1. CVPR Winter seminar Jaemin Kim
Bayesian Network
CVPR Winter seminar
Jaemin Kim

2. Outline Concepts in Probability Bayesian Networks Inference
Random variables
Basic properties (Bayes rule)
Bayesian Networks
Inference
Decision making
Learning networks from data
Reasoning over time
Applications

3. Probability distribution P(X|x)
Probabilities
Probability distribution P(X|x)
X is a random variable
Discrete
Continuous
x is background state of information

4. Discrete Random Variables
Finite set of possible outcomes
X binary:

5. Continuous Random Variables
Probability distribution (density function) over continuous values

6. Joint Conditional More Probabilities Probability that both X=x and Y=y
Probability that X=x given we know that Y=y

7. Rule of Probabilities
Product Rule
Marginalization
X binary:

8. Bayes Rule

9. Graph Model 목적: Definition:
특정 variable에 관한 정보 (probability distribution)를 상관관계가 있는 다른 variables 관한 정보부터 추출
Definition:
A collection of variables (nodes) with a set of dependencies (edges) between the variables, and
a set of probability distribution functions for each variable
A Bayesian network is a special type of graph model which is a directed acyclic graph (DAG)

10. Bayesian Networks Conditional probability specifications A Graph
nodes represent the random variables
directed edges (arrows) between pairs of nodes
it must be a Directed Acyclic Graph (DAG)
the graph represents relationships between variables
Conditional probability specifications
the conditional probability distribution (CPD) of each variable
given its parents
discrete variable: table (CPT)

11. Bayesian Networks (Belief Networks)
A Graph
directed edges (arrows) between pairs of nodes
causality: A “causes” B
AI an statistics communities
Markov Random fields (MRF)
A Graph
undirected edges (arrows) between pairs of nodes
a simple definition of independence:
If all paths between the nodes in A and B are separated by a node c
A and B are conditionally independent given a third set C
physics and vision communities

12. Bayesian Networks

13. Basics Bayesian networks Structured representation
Conditional independence
Naïve Bayes model
Independence facts

14. Bayesian networks P( S=no) 0.80 S=light) 0.15 S=heavy) 0.05 Smoking=
Cancer P(
S=no)
0.80
S=light)
0.15
S=heavy)
0.05
P(S):
Smoking= no
light
heavy P(
C=none)
0.96
0.88
0.60
C=benign)
0.03
0.08
0.25
C=malig)
0.01
0.04
0.15
P(C|S):

15. P(C,S) = P(C|S) P(S) Product Rule
P(C=none ^ S=no) = P(C=none | S=no)P(S=no) = 0.96*0.8 = 0.768

16. P(C,S) = P(C|S) P(S) Product Rule
P(C=none ^ S=no) = P(C=none | S=no)P(S=no) = 0.96*0.8 = 0.768

17. Marginalization P(Smoke) P(Cancer)
P(S=no) = P(S=no ^ C=no) + P(S=no ^ C=be) P(S=no & C=mal)
P(C=mal) = P(C=mal ^ S=no) + P(C=mal ^ S=light) + P(C=mal | S=heavy)

18. Bayes Rule Revisited Cancer= none benign malignant P( S=no) 0.821
0.522
0.421
S=light)
0.141
0.261
0.316
S=heavy)
0.037
0.217
0.263
P(S|C):

19. A Bayesian Network Age Gender Exposure to Toxics Smoking Cancer Serum
Calcium
Lung
Tumor

20. Problems with Large Instances
The joint probability distribution, P(A,G,E,S,C,L,SC)
For five binary variables there are 27 = 128 values in the joint distribution (for 100 variables there are over 1030 values)
How are these values to be obtained?
Inference
To obtain posterior distributions once some evidence is available requires summation over an exponential number of terms eg 22 in the calculation of
which increases to 297 if there are 100 variables.

21. Independence Age and Gender are independent. Age Gender
P(A,G) = P(G)P(A)
P(A|G) = P(A) A ^ G
P(G|A) = P(G) G ^ A
P(A,G) = P(G|A) P(A) = P(G)P(A)
P(A,G) = P(A|G) P(G) = P(A)P(G)

22. Conditional Independence
Cancer is independent of Age and Gender given Smoking.
Age
Gender
Smoking
P(C|A,G,S) = P(C|S) C ^ A,G | S
Cancer
(Smoking=heavy)조건은 Age와 Gender의 확률분포를 제한
(Smoking=heavy)조건은 cancer의 확률분포를 제한
(Smoking=heavy)조건하에서 cancer는 age와 gender에 독립

23. More Conditional Independence: Naïve Bayes
Serum Calcium and Lung Tumor are dependent
Cancer
Serum Calcium is independent of Lung Tumor, given Cancer
P(L|SC,C) = P(L|C)
Serum
Calcium
Lung
Tumor 혈청

24. More Conditional Independence: Explaining Away
Exposure to Toxics and Smoking are independent
Exposure
to Toxics
Smoking
E ^ S
Cancer
Exposure to Toxics is dependent on Smoking, given Cancer
P(E = heavy | C = malignant) >
P(E = heavy | C = malignant, S=heavy)

25. More Conditional Independence: Explaining Away
Exposure
to Toxics
Exposure
to Toxics
Smoking
Smoking
Cancer
Cancer
Exposure to Toxics is dependent on Smoking,
given Cancer
Moralize the graph.

26. Put it all together Smoking Gender Age Cancer Lung Tumor Serum Calcium
Exposure
to Toxics

27. General Product (Chain) Rule for Bayesian Networks
Pai=parents(Xi)

28. Conditional Independence
A variable (node) is conditionally independent of its non-descendants given its parents.
Age
Gender
Non-Descendants
Exposure
to Toxics
Smoking
Parents
Cancer is independent of Age and Gender given Exposure to Toxics and Smoking.
Cancer
Serum
Calcium
Lung
Tumor
Descendants

29. Another non-descendant
Age
Gender
Cancer is independent of Diet given Exposure to Toxics and Smoking.
Exposure
to Toxics
Smoking
Diet
Cancer
Serum
Calcium
Lung
Tumor

30. Representing the Joint Distribution
In general, for a network with nodes X1, X2, …, Xn then
An enormous saving can be made regarding the number of values required for the joint distribution.
To determine the joint distribution directly for n binary variables 2n – 1 values are required.
For a BN with n binary variables and each node has at most k parents then less than 2kn values are required.

31. An Example P(s1)=0.2 Smoking history Fatigue Bronchitis Lung Cancer
X-ray
P(b1|s1)=0.25 P(b1|s2)=0.05
P(l1|s1)=0.003 P(l1|s2)=
P(f1|b1,l1)=0.75 P(f1|b1,l2)=0.10 P(f1|b2,l1)=0.5 P(f1|b2,l2)=0.05
P(x1|l1)=0.6 P(x1|l2)=0.02

32. Solution
Note that our joint distribution with 5 variables can be represented as:
Consequently the joint probability distribution can now be expressed as
For example, the probability that someone has a smoking history, lung cancer but not bronchitis, suffers from fatigue and tests positive in an X-ray test is

33. Independence and Graph Separation
Given a set of observations, is one set of variables dependent on another set?
Observing effects can induce dependencies.
d-separation (Pearl 1988) allows us to check conditional independence graphically.

34. Bayesian networks Additional structure Nodes as functions
Causal independence
Context specific dependencies
Continuous variables
Hierarchy and model construction

35. A BN node is conditional distribution function
Nodes as funtions
A BN node is conditional distribution function
its parent values are the inputs
its output is a distribution over its values
lo : 0.7
med : 0.1
hi : 0.2 b a
a b
a b A
0.1
0.3
0.6
0.7
0.1
0.2 lo
med hi
0.4
0.2
0.7
0.1
0.2
0.5 X X
0.3
0.2 B

36. Nodes as funtions Any type of function from Val(A,B) to distributions
lo : 0.7
med : 0.1
hi : 0.2 b a A
Any type of function
from Val(A,B)
to distributions
over Val(X) X X B

37. Continuous variables A/C Setting Outdoor Temperaturehi
97o
Function from Val(A,B)
to density functions
over Val(X)
P(x) x
Indoor
Temperature
Indoor
Temperature

38. Gaussian (normal) distributions
N(m, s)
different mean
different variance

39. Each variable is a linear function of its parents,
Gaussian networks X Y
Each variable is a linear function of its parents,
with Gaussian noise
Joint probability density functions: X Y X Y

40. Composing functions
Recall: a BN node is a function
We can compose functions to get more complex functions.
The result: A hierarchically structured BN.
Since functions can be called more than once, we can reuse a BN model fragment in multiple contexts.

41. Owner Owner Car: Maintenance Age Original-value Mileage Brakes: Brakes
Income
Owner
Maintenance
Age
Original-value
Mileage
Brakes:
Power
Brakes
Tires:
RF-Tire
LF-Tire
Traction
Pressure
Car:
Engine
Engine:
Power
Engine
Tires
Fuel-efficiency
Braking-power

42. Knowledge acquisition
Bayesian Networks
Knowledge acquisition
Variables
Structure
Numbers

43. Values versus Probabilities
What is a variable?
Collectively exhaustive, mutually exclusive values
Error Occured
No Error
Risk of Smoking
Smoking
Values versus Probabilities

44. Clarity Test: Knowable in Principle
Weather {Sunny, Cloudy, Rain, Snow}
Gasoline: Cents per gallon
Temperature {  100F , < 100F}
User needs help on Excel Charting {Yes, No}
User’s personality {dominant, submissive}

45. Structuring Network structure corresponding
Gender
Age
Network structure corresponding
to “causality” is usually good.
Smoking
Exposure
to Toxic
Cancer
Genetic
Damage
Extending the conversation.
Lung
Tumor

46. Course Contents
Concepts in Probability
Bayesian Networks
Inference
Decision making
Learning networks from data
Reasoning over time
Applications

47. Inference
Patterns of reasoning
Basic inference
Exact inference
Exploiting structure
Approximate inference

48. How likely are elderly males to get malignant cancer?
Predictive Inference
Age
Gender
How likely are elderly males
to get malignant cancer?
Exposure
to Toxics
Smoking
Cancer
P(C=malignant | Age>60, Gender= male)
Serum
Calcium
Lung
Tumor

49. Combined
Age
Gender
How likely is an elderly male patient with high Serum Calcium to have malignant cancer?
Exposure
to Toxics
Smoking
Cancer
P(C=malignant | Age>60,
Gender= male, Serum Calcium = high)
Serum
Calcium
Lung
Tumor

50. Explaining away
Age
Gender
If we see a lung tumor, the probability of heavy smoking and of exposure to toxics both go up.
Exposure
to Toxics
Smoking
If we then observe heavy smoking, the probability of exposure to toxics goes back down.
Smoking
Cancer
Serum
Calcium
Lung
Tumor

51. Inference in Belief Networks
Find P(Q=q|E= e)
Q the query variable
E set of evidence variables
P(q | e) =
P(q, e)
P(e)
X1,…, Xn are network variables except Q, E
P(q, e) =
S P(q, e, x1,…, xn)
x1,…, xn

52. A B C Basic Inference P(b) = S P(a, b) = S P(b | a) P(a)
P(c) = S P(c | b) P(b)
P(c) = S P(a, b, c)
b,a
b,a
= S P(c | b) P(b | a) P(a)
= S P(c | b) S P(b | a) P(a) b a
P(b)

53. because of independence of Y1, Y2:
Inference in trees Y1 Y2 X X
P(x) = S P(x | y1, y2) P(y1, y2)
y1, y2
because of independence of Y1, Y2:
y1, y2
= S P(x | y1, y2) P(y1) P(y2)

54. Polytrees
A network is singly connected (a polytree) if it contains no undirected loops. D C
Theorem: Inference in a singly connected network can be done in linear time*.
Main idea: in variable elimination, need only maintain distributions over single nodes.
* in network size including table sizes.

55. The problem with loops Cloudy Rain Sprinkler Grass-wet
P(c)
0.5
Cloudy c c c c
Rain
Sprinkler
P(s)
P(r)
0.01
0.99
0.99
0.01
Grass-wet
deterministic or
The grass is dry only if no rain and no sprinklers.
P(g) = P(r, s) ~ 0

56. The problem with loops contd.1
P(g) =
P(g | r, s) P(r, s) P(g | r, s) P(r, s)
+ P(g | r, s) P(r, s) P(g | r, s) P(r, s)
= P(r, s)
~ 0
= P(r) P(s) ~ 0.5 ·0.5 = 0.25
problem

57. S S A P(c) = S P(c | b) S P(b | a) P(a) P(b) B C Variable eliminationx
P(A)
P(B | A)
P(B, A) C S A
P(B) x
P(C | B)
P(C, B) S B
P(C)

58. Inference as variable elimination
A factor over X is a function from val(X) to numbers in [0,1]:
A CPT is a factor
A joint distribution is also a factor
BN inference:
factors are multiplied to give new ones
variables in factors summed out
A variable can be summed out as soon as all factors mentioning it have been multiplied.

59. Variable Elimination with loopsx
P(A,G,S)
P(A)
P(S | A,G)
P(G)
Age
Gender
P(A,E,S)
P(E | A) x
P(A,S) S G
Exposure
to Toxics
Smoking
P(E,S) S A
P(C | E,S)
P(E,S,C) x
Cancer S
E,S
P(C)
P(L | C) x
P(C,L) S C
P(L)
Serum
Calcium
Lung
Tumor
Complexity is exponential in the size of the factors

60. Inference in BNs and Junction Tree
The main point of BNs is to enable probabilistic inference to be performed. Inference is the task of computing the probability of each value of a node in BNs when other variables’ values are know.
The general idea is doing inference by representing the joint probability distribution on an undirected graph called the Junction tree
The junction tree has the following characteristics:
it is an undirected tree, its nodes are clusters of variables
given two clusters, C1 and C2, every node on the path between them contains their intersection C1  C2
a Separator, S, is associated with each edge and contains the variables in the intersection between neighbouring nodes
ABC
BCD
CDE BC CD

61. Inference in BNs Moralize the Bayesian network
Triangulate the moralized graph
Let the cliques of the triangulated graph be the nodes of a tree, and construct the junction tree
Belief propagation throughout the junction tree to do inference

62. Constructing the Junction Tree (1)
Step 1. Form the moral graph from the DAG
Consider BN in our example S F B L X S F B L X
Moral Graph – marry parents and remove arrows
DAG

63. Constructing the Junction Tree (2)
Step 2. Triangulate the moral graph
An undirected graph is triangulated if every cycle of length greater than 3 possesses a chord S F B L X

64. Constructing the Junction Tree (3)
Step 3. Identify the Cliques
A clique is a subset of nodes which is complete (i.e. there is an edge between every pair of nodes) and maximal. S F B L X
Cliques
{B,S,L} {B,L,F} {L,X} 

65. Constructing the Junction Tree (4)
Step 4. Build Junction Tree
The cliques should be ordered (C1,C2,…,Ck) so they possess the running intersection property: for all 1 < j ≤ k, there is an i < j such that Cj  (C1… Cj-1)  Ci.
To build the junction tree choose one such I for each j and add an edge between Cj and Ci.
Junction Tree
Cliques
{B,S,L} {B,L,F} {L,X}
BSL
BLF LX  BL L

66. Potentials Initialization
To initialize the potential functions:
1. set all potentials to unity
2. for each variable, Xi, select one node in the junction tree (i.e. one clique) containing both that variable and its parents, pa(Xi), in the original DAG
3. multiply the potential by P(xi|pa(xi))
BSL
BLF LX BL L

67. Potential Representation
The joint probability distribution can now be represented in terms of potential functions, ϕ, defined on each clique and each separator of the junction tree. The joint distribution is given by
The idea is to transform one representation of the joint distribution to another in which for each clique, C, the potential function gives the marginal distribution for the variables in C, i.e.
This will also apply for the separators, S.

68. Triangulation
Given a numbered graph, proceed from node n, decrease to 1
Determine the lower-numbered nodes which are adjacent to the current node, including those which may have been made adjacent to this node earlier in this algorithm
Connects these nodes to each other.

69. Triangulation Numbering the nodes Arbitrarily number the nodes
Maximum cardinality search
Give any node a value of 1
For each subsequent number, pick an new unnumbered node that neighbors the most already numbered nodes

70. Triangulation BN
Moralized graph

71. Triangulation 6 7 8 2 5 4 1 3 5 3 8 4 6 7 2 1
Arbitrary numbering

72. Triangulation Maximum cardinality search 6 7 8 2 5 4 1 3 6 7 8 2 5 4 1

73. Concepts in Probability Bayesian Networks Inference Decision making
Course Contents
Concepts in Probability
Bayesian Networks
Inference
Decision making
Learning networks from data
Reasoning over time
Applications

74. Decision - an irrevocable allocation of domain resources
Decision making
Decision - an irrevocable allocation of domain resources
Decision should be made so as to maximize expected utility.
View decision making in terms of
Beliefs/Uncertainties
Alternatives/Decisions
Objectives/Utilities

75. Course Contents Concepts in Probability Bayesian Networks Inference
Decision making
Learning networks from data
Reasoning over time
Applications

76. Learning networks from data
The learning task
Parameter learning
Fully observable
Partially observable
Structure learning
Hidden variables

77. Input: fully or partially observable data cases?
The learning task
B E A C N
Call
Alarm
Burglary
Earthquake
Newscast
Output: BN modeling data e a c b n b e a c n
...
Input: training data
Input: fully or partially observable data cases?
Output: parameters or also structure?

78. Parameter learning: one variable
Unfamiliar coin:
Let q = bias of coin (long-run fraction of heads)
If q known (given), then
P(X = heads | q ) = q
Different coin tosses independent given q
P(X1, …, Xn | q ) =
q h (1-q)t
h heads, t tails

79. Input: a set of previous coin tosses
Maximum likelihood
Input: a set of previous coin tosses
X1, …, Xn = {H, T, H, H, H, T, T, H, . . ., H}
h heads, t tails
Goal: estimate q
The likelihood P(X1, …, Xn | q ) = q h (1-q )t
The maximum likelihood solution is:
q* = h
h+t

80. Conditioning on data D P(q ) P(q | D)  P(q ) P(D | q )
h heads, t tails
P(q )
P(q | D)
 P(q ) P(D | q )
= P(q ) q h (1-q )t
1 head
1 tail

81. Conditioning on data Good parameter distribution:
* Dirichlet distribution generalizes Beta to non-binary variables.

82. General parameter learning
A multi-variable BN is composed of several independent parameters (“coins”).
Three parameters: A B
qA, qB|a, qB|a
Can use same techniques as one-variable case to learn each one separately
Max likelihood estimate of qB|a would be:
#data cases with b, a
#data cases with a
q*B|a =

83. Partially observable data
B E A C N
Burglary
Earthquake b ? a c ?
Alarm b ? a ? n
Newscast
Call
...
Fill in missing data with “expected” value
expected = distribution over possible values
use “best guess” BN to estimate distribution

84. Sj I(n,e | dj) S j I(e | dj) Intuition In fully observable case:
1 if E=e in data case dj
0 otherwise =
q*n|e =
#data cases with n, e
#data cases with e
In partially observable case I is unknown.
Best estimate for I is:
Problem: q* unknown.

85. Expectation Maximization (EM)
Repeat :
until convergence.
Expectation (E) step
Use current parameters q to estimate filled in data.
Maximization (M) step
Use filled in data to do max likelihood estimation
Set:

86. Structure learning
Goal:
find “good” BN structure (relative to data)
Solution:
do heuristic search over space of network
structures.

87. Search space Space = network structures
Operators = add/reverse/delete edges

88. Heuristic search
Use scoring function to do heuristic search (any algorithm).
Greedy hill-climbing with randomness works pretty well.
score

89. Scoring
Fill in parameters using previous techniques & score completed networks.
One possibility for score: D
likelihood function: Score(B) = P(data | B)
Example: X, Y independent coin tosses
typical data = (27 h-h, 22 h-t, 25 t-h, 26 t-t)
Maximum likelihood network structure: X Y
Max. likelihood network typically fully connected
This is not surprising: maximum likelihood always overfits…

90. Better scoring functions
MDL formulation: balance fit to data and model complexity (# of parameters)
Score(B) = P(data | B) - model complexity
Full Bayesian formulation
prior on network structures & parameters
more parameters  higher dimensional space
get balance effect as a byproduct*
* with Dirichlet parameter prior, MDL is an approximation
to full Bayesian score.

91. There may be interesting variables that we never get to observe:
Hidden variables
There may be interesting variables that we never get to observe:
topic of a document in information retrieval;
user’s current task in online help system.
Our learning algorithm should
hypothesize the existence of such variables;
learn an appropriate state space for them.

92. E1 E3 E2
randomly
scattered data

93. E1 E3 E2
actual data

94. Bayesian clustering (Autoclass)
naïve Bayes model:
…... E1 E2 En
(hypothetical) class variable never observed
if we know that there are k classes, just run EM
learned classes = clusters
Bayesian analysis allows us to choose k, trade off fit to data with model complexity

95. E1 E3 E2
Resulting cluster
distributions

96. Detecting hidden variables
Unexpected correlations hidden variables.
Cholesterolemia
Test1
Test2
Test3
Hypothesized model
Cholesterolemia
Test1
Test2
Test3
Data model
Cholesterolemia
Test1
Test2
Test3
“Correct” model
Hypothyroid

97. Course Contents Concepts in Probability Bayesian Networks Inference
Decision making
Learning networks from data
Reasoning over time
Applications

98. Dynamic Bayesian networks Hidden Markov models
Reasoning over time
Dynamic Bayesian networks
Hidden Markov models
Decision-theoretic planning
Markov decision problems
Structured representation of actions
The qualification problem & the frame problem
Causality (and the frame problem revisited)

99. Dynamic environments Markov property:
State(t+2)
State(t+1)
State(t)
Markov property:
past independent of future given current state;
a conditional independence assumption;
implied by fact that there are no arcs t t+2.

100. Dynamic Bayesian networks
State described via random variables.
Velocity(t)
Position(t)
Weather(t)
Drunk(t)
Velocity(t+1)
Position(t+1)
Weather(t+1)
Drunk(t+1)
Velocity(t+2)
Position(t+2)
Weather(t+2)
Drunk(t+2)
...

101. An HMM is a simple model for a partially observable stochastic domain.
Hidden Markov model
An HMM is a simple model for a partially observable stochastic domain.
State transition
model
Observation
State(t)
State(t+1)
Obs(t)
Obs(t+1)

102. Partially observable stochastic environment:
Hidden Markov model
Partially observable stochastic environment:
0.8
0.15
0.05
Mobile robots:
states = location
observations = sensor input
Speech recognition:
states = phonemes
observations = acoustic signal
Biological sequencing:
states = protein structure
observations = amino acids

103. Acting under uncertainty
Markov Decision Problem (MDP)
action model
agent
observes
state
State(t+2)
Action(t+1)
Action(t)
Reward(t+1)
Reward(t)
State(t)
State(t+1)
Overall utility = sum of momentary rewards.
Allows rich preference model, e.g.:
rewards corresponding
to “get to goal asap” =
goal states
other states

104. Partially observable MDPs
agent observes
Obs, not state
State(t+2)
State(t)
State(t+1)
Action(t)
Action(t+1)
Reward(t+1)
Reward(t)
Obs(t)
Obs(t+1)
Obs depends
on state
The optimal action at time t depends on the entire history of previous observations.
Instead, a distribution over State(t) suffices.

105. Structured representation
Position(t+1)
Holding(t+1)
Direction(t+1)
Move:
Position(t)
Holding(t)
Direction(t)
Preconditions
Effects
Position(t)
Holding(t)
Direction(t)
Position(t+1)
Holding(t+1)
Direction(t+1)
Turn:
Probabilistic action model
allows for exceptions & qualifications;
persistence arcs: a solution to the frame problem.

106. Medical expert systems
Applications
Medical expert systems
Pathfinder
Parenting MSN
Fault diagnosis
Ricoh FIXIT
Decision-theoretic troubleshooting
Vista
Collaborative filtering