1. Russell and Norvig: Chapter 14 CMCS424 Fall 2005
Bayesian Networks
Russell and Norvig: Chapter 14
CMCS424 Fall 2005

2. Probabilistic Agent ? sensors environment agent actuators
I believe that the sun
will still exist tomorrow with probability
and that it will be a sunny
with probability 0.6

3. Problem
At a certain time t, the KB of an agent is some collection of beliefs
At time t the agent’s sensors make an observation that changes the strength of one of its beliefs
How should the agent update the strength of its other beliefs?

4. Purpose of Bayesian Networks
Facilitate the description of a collection of beliefs by making explicit causality relations and conditional independence among beliefs
Provide a more efficient way (than by using joint distribution tables) to update belief strengths when new evidence is observed

5. Other Names
Belief networks
Probabilistic networks
Causal networks

6. Bayesian Networks
A simple, graphical notation for conditional independence assertions resulting in a compact representation for the full joint distribution
Syntax:
a set of nodes, one per variable
a directed, acyclic graph (link = ‘direct influences’)
a conditional distribution for each node given its parents: P(Xi|Parents(Xi))

7. Example
Topology of network encodes conditional independence assertions:
Cavity
Weather
Toothache
Catch
Weather is independent of other variables
Toothache and Catch are independent given Cavity

8. Example I’m at work, neighbor John calls to say my alarm is
ringing, but neighbor Mary doesn’t call. Sometime it’s set off by a minor earthquake. Is there a burglar?
Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls
Network topology reflects “causal” knowledge: - A burglar can set the alarm off - An earthquake can set the alarm off - The alarm can cause Mary to call - The alarm can cause John to call

9. A Simple Belief Network
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
causes
effects
Intuitive meaning of arrow
from x to y: “x has direct influence on y”
Directed acyclic graph (DAG)
Nodes are random variables

10. Assigning Probabilities to Roots
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls

11. Conditional Probability Tables
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|B,E)
TTFF
TFTF
Size of the CPT for a node with k parents: ?

12. Conditional Probability Tables
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|B,E)
TTFF
TFTF A
P(J|A) TF A
P(M|A) TF

13. What the BN Means P(x1,x2,…,xn) = Pi=1,…,nP(xi|Parents(Xi)) Burglary
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF
P(x1,x2,…,xn) = Pi=1,…,nP(xi|Parents(Xi)) A
P(J|A) TF A
P(M|A) TF

14. Calculation of Joint Probability
P(B)
0.001
P(E)
0.002
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls B E
P(A|…)
TTFF
TFTF
P(JMABE) = P(J|A)P(M|A)P(A|B,E)P(B)P(E) = 0.9 x 0.7 x x x = A
P(J|…) TF A
P(M|…) TF

15. What The BN Encodes
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
For example, John does not observe any burglaries directly
Each of the beliefs JohnCalls and MaryCalls is independent of Burglary and Earthquake given Alarm or Alarm
The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm

16. What The BN Encodes
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
For instance, the reasons why John and Mary may not call if there is an alarm are unrelated
Note that these reasons could be other beliefs in the network. The probabilities summarize these non-explicit beliefs
Each of the beliefs JohnCalls and MaryCalls is independent of Burglary and Earthquake given Alarm or Alarm
The beliefs JohnCalls and MaryCalls are independent given Alarm or Alarm

17. Structure of BN
The relation: P(x1,x2,…,xn) = Pi=1,…,nP(xi|Parents(Xi)) means that each belief is independent of its predecessors in the BN given its parents
Said otherwise, the parents of a belief Xi are all the beliefs that “directly influence” Xi
Usually (but not always) the parents of Xi are its causes and Xi is the effect of these causes
E.g., JohnCalls is influenced by Burglary, but not directly. JohnCalls is directly influenced by Alarm

18. Construction of BN
Choose the relevant sentences (random variables) that describe the domain
Select an ordering X1,…,Xn, so that all the beliefs that directly influence Xi are before Xi
For j=1,…,n do:
Add a node in the network labeled by Xj
Connect the node of its parents to Xj
Define the CPT of Xj
The ordering guarantees that the BN will have no cycles

19. Cond. Independence Relations
Ancestor X Y1 Y2
Non-descendent
1. Each random variable X, is conditionally independent of its non-descendents, given its parents Pa(X)
Formally, I(X; NonDesc(X) | Pa(X))
2. Each random variable is conditionally independent of all the other nodes in the graph, given its neighbor
Parent
Non-descendent
Descendent

20. Distribution conditional to the observations made
Inference In BN
Set E of evidence variables that are observed, e.g., {JohnCalls,MaryCalls}
Query variable X, e.g., Burglary, for which we would like to know the posterior probability distribution P(X|E)
Distribution conditional to the observations made ? T
P(B|…) M J

21. Inference Patterns Basic use of a BN: Given new
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Diagnostic
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Causal
Basic use of a BN: Given new
observations, compute the new strengths of some (or all) beliefs
Other use: Given the strength of
a belief, which observation should
we gather to make the greatest
change in this belief’s strength
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Intercausal
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls
Mixed

22. Types Of Nodes On A Path diverging Radio Battery SparkPlugs Starts Gas
Moves
linear
converging

23. Independence Relations In BN
diverging
Radio
Battery
SparkPlugs
Starts
Gas
Moves
linear
Given a set E of evidence nodes, two beliefs connected by an undirected path are independent if one of the following three
conditions holds:
1. A node on the path is linear and in E
2. A node on the path is diverging and in E
3. A node on the path is converging and neither this node, nor any descendant is in E
converging

24. Independence Relations In BN
diverging
Radio
Battery
SparkPlugs
Starts
Gas
Moves
linear
Given a set E of evidence nodes, two beliefs connected by an undirected path are independent if one of the following three
conditions holds:
1. A node on the path is linear and in E
2. A node on the path is diverging and in E
3. A node on the path is converging and neither this node, nor any descendant is in E
converging
Gas and Radio are independent given evidence on SparkPlugs

25. Independence Relations In BN
diverging
Radio
Battery
SparkPlugs
Starts
Gas
Moves
linear
Given a set E of evidence nodes, two beliefs connected by an undirected path are independent if one of the following three
conditions holds:
1. A node on the path is linear and in E
2. A node on the path is diverging and in E
3. A node on the path is converging and neither this node, nor any descendant is in E
Gas and Radio are independent given evidence on Battery
converging

26. Independence Relations In BN
diverging
Radio
Battery
SparkPlugs
Starts
Gas
Moves
linear
Given a set E of evidence nodes, two beliefs connected by an undirected path are independent if one of the following three
conditions holds:
1. A node on the path is linear and in E
2. A node on the path is diverging and in E
3. A node on the path is converging and neither this node, nor any descendant is in E
Gas and Radio are independent given no evidence, but they are dependent given evidence on
Starts or Moves
converging

27. BN Inference Simplest Case: A B P(B) = P(a)P(B|a) + P(~a)P(B|~a) B A C
P(C) = ???

28. BN Inference Chain: … X1 X2 Xn
What is time complexity to compute P(Xn)?
What is time complexity if we computed the full joint?

29. Inference Ex. 2 Cloudy Rain Algorithm is computing not individual
Sprinkler
Cloudy
WetGrass
Algorithm is computing not individual
probabilities, but entire tables
Two ideas crucial to avoiding exponential blowup:
because of the structure of the BN, some subexpression in the joint depend only on a small number of variable
By computing them once and caching the result, we can avoid generating them exponentially many times

30. Variable Elimination General idea: Write query in the form Iteratively
Move all irrelevant terms outside of innermost sum
Perform innermost sum, getting a new term
Insert the new term into the product

31. A More Complex Example “Asia” network: Visit to Asia Smoking
Lung Cancer
Tuberculosis
Abnormality
in Chest
Bronchitis
X-Ray
Dyspnea

32. Need to eliminate: v,s,x,t,l,a,b Initial factors
We want to compute P(d)
Need to eliminate: v,s,x,t,l,a,b
Initial factors

33. Need to eliminate: v,s,x,t,l,a,b Initial factors
We want to compute P(d)
Need to eliminate: v,s,x,t,l,a,b
Initial factors
Eliminate: v
Compute:
Note: fv(t) = P(t)
In general, result of elimination is not necessarily a probability term

34. Need to eliminate: s,x,t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: s,x,t,l,a,b
Initial factors
Eliminate: s
Compute:
Summing on s results in a factor with two arguments fs(b,l)
In general, result of elimination may be a function of several variables

35. Need to eliminate: x,t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: x,t,l,a,b
Initial factors
Eliminate: x
Compute:
Note: fx(a) = 1 for all values of a !!

36. Need to eliminate: t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: t,l,a,b
Initial factors
Eliminate: t
Compute:

37. We want to compute P(d) Need to eliminate: l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: l,a,b
Initial factors
Eliminate: l
Compute:

38. We want to compute P(d) Need to eliminate: b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: b
Initial factors
Eliminate: a,b
Compute:

39. Variable Elimination
We now understand variable elimination as a sequence of rewriting operations
Actual computation is done in elimination step
Computation depends on order of elimination

40. Dealing with evidence How do we deal with evidence?S L T A B X D
Dealing with evidence
How do we deal with evidence?
Suppose get evidence V = t, S = f, D = t
We want to compute P(L, V = t, S = f, D = t)

41. Dealing with Evidence We start by writing the factors:X D
Dealing with Evidence
We start by writing the factors:
Since we know that V = t, we don’t need to eliminate V
Instead, we can replace the factors P(V) and P(T|V) with
These “select” the appropriate parts of the original factors given the evidence
Note that fp(V) is a constant, and thus does not appear in elimination of other variables

42. Dealing with Evidence Given evidence V = t, S = f, D = tB X D
Given evidence V = t, S = f, D = t
Compute P(L, V = t, S = f, D = t )
Initial factors, after setting evidence:

43. Dealing with Evidence Given evidence V = t, S = f, D = tB X D
Given evidence V = t, S = f, D = t
Compute P(L, V = t, S = f, D = t )
Initial factors, after setting evidence:
Eliminating x, we get

44. Dealing with Evidence Given evidence V = t, S = f, D = tB X D
Given evidence V = t, S = f, D = t
Compute P(L, V = t, S = f, D = t )
Initial factors, after setting evidence:
Eliminating x, we get
Eliminating t, we get

45. Dealing with Evidence Given evidence V = t, S = f, D = tB X D
Given evidence V = t, S = f, D = t
Compute P(L, V = t, S = f, D = t )
Initial factors, after setting evidence:
Eliminating x, we get
Eliminating t, we get
Eliminating a, we get

46. Dealing with Evidence Given evidence V = t, S = f, D = tB X D
Given evidence V = t, S = f, D = t
Compute P(L, V = t, S = f, D = t )
Initial factors, after setting evidence:
Eliminating x, we get
Eliminating t, we get
Eliminating a, we get
Eliminating b, we get

47. Variable Elimination Algorithm
Let X1,…, Xm be an ordering on the non-query variables
For I = m, …, 1
Leave in the summation for Xi only factors mentioning Xi
Multiply the factors, getting a factor that contains a number for each value of the variables mentioned, including Xi
Sum out Xi, getting a factor f that contains a number for each value of the variables mentioned, not including Xi
Replace the multiplied factor in the summation

48. Complexity of variable elimination
Suppose in one elimination step we compute
This requires
multiplications
For each value for x, y1, …, yk, we do m multiplications
additions
For each value of y1, …, yk , we do |Val(X)| additions
Complexity is exponential in number of variables in the intermediate factor!

49. Understanding Variable Elimination
We want to select “good” elimination orderings that reduce complexity
This can be done be examining a graph theoretic property of the “induced” graph; we will not cover this in class.
This reduces the problem of finding good ordering to graph-theoretic operation that is well-understood—unfortunately computing it is NP-hard!

50. Exercise: Variable elimination
p(study)=.6
smart
study
p(smart)=.8
p(fair)=.9
prepared
fair Sm St
P(Pr|…) T F .9 .5 .7 .1
pass Sm Pr F
P(Pa|…) T .9 .1 .7 .2
Query: What is the probability that a student is smart, given that they pass the exam?

51. Approaches to inference
Exact inference
Inference in Simple Chains
Variable elimination
Clustering / join tree algorithms
Approximate inference
Stochastic simulation / sampling methods
Markov chain Monte Carlo methods

52. Stochastic simulation - direct
Suppose you are given values for some subset of the variables, G, and want to infer values for unknown variables, U
Randomly generate a very large number of instantiations from the BN
Generate instantiations for all variables – start at root variables and work your way “forward”
Rejection Sampling: keep those instantiations that are consistent with the values for G
Use the frequency of values for U to get estimated probabilities
Accuracy of the results depends on the size of the sample (asymptotically approaches exact results)

53. Direct Stochastic Simulation
P(WetGrass|Cloudy)?
Rain
Sprinkler
Cloudy
WetGrass
P(WetGrass|Cloudy)
= P(WetGrass  Cloudy) / P(Cloudy)
1. Repeat N times: Guess Cloudy at random For each guess of Cloudy, guess Sprinkler and Rain, then WetGrass
2. Compute the ratio of the # runs where WetGrass and Cloudy are True over the # runs where Cloudy is True

54. Exercise: Direct sampling
p(study)=.6
smart
study
p(smart)=.8
p(fair)=.9
prepared
fair Sm St
P(Pr|…) T F .9 .5 .7 .1
pass Sm Pr F
P(Pa|…) T .9 .1 .7 .2
Topological order = …?
Random number generator: .35, .76, .51, .44, .08, .28, .03, .92, .02, .42

55. Likelihood weighting
Idea: Don’t generate samples that need to be rejected in the first place!
Sample only from the unknown variables Z
Weight each sample according to the likelihood that it would occur, given the evidence E

56. Markov chain Monte Carlo algorithm
So called because
Markov chain – each instance generated in the sample is dependent on the previous instance
Monte Carlo – statistical sampling method
Perform a random walk through variable assignment space, collecting statistics as you go
Start with a random instantiation, consistent with evidence variables
At each step, for some nonevidence variable, randomly sample its value, consistent with the other current assignments
Given enough samples, MCMC gives an accurate estimate of the true distribution of values

57. Applications http://excalibur.brc.uconn.edu/~baynet/researchApps.html
Medical diagnosis, e.g., lymph-node deseases
Fraud/uncollectible debt detection
Troubleshooting of hardware/software systems

58. Learning Bayesian Networks

59. Learning Bayesian networks
Inducer E R B A C
Data +
Prior information .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B)

60. Known Structure -- Complete Data
E, B, A
<Y,N,N>
<Y,Y,Y>
<N,N,Y>
<N,Y,Y> . E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is specified
Inducer needs to estimate parameters
Data does not contain missing values

61. Unknown Structure -- Complete Data
E, B, A
<Y,N,N>
<Y,Y,Y>
<N,N,Y>
<N,Y,Y> . E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is not specified
Inducer needs to select arcs & estimate parameters
Data does not contain missing values

62. Known Structure -- Incomplete Data
E, B, A
<Y,N,N>
<Y,?,Y>
<N,N,Y>
<N,Y,?> .
<?,Y,Y> E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is specified
Data contains missing values
We consider assignments to missing values

63. Known Structure / Complete Data
Given a network structure G
And choice of parametric family for P(Xi|Pai)
Learn parameters for network
Goal
Construct a network that is “closest” to probability that generated the data

64. Learning Parameters for a Bayesian NetworkC
Training data has the form:

65. Unknown Structure -- Complete Data
E, B, A
<Y,N,N>
<Y,Y,Y>
<N,N,Y>
<N,Y,Y> . E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is not specified
Inducer needs to select arcs & estimate parameters
Data does not contain missing values

66. Benefits of Learning Structure
Discover structural properties of the domain
Ordering of events
Relevance
Identifying independencies  faster inference
Predict effect of actions
Involves learning causal relationship among variables

67. Why Struggle for Accurate Structure?
Earthquake
Alarm Set
Sound
Burglary
Truth
Adding an arc
Missing an arc
Earthquake
Alarm Set
Sound
Burglary
Earthquake
Alarm Set
Sound
Burglary
Cannot be compensated by accurate fitting of parameters
Also misses causality and domain structure
Increases the number of parameters to be fitted
Wrong assumptions about causality and domain structure

68. Approaches to Learning Structure
Constraint based
Perform tests of conditional independence
Search for a network that is consistent with the observed dependencies and independencies
Pros & Cons
Intuitive, follows closely the construction of BNs
Separates structure learning from the form of the independence tests
Sensitive to errors in individual tests

69. Approaches to Learning Structure
Score based
Define a score that evaluates how well the (in)dependencies in a structure match the observations
Search for a structure that maximizes the score
Pros & Cons
Statistically motivated
Can make compromises
Takes the structure of conditional probabilities into account
Computationally hard

70. Heuristic Search Define a search space:
nodes are possible structures
edges denote adjacency of structures
Traverse this space looking for high-scoring structures
Search techniques:
Greedy hill-climbing
Best first search
Simulated Annealing
...

71. Heuristic Search (cont.)D
Typical operations:
Add C D S C E D
Reverse C E
Delete C E S C E D S C E D

72. Exploiting Decomposability in Local Search
Caching: To update the score of after a local change, we only need to re-score the families that were changed in the last move S C E D S C E D S C E D S C E D

73. Greedy Hill-Climbing Simplest heuristic local search
Start with a given network
empty network
best tree
a random network
At each iteration
Evaluate all possible changes
Apply change that leads to best improvement in score
Reiterate
Stop when no modification improves score
Each step requires evaluating approximately n new changes

74. Greedy Hill-Climbing: Possible Pitfalls
Greedy Hill-Climbing can get struck in:
Local Maxima:
All one-edge changes reduce the score
Plateaus:
Some one-edge changes leave the score unchanged
Happens because equivalent networks received the same score and are neighbors in the search space
Both occur during structure search
Standard heuristics can escape both
Random restarts
TABU search

75. Summary Belief update Role of conditional independence Belief networks
Causality ordering
Inference in BN
Stochastic Simulation
Learning BNs