Introduction to Bayesian Belief Nets

1
Introduction toBayesian Belief Nets

• Russ Greiner
• Dept of Computing Science
• Alberta Ingenuity Centre for Machine Learning
• University of Alberta
• http//www.cs.ualberta.ca/greiner/bn.html

2
1996
1990
1980
3
 
 
4
Motivation

• Gates says LATimes, 28/Oct/96
• Microsofts competitive advantages is its
  expertise in Bayesian networks

• Current Products
• Microsoft Pregnancy and Child Care (MSN)
• Answer Wizard (Office, )
• Print Troubleshooter
• Excel Workbook Troubleshooter
• Office 95 Setup Media Troubleshooter
• Windows NT 4.0 Video Troubleshooter
• Word Mail Merge Troubleshooter

5
Motivation (II)

• US Army SAIP (Battalion Detection from SAR, IR
  GulfWar)

• NASA Vista (DSS for Space Shuttle)
• GE Gems (real-time monitor for utility
  generators)
• Intel (infer possible processing problems from
  end-of-line tests on semiconductor chips)
• KIC
• medical sleep disorders, pathology, trauma care,
  hand and wrist evaluations, dermatology,
  home-based health evaluations
• DSS for capital equipment locomotives,
  gas-turbine engines, office equipment

6
Motivation (III)

• Lymph-node pathology diagnosis
• Manufacturing control
• Software diagnosis
• Information retrieval
• Types of tasks
• Classification/Regression
• Sensor Fusion
• Prediction/Forecasting

7
Outline

• Existing uses of Belief Nets (BNs)
• How to reason with BNs
• Specific Examples of BNs
• Contrast with Rules, Neural Nets,
• Possible applications of BNs
• Challenges
• How to reason efficiently
• How to learn BNs

8

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Objectives Decision Support System

• Determine
• which tests to perform
• which repair to suggest
• based on costs, sensitivity/specificity,

• Use all sources of information
• symbolic (discrete observations, history, )
• signal (from sensors)
• Handle partial information
• Adapt to track fault distribution

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Underlying Task

• Situation Given observations O1v1, Okvk
• (symptoms, history, test results, )
• what is best DIAGNOSIS Dxi for patient?

• Seldom Completely Certain

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Underlying Task, II

• Situation Given observations O1v1, Okvk
• (symptoms, history, test results, )
• what is best DIAGNOSIS Dxi for patient?

• Challenge How to express Probabilities?

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How to deal with Probabilities

• Sufficient atomic events
• for all 21N values u ? T, F, vj ?T, F

P( Dx u, O1v1,..., Ok vk,, ONvN )

• But even if binary Dx, 20 binary obs.s. ?
  gt2,097,000 numbers!

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Problems with Atomic Events

• Representation is not intuitive
• ? Should make connections explicit
• use local information

P(Jaundice Hepatitis), P(LightDim
BadBattery),

• Too many numbers O(2N)
• Hard to store
• Hard to use
• Must add 2r values to marginalize r
  variables
• Hard to learn
• Takes O(2N) samples to learn 2N parameters
• ? Include only necessary connections

14

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Hepatitis Example
H Hepatitis J Jaundice B (positive) Blood
test

• (Boolean) Variables

• Want P( H1 J0, B1 )
• , P(H1 B1, J1), P(H1 B0,J0),

• Alternatively

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Encoding Causal Links

• Simple Belief Net

• Node Variable
• Link Causal dependency

• CPTable P(child parents)

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Encoding Causal Links
P(H1)
0.05
H
h P(B1 Hh)
1 0.95
0 0.03
h b P(J1h , b )
1 1 0.8
1 0 0.8
0 1 0.3
0 0 0.3
B
J

• P(J H, B0) P(J H, B1) ? J, H ! ?
  P( J H, B) P(J H)
• J is INDEPENDENT of B, once we know H
• Dont need B? J arc!

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Encoding Causal Links
P(H1)
0.05
H
h P(B1 Hh)
1 0.95
0 0.03
h P(J1h )
1 0.8
1
0 0.3
0
B
J

• P(J H, B0) P(J H, B1) ? J, H ! ?
  P( J H, B) P(J H)
• J is INDEPENDENT of B, once we know H
• Dont need B? J arc!

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Encoding Causal Links
P(H1)
0.05
H
h P(B1 Hh)
1 0.95
0 0.03
h P(J1h )
1 0.8
0 0.3
B
J

• P(J H, B0) P(J H, B1) ? J, H ! ?
  P( J H, B) P(J H)
• J is INDEPENDENT of B, once we know H
• Dont need B? J arc!

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Sufficient Belief Net
P(H1)
0.05
h P(B1 Hh)
1 0.95
0 0.03
h P(J1h )
1 0.8
0 0.3

• Requires P(H1) known
• P(J1 H1) known
• P(B1 H1) known
• (Only 5 parameters, not 7)

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Factoring

• B does depend on J
• If J1, then likely that H1 ? B 1

N.b., B and J ARE correlated a priori P(J B
) ? P(J) GIVEN H, they become uncorrelated
P(J B, H) P(J H)
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Factored Distribution

• Symptoms independent, given Disease

• ReadingAbility and ShoeSize are dependent,
• P(ReadAbility ShoeSize ) ? P(ReadAbility )
• but become independent, given Age
• P(ReadAbility ShoeSize, Age ) P(ReadAbility
  Age)

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Naïve Bayes

• Classification Task
• Given O1 v1, , On vn
• Find hi that maximizes (H hi O1 v1,
  , On vn)

• Find argmax hi

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Naïve Bayes (cont)

• Normalizing term
• (No need to compute, as same for all hi)

• Easy to use for Classification

• Can use even if some vjs not specified

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Bigger Networks
P(I1)
0.20
P(H1)
0.32
g lt P(H1g ,lt )
1 1 0.82
1 0 0.10
0 1 0.45
0 0 0.04
h P(J1 h )
1 0.8
0 0.3
h P(B1 h )
1 0.98
0 0.01

• Intuition Show CAUSAL connections
• GeneticPH CAUSES Hepatitis Hepatitis CAUSES
  Jaundice

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Belief Nets

• DAG structure
• Each node ? Variable v
• v depends (only) on its parents
• conditional prob P(vi parenti ?0,1,?
  )

• v is INDEPENDENT of non-descendants,
• given assignments to its parents

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Less Trivial Situations

• N.b., obs1 is not always independent of obs2
  given H
• Eg, FamilyHistoryDepression causes
  MotherSuicide and Depression
• MotherSuicide causes Depression (w/ or w/o
  F.H.Depression)

• Here, P( D MS, FHD ) ? P( D FHD ) !

• Can be done using Belief Network,
• but need to specify
• P( FHD ) 1
• P( MS FHD ) 2
• P( D MS, FHD ) 4

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Example Car Diagnosis
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MammoNet
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ALARM

• A Logical Alarm Reduction Mechanism
• 8 diagnoses, 16 findings,

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Troup Detection
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ARCO1 Forecasting Oil Prices
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ARCO1 Forecasting Oil Prices
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Forecasting Potato Production
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Warning System
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Extensions

• Find best values (posterior distr.) for
• SEVERAL (gt 1) output variables
• Partial specification of input values
• only subset of variables
• only distribution of each input variable
• General Variables
• Discrete, but domain gt 2
• Continuous (Gaussian x ?i bi yi for
  parents Y )
• Decision Theory ? Decision Nets (Influence
  Diagrams) Making Decisions, not just assigning
  probs
• Storing P( v p1, p2,,pk) General CP Tables
  0(2k) Noisy-Or, Noisy-And, Noisy-Max Decision
  Trees

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Outline

• Existing uses of Belief Nets (BNs)
• How to reason with BNs
• Specific Examples of BNs
• Contrast with Rules, Neural Nets,
• Possible applications of BNs
• Challenges
• How to reason efficiently
• How to learn BNs

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Belief Nets vs Rules

• Both have Locality
• Specific clusters (rules / connected nodes)

WHY? Easier for people to reason CAUSALLY even
if use is DIAGNOSTIC

• BN provide OPTIMAL way to deal with
• Uncertainty
• Vagueness (var not given, or only dist)
• Error
• Signals meeting Symbols

• BN permits different directions of inference

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Belief Nets vs Neural Nets

• Both have graph structure but

• BN Nodes have SEMANTICs
• Combination Rules Sound Probability
• NN Nodes arbitrary
• Combination Rules Arbitrary

• So harder to
• Initialize NN
• Explain NN
• (But perhaps easier to learn NN from examples
  only?)

• BNs can deal with
• Partial Information
• Different directions of inference

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Belief Nets vs Markov Nets

• Each uses graph structure
• to FACTOR a distribution
• explicitly specify dependencies, implicitly
  independencies

• but subtle differences
• BNs capture causality, hierarchies
• MNs capture temporality

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Uses of Belief Nets 1

• Medical Diagnosis Assist/Critique MD
• identify diseases not ruled-out
• specify additional tests to perform
• suggest treatments appropriate/cost-effective
• react to MDs proposed treatment

• Decision Support Find/repair faults in complex
  machines
• Device, or Manufacturing Plant, or
• based on sensors, recorded info, history,

• Preventative Maintenance
• Anticipate problems in complex machines
• Device, or Manufacturing Plant, or
• based on sensors, statistics, recorded info,
  device history,

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Uses (cont)

• Logistics Support Stock warehouses
  appropriatelybased on (estimated) freq. of
  needs, costs,
• Diagnose Software Find most probable bugs,
  given program behavior, core dump, source code,
  
• Part Inspection/Classification based on
  multiple sensors, background, model of
  production,
• Information Retrieval Combine information from
  various sources, based on info from various
  agents,

General Partial Info, Sensor fusion -Classificati
on -Interpretation -Prediction -
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Challenge 1Computational Efficiency
For given BN General problem is Given
Compute If BN is poly tree, ? efficient
alg. - If BN is genl DAG (gt1 path from X to
Y) - NP-hard in theory - slow in
practice Tricks Get approximate answer
(quickly) Use abstraction of BN Use
abstraction of query (range)
O1 v1, , On vn
D
I
P(H O1 v1, , On vn)
H
J
B
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2aObtaining Accurate BN

• BN encodes distribution over n variables
• Not O(2n) values, but only ?i 2k_i
• (Node ni binary, with ki parents)
• Still lots of values! structure ..
• ? Qualitative Information
• Structure What depends on what?
• Easy for people (background knowledge)
• But NP-hard to learn from samples
• ? Quantitative Information
• Actual CP-tables
• Easy to learn, given lots of examples.
• But people have hard time

Knowledge acquisition from human experts
Simple learning algorithm
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Notes on Learning

• Mixed Sources Person provides structure
• Algorithm fills-in numbers.

• Just Human Expert People produce CP-table, as
  well as structure
• Relatively few values really required
• Esp. if NoisyOr, NoisyAnd, NaiveBayes,
• Actual values not that important
• Sensitivity studies

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My Current Work

• Learning Belief Nets
• Model selection
• Challenging myth that MDL is appropriate criteria
• Learning performance system, not model
• Validating Belief Nets
• Error bars around answers
• Adaptive User Interfaces
• Efficient Vision Systems
• Foundations of Learnability
• Learning Active Classifiers
• Sequential learners
• Condition Based maintenance, Bio-signal
  interpretation,

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2b Maintaining Accurate BN

• The world changes.
• Information in BN may be
• perfect at time t
• sub-optimal at time t 20
• worthless at time t 200

• Need to MAINTAIN a BN over time
• using on-going human consultant

• Adaptive BN
• Dirichlet distribution (variables)
• Priors over BNs

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Conclusions

• Provide effective way to
• Represent complicated, inter-related events
• Reason about such situations
• Diagnosis, Explanation, ValueOfInfo
• Explain conclusions
• Mix Symbolic and Numeric observations

• Challenges
• Efficient ways to use BNs
• Ways to create BNs
• Ways to maintain BNs
• Reason about time

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Extra Slides

• AI Seminar
• Friday, noon, CSC3-33
• Free PIZZA!
• http//www.cs.ualberta.ca/ai/ai-seminar.html
• References
• http//www.cs.ualberta.ca/greiner/bn.html
• Crusher Controller
• Formal Framework
• Decision Nets
• Developing the Model
• Why Reasoning is Hard
• Learning Accurate Belief Nets

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References

• http//www.cs.ualberta.ca/greiner/bn.html
• Overview textbooks
• Judea Pearl, Probabilistic Reasoning in
  Intelligent Systems Networks of Plausible
  Inference, Morgan Kaufmann, 1988.
• Stuart Russell and Peter Norvig, Artificial
  Intelligence A Modern Approach, Prentice Hall,
  1995. (See esp Ch 14, 15, 19.)
• General info re BayesNets
• http//www.afit.af.mil80/Schools/EN/ENG/LABS/AI/B
  ayesianNetworks
• Proceedings http//www.sis.pitt.edu/dsl/uai.html
  
• Assoc for Uncertainty in AI http//www.auai.org/
• Learning
• David Heckerman, A tutorial on learning with
  Bayesian networks, 1995,
• http//www.research.microsoft.com/research/dtg/he
  ckerma/TR-95-06.htm
• Software
• General http//bayes.stat.washington.edu/almon
  d/belief.html
• JavaBayes http//www.cs.cmu.edu/fgcozman/Research
  /JavaBaye
• Norsys http//www.norsys.com/

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Decision Net Test/Buy a Car
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Utility Decision Nets

• Given c( action, state) ? R (cost function)
• Cp(a) Es c(a,s) ? s?S p(s obs) c(a, s)
• Best (immediate) action a argmina ?A Cp(a)
  
• Decision Net (like Belief Net) but
• 3 types of nodes
• chance (like Belief net)
• action repair, sensing
• cost/utility
• Links for dependency
• Given observations, obs, computes best action, a
• Sequence of Actions MDPs, POMDPs,

Go Back
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Decision Net Drill for Oil?
Go Back
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Formal Framework

• Always true
• P(x1, ,xn) P(x1) P(x2 x1) P (x3 x2,
  x1) P (xn xn-1,,x1)
• Given independencies,
• P(xk x1,,xk-1) P (xk pak) for some
  pak ?x1, , xk-1
• Hence
• So just connect each y ? pai to xi ? DAG
  structure

Note -Size of BN is
.
so better to use small
pai. -pai 1,,i 1 is never incorrect
but seldom minl (so hard to store,
learn, reason with,) - Order of variables can
make HUGE difference Can have pai 1
for one ordering pai i 1 for another
Go Back
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Developing the Model

• Source of information
• (Human) Expert (s)
• Data from earlier Runs
• Simulator

• Typical Process
• 1. Develop / Refine Initial Prototype
• 2. Test Prototype ? Accurate System
• 3. Deploy System
• 4. Update / Maintain System

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Develop/Refine Prototype

• Requires expert
• useful to have data

• Initial Interview(s)
• To establish what relates to what
• Expert time ½ - day

• Iterative process (Gradual refinement)
• To refine qualitative connections
• To establish correct operations

Expert presents Good Performance KE implements
Experts claims KE tests on examples (real data
or expert), and reports to Expert
Expert time 1 2 hours / week for ??
Weeks (Depends on complexity of device, and
accuracy of model)
Go Back
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Why Reasoning is Hard

• BN reasoning may look easy
• Just propagate information from node to node

P(Zt)
0.5
z P(BtZz)
t 0.0
f 1.0
z P(AtZz)
t 1.0
f 0.0
a b P(Cta,b)
t t 1.0
t f 0.0
f t 0.0
f f 0.0

• Challenge What is P(Ct)?

A Z B P ( A t ) P ( B f ) ½
So ? P ( C t ) P ( A t, B t)
P ( A t) P( B t) ½ ½ ¼
Wrong P ( C t ) 0 !

• Need to maintain dependencies! P ( A t, B t
  ) P ( A t ) P ( B t A t)

Go Back
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Crusher Controller

• Given observations
• History, sensor readings, schedule,
• Specify best action for crusher
• stop immediately, increase roller speed by ?
• Best minimize expected cost

• Initially just recommendation to human operator
• Later Directly implement (some) actions
• ?Request values of other sensors?

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Approach

• For each state s
• (Good flow, tooth about to enter, )
• for each action a
• (Stop immediately, Change p7 0.32, )
• determine utility of performing a in s
• (Cost of lost production if stopped
• of reduced production efficient if continue )
• Use observations to estimate (dist over) current
  states
• Infer EXPECTED UTILITY of each action, based on
  distr.
• Return action with highest Expected Utility

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Details

• Inputs
• Sensor Readings (history)
• Camera, microphone, power-draw
• Parameter settings
• Log files, Maintenance records
• Schedule (maintenance, anticipated load, )
• Outputs
• Continue as is
• Adjust parameters
• GapSize, ApronFeederSpeed, 1J_ConveyorSpeed
• Shut down immediately
• Step adding new material
• Tell operator to look

• State CrusherEnvironment
• UncrushableThingsNowInCrusher
• TeethMissing
• NextUncrushableEntry
• Control Parameters

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Benefits

• Increase Crusher Effectiveness
• Find best settings for parameters
• To maximize production of well-sized chunks
• Reduce Down Time
• Know when maintain/repair is critical
• Reduce Damage to Crusher
• Usable Model of Crusher
• Easy to modify when needed
• Training
• Design of next generation
• Prototype for design of control, diagnostician
  of other machines

Go Back
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My Background

• PhD, Stanford (Computer Science)
• Representational issues, Analogical Inference
• everything in Logic
• PostDoc at UofToronto (CS)
• Foundations of learnability, logical inference,
  DB, control theory,
• everything in Logic
• Industrial research (Siemens Corporate Research)
• Need to solve REAL problems
• Theory Revision, Navigational systems,
• logic is not be-all-and-end-all!
• Prof at UofAlberta (CS)
• Industrial problems (Siemens, BioTools, Syncrude)
• Foundations of learnability, probabilistic
  inference

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Less Trivial Situations

• N.b., obs1 is not always independent of obs2
  given H
• Eg, FamilyHistoryDepression causes
  MotherSuicide and Depression
• MotherSuicide causes Depression (w/ or w/o
  F.H.Depression)

FHD
f P(MS1 FHDf)
1 0.10
0 0.03
MS
f m P(D1 FHDf, MSm)
1 1 0.97
1 0 0.90
0 1 0.08
0 0 0.04
D

• Here, P( D MS, FHD ) ? P( D FHD ) !

• Can be done using Belief Network,
• but need to specify
• P( FHD ) 1
• P( MS FHD ) 2
• P( D MS, FHD ) 4