parameters: P ( | D )  P ( D ) · P ( D | )  P ( D ) · L ( : D )
  • Example: Sprinkler
    • Scenarios D = {( Season ( i ), Sprinkler ( i ), Rain ( i ), Moisture ( i ), Slipperiness ( i ))}
    • P ( Su , Off , Dr , Wet , NS ) = P ( S ) · P ( O | S ) · P ( D | S ) · P ( W | O , D ) · P ( N | W )
  • MLE for multinomial distribution (e.g., {Spring, Summer, Fall, Winter}):
  • Likelihood for multinomials
  • Binomial case: N 1 = # heads, N 2 = # tails (“frequency is ML estimator”)

l

l

k k N

Θ ˆ N



K

k

N LΘ D Θk k 1

Scores for Learning Structure:

Prior over Parameters

Learning Structure:

K2 Algorithm and ALARM

• Algorithm Learn-BBN-Structure-K2 ( D, Max-Parents )

FOR i  1 to n DO // arbitrary ordering of variables { x 1 , x 2 , …, xn } WHILE ( Parents [ xi ]. Size < Max-Parents ) DO // find best candidate parent Best  argmaxj>i ( P ( D | xj  Parents [ xi ]) // max Dirichlet score IF ( Parents [ xi ] + Best ). Score > Parents [ xi ]. Score ) THEN Parents [ xi ] += Best

RETURN ({ Parents [ xi ] | i  {1, 2, …, n }})

• A Logical Alarm Reduction Mechanism [Beinlich et al , 1989]
  • BBN model for patient monitoring in surgical anesthesia
  • Vertices (37): findings (e.g., esophageal intubation ), intermediates, observables
  • K2 : found BBN different in only 1 edge from gold standard (elicited from expert)

17

6 5 4

19

10 21

27 11 31

20

22 15 34

32 29 12 9

28 7 8 30

25 18 26 1 2 3

33 14

35

23

13

36

24

16

3 7

Learning Structure:

(Score-Based) Hypothesis Space Search

• Learning Structure: Beyond Trees
  • Problem not as easy for more complex networks
  • Example
    • Allow two parents (even singly-connected case, aka polytree)
    • Greedy algorithms no longer guaranteed to find optimal network
    • In fact, no efficient algorithm exists
  • Theorem: finding network structure with maximal score, where H restricted to BBNs with at most k parents for each variable, is NP -hard for k > 1
• Heuristic Search of Search Space H
  • Define H : elements denote possible structures, adjacency relation denotes transformation (e.g., arc addition, deletion, reversal)
  • Traverse this space looking for high-scoring structures
  • Algorithms
    • Greedy hill-climbing
    • Best-first search
    • Simulated annealing

In-Class Exercise:

Hugin Demo

• Hugin
  • Commercial product for BBN inference: http://www.hugin.com
  • First developed at University of Aalborg, Denmark
• Applications
  • Popular research tool for inference and learning
  • Used for real-world decision support applications
    • Safety and risk evaluation: http://www.hugin.com/serene/
    • Diagnosis and control in unmanned subs: http://advocate.e-motive.com
    • Customer support automation: http://www.cs.auc.dk/research/DSS/SACSO/
• Capabilities
  • Lauritzen-Spiegelhalter algorithm for inference (clustering aka clique reduction)
  • Object Oriented Bayesian Networks (OOBNs): structured learning and inference
  • Influence diagrams for decision-theoretic inference (utility + probability)
  • See: http://www.hugin.com/doc.html

In-Class Exercise:

Hugin and CPT Elicitation

• Hugin Tutorials
  • Introduction: causal reasoning for diagnosis in decision support (toy problem)
    • http://www.hugin.com/hugintro/bbn_pane.html
    • Example domain: explaining low yield (drought versus disease)
  • Tutorial 1: constructing a simple BBN in Hugin
    • http://www.hugin.com/hugintro/bbn_tu_pane.html
    • Eliciting CPTs (or collecting from data) and entering them
  • Tutorial 2: constructing a simple influence diagram (decision network) in Hugin
    • http://www.hugin.com/hugintro/id_tu_pane.html
    • Eliciting utilities (or collecting from data) and entering them
• Other Important BBN Resources
  • Microsoft Bayesian Networks: http://www.research.microsoft.com/dtas/msbn/
  • XML BN (Interchange Format): http://www.research.microsoft.com/dtas/bnformat/
  • BBN Repository (more data sets) http://www-nt.cs.berkeley.edu/home/nir/public_html/Repository/index.htm

Bayesian Network Learning:

Related Fields and References

• ANNs: BBNs as Connectionist Models
• GAs: BBN Inference, Learning as Genetic Optimization, Programming
• Hybrid Systems (Symbolic / Numerical AI)
• Conferences
  • General (with respect to machine learning)
    • International Conference on Machine Learning (ICML)
    • American Association for Artificial Intelligence (AAAI)
    • International Joint Conference on Artificial Intelligence (IJCAI, biennial)
  • Specialty
    • International Joint Conference on Neural Networks (IJCNN)
    • Genetic and Evolutionary Computation Conference (GECCO)
    • Neural Information Processing Systems (NIPS)
    • Uncertainty in Artificial Intelligence (UAI)
    • Computational Learning Theory (COLT)
• Journals
  • General: Artificial Intelligence , Machine Learning , Journal of AI Research
  • Specialty: Neural Networks , Evolutionary Computation , etc.

Learning Bayesian Networks:

Missing Observations

• Problem Definition
  • Given: data ( n -tuples) with missing values, aka partially observable (PO) data
  • Kinds of missing values
    • Undefined, unknown (possible new )
    • Missing, corrupted (not properly collected)
  • Second case (“truly missing”): want to fill in****? with expected value
• Solution Approaches
  • Expected = distribution over possible values
  • Use “best guess” BBN to estimate distribution
  • Expectation-Maximization (EM) algorithm can be used here
• Intuitive Idea
  • Want to find hML in PO case ( D  unobserved variables  observed variables )
  • Estimation step: calculate E [ unobserved variables | h ], assuming current h
  • Maximization step: update wijk to maximize E [lg P ( D | h )], D  all variables