1. Introduction to Bayesian Belief Nets
Russ Greiner
Dep’t of Computing Science
Alberta Ingenuity Centre for Machine Learning
University of Alberta

3. Motivation Gates says [LATimes, 28/Oct/96]: Current Products
Microsoft’s competitive advantages is its expertise in “Bayesian networks”
Current Products
Microsoft Pregnancy and Child Care (MSN)
Answer Wizard (Office 95, Office 2000)
Print Troubleshooter
Excel Workbook Troubleshooter
Office 95 Setup Media Troubleshooter
Windows NT 4.0 Video Troubleshooter
Word Mail Merge Troubleshooter

4. 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

5. Motivation (III) Lymph-node pathology diagnosis Manufacturing control
Software diagnosis
Information retrieval
Types of tasks
Classification/Regression
Sensor Fusion
Prediction/Forecasting

6. Outline How to reason with BNs Specific Examples of BNs
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

7. Symptoms Signs Plan Diagnosis Chief complaint History, … Physical Exam
Blah blah ouch yak
ouch blah ouch blah
blah ouch blah
Symptoms
Chief complaint
History, …
Signs
Physical Exam
Test results, …
Diagnosis
Plan
Treatment, …

8. 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

9. Underlying Task Seldom Completely Certain
Situation: Given observations {O1=v1, … Ok=vk}
(symptoms, history, test results, …)
what is best DIAGNOSIS Dxi for patient?
Approach1: Use set of obs1 & … & obsm  Dxi rules
but… Need rule for each situation
for each diagnosis Dxr
for each set of possible values vj for Oj
for each subset of obs. {Ox1, Ox2, … }  {Oj}
Can’t use
if only know Temp and BP
If Temp>100 & BP = High & Cough = Yes  DiseaseX
Seldom Completely Certain

10. Underlying Task, II Approach 2: Compute Probabilities of Dxi
Situation: Given observations {O1=v1, … Ok=vk}
(symptoms, history, test results, …)
what is best DIAGNOSIS Dxi for patient?
Approach 2: Compute Probabilities of Dxi
given observations { obsj }
P( Dx = u | O1= v1, …, Ok= vk )
Challenge: How to express Probabilities?

11. How to deal with Probabilities
Sufficient: “atomic events”:
for all 21+N values u  {T, F}, vj {T, F}
P( Dx = u, O1=v1,..., Ok= vk,…, ON=vN )
P( Dx=T, O1=T, O2=T, …, ON=T ) = 0.03
P( Dx=T, O1=T, O2=T, …, ON=F ) = 0.4 …
P( Dx=T, O1=F, O2=F, … , ON=T ) = 0
P( Dx=F, O1=F, O2=F, …, ON=F ) = 0.01 
Then: Marginalize:
Conditionalize:
P( Dx = u, O1= v1,…,Ok= vk ) = Σ P( Dx = u , O1= v1 , …, Ok= vk, …, ON= vN )
P( Dx = u | O1 = v1,…, Ok = vk) P( Dx = u, O1 = v1,…,Ok = vk )
P( O1 = v1,…,Ok = vk)
But… even if binary Dx, 20 binary obs.’s.  >2,097,000 numbers!

12. 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”
Belief Nets 

13. ? Hepatitis?
? Hepatitis,
not Jaunticed
but +BloodTest ?
Jaunticed
BloodTest

14. Hepatitis Example (Boolean) Variables: Alternatively… H Hepatitis
J Jaundice
B (positive) Blood test
(Boolean) Variables:
Want P( H=1 | J=0, B=1 )
…, P(H=1 | B=1, J=1), P(H=1 | B=0,J=0), …
Option 1:
J B H P(J, B, H)
…Marginalize/Conditionalize, to get P( H=1 | J=0, B=1 ) …
Alternatively…

15. Encoding Causal Links Simple Belief Net: Node ~ Variable
0.95
0.05
P(H=0)
P(H=1)
0.97
0.03 1
P(B=0 | H=h)
P(B=1 | H=h) h
0.7
0.3
0.2
P(J=0|h,b)
0.8
P(J=1|h,b) b H B J
Node ~ Variable
Link ~ “Causal dependency”
“CPTable” ~ P(child | parents)

16. Encoding Causal Links
P(H=1)
0.05 H h
P(B=1 | H=h) 1
0.95
0.03 h b
P(J=1|h , b ) 1
0.8
0.3 B J
P(J | H, B=0) = P(J | H, B=1)  J, H !  P( J | H, B) = P(J | H)
J is INDEPENDENT of B, once we know H
Don’t need B J arc!

17. Encoding Causal Links
P(H=1)
0.05 H h
P(B=1 | H=h) 1
0.95
0.03 h
P(J=1|h ) 1
0.8
0.3 B J
P(J | H, B=0) = P(J | H, B=1)  J, H !  P( J | H, B) = P(J | H)
J is INDEPENDENT of B, once we know H
Don’t need B J arc!

18. Encoding Causal Links
P(H=1)
0.05 H h
P(B=1 | H=h) 1
0.95
0.03 h
P(J=1|h ) 1
0.8
0.3 B J
P(J | H, B=0) = P(J | H, B=1)  J, H !  P( J | H, B) = P(J | H)
J is INDEPENDENT of B, once we know H
Don’t need B J arc!

19. Sufficient Belief Net Requires: P(H=1) known H B J P(J=1 | H=1) known
0.05 H B J h
P(B=1 | H=h) 1
0.95
0.03 h
P(J=1|h ) 1
0.8
0.3
Requires: P(H=1) known
P(J=1 | H=1) known
P(B=1 | H=1) known
(Only 5 parameters, not 7)
Hence: P(H=1 | J=0, B=1) = P(H=1) P(J=0 | H=1) P(B=1 | J=0,H=1)
P(B=1 | H=1)

20. “Factoring” B does depend on J: but… ONLY THROUGH H:
If J=1, then likely that H=1  B =1
but… ONLY THROUGH H:
If know H=1, then likely that B=1
… doesn’t matter whether J=1 or J=0 ! 
P(B=1 | J=0, H=1) = P(B=1 | H=1)
N.b., B and J ARE correlated a priori P(B | J )  P(B)
GIVEN H, they become uncorrelated P(B | J, H) = P(B | H)

21. Factored Distribution
Symptoms independent, given Disease
H Hepatitis
J Jaundice
B (positive) Blood test
P( B | J )  P ( B ) but
P( B | J,H ) = P ( B | H )
ReadingAbility and ShoeSize are dependent,
P(ReadAbility | ShoeSize )  P(ReadAbility )
but become independent, given Age
P(ReadAbility | ShoeSize, Age ) = P(ReadAbility | Age)
Age
ShoeSize
Reading

22. “Naïve Bayes” Classification Task: Given Find argmax {hi} ...
Given { O1 = v1, …, On = vn }
Find hi that maximizes (H = hi | O1 = v1, …, On = vn)
Given
P(H = hi )
P(Oj = vj | H = hi)
Independent: P(Oj | H, Ok,…) = P(Oj | H) H O2 O1 On
...
Find argmax {hi}

23. Naïve Bayes (con’t) Easy to use for ClassificationH O2 O1 On
...
Naïve Bayes (con’t)
Normalizing term
(No need to compute, as same for all hi)
Easy to use for Classification
Can use even if some vjs not specified
If k Dx’s and n Ois,
requires only k priors, n * k pairwise-conditionals
(Not 2n+k… relatively easy to learn)
2,147,438,647 61 30
2,047 21 10
2n+1 – 1
1+2n n

24. Bigger Networks Jaundice Hepatitis Bloodtest
P(I=1)
0.20
P(H=1)
0.32
LiverTrauma
Jaundice
GeneticPH
Hepatitis
Bloodtest d i
P(H=1|d ,i ) 1
0.82
0.10
0.45
0.04 h
P(J=1| h ) 1
0.8
0.3 h
P(B=1| h ) 1
0.98
0.01
Intuition: Show CAUSAL connections:
GeneticPH CAUSES Hepatitis; Hepatitis CAUSES Jaundice
If GeneticPH, then expect Jaundice:
GeneticPH  Hepatitis  Jaundice
But only via Hepatitis: GeneticPH and not Hepatitis  Jaundice
P( J | D )  P( J ) but
P( J | D,H ) = P( J | H)

25. Belief Nets DAG structure v is INDEPENDENT of non-descendants,
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
Given H = 1,
- D has no influence on J
- J has no influence on B
- etc. D I H J B

26. 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 MS D
0.001
P(FHD=1)
0.10 1
0.03
P(MS=1 | FHD=f) f
0.04
0.08
0.90
0.97
P(D=1 | FHD=f, MS=m) m
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

27. Example: Car Diagnosis

28. MammoNet

29. ALARM
A Logical Alarm Reduction Mechanism
8 diagnoses, 16 findings, …

30. Troup Detection

31. ARCO1: Forecasting Oil Prices

32. ARCO1: Forecasting Oil Prices

33. Forecasting Potato Production

34. Warning System

35. Extensions Find best values (posterior distr.) for
SEVERAL (> 1) “output” variables
Partial specification of “input” values
only subset of variables
only “distribution” of each input variable
General Variables
Discrete, but domain > 2
Continuous (Gaussian: x = i bi yi for parents {Y} )
Decision Theory  Decision Nets (Influence Diagrams) Making Decisions, not just assigning prob’s
Storing P( v | p1, p2,…,pk) General “CP Tables” 0(2k) Noisy-Or, Noisy-And, Noisy-Max “Decision Trees”

36. Outline Contrast with Rules, Neural Nets, … How to reason with BNs
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

37. Belief Nets vs Rules Both have “Locality”
Specific clusters (rules / connected nodes)
Often same nodes (rep’ning Propositions) but
BN: Cause  Effect
“Hep  Jaundice” P(J | H )
Rule: Effect  Cause
“Jaundice  Hep”
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 “direction”s of inference

38. 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 “direction”s of inference

39. 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” C B A
Technical: BNs use DIRECTRED arcs
 allow “induced dependencies”
I (A, {}, B) “A independent of B, given {}”
¬ I (A, C, B) “A dependent on B, given C”
MNs use UNDIRECTED arcs
 allow other independencies
I(A, BC, D) A independent of D, given B, C
I(B, AD, C) B independent of C, given A, D D C B A

40. 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 MD’s 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,…

41. Uses (con’t)
Logistics Support: Stock warehouses appropriately …based 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
-Classification -Interpretation
-Prediction -…

42. Challenge #1 Computational Efficiency
For given BN:
General problem is
Given
Compute
+ If BN is “poly tree”,  efficient alg.
- If BN is gen’l DAG (>1 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

43. # 2a:Obtaining 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

44. Notes on Learning Mixed Sources: Person provides structure;
Algorithm fills-in numbers.
Just Learning Algorithm:  algorithms that
learn from sample
structure
values
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

45. My Current Work Learning Belief Nets Validating 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, …

46. # 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

47. Conclusions Belief Nets are PROVEN TECHNOLOGY Provide effective way to
Medical Diagnosis
DSS for complex machines
Forecasting, Modeling, InfoRetrieval…
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

48. Extra Slides AI Seminar References Crusher Controller Formal Framework
Friday, noon, CSC3-33
Free PIZZA!
References
Crusher Controller
Formal Framework
Decision Nets
Developing the Model
Why Reasoning is Hard
Learning Accurate Belief Nets

49. 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, (See esp Ch 14, 15, 19.)
General info re BayesNets
Proceedings:
Assoc for Uncertainty in AI
Learning:
David Heckerman, A tutorial on learning with Bayesian networks, 1995,
Software:
General:
JavaBayes
Norsys

50. Decision Net: Test/Buy a Car

51. 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

52. Decision Net: Drill for Oil?
Go Back

53. Formal Framework Always true: Given independencies, Hence
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 min’l…
(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

54. Developing the Model Source of information Typical Process
+ (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

55. 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 Expert’s 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

56. Why Reasoning is Hard BN reasoning may look easy:
Just “propagate” information from node to node
P(Z=t)
0.5 Z z
P(B=t|Z=z) t
0.0 f
1.0 z
P(A=t|Z=z) t
1.0 f
0.0 A B a b
P(C=t|a,b) t
1.0 f
0.0 C
Challenge: What is P(C=t)?
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

57. Crusher Controller Given observations Specify best action for crusher
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?

58. 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

59. Details State “CrusherEnvironment” Inputs Outputs
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

60. Benefits Increase Crusher Effectiveness Reduce Down Time
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

61. My Background Prof at UofAlberta (CS) PhD, Stanford (Computer Science)
Representational issues, Analogical Inference
… everything in Logic
PostDoc at UofToronto (CS)
Foundations of learnability, logical inference, DB, control theory, …
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 …

62. 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)
0.001
P(FHD=1)
FHD f
P(MS=1 | FHD=f) 1
0.10
0.03 MS f m
P(D=1 | FHD=f, MS=m) 1
0.97
0.90
0.08
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