1. Conditional Independence Creating Tables
Used in Spring 2012, Spring 2013, Winter 2014 (partially)
Bayesian Networks
Conditional Independence
Creating Tables
Notations for Bayesian Networks
Calculating conditional probabilities from the tables
Calculating conditional independence
Markov Chain Monte Carlo
Markov Models.
Markov Models and Probabilistic methods in vision

2. Introduction to Probabilistic Robotics
Probabilities
Bayes rule
Bayes filters
Bayes networks
Markov Chains
new
next

3. Bayesian Networks and Markov Models
Applications in User Modeling
Applications in Natural Language Processing
Applications in robotic control
Applications in robot Vision

4. Bayesian Networks (BNs) – Overview
Introduction to BNs
Nodes, structure and probabilities
Reasoning with BNs
Understanding BNs
Extensions of BNs
Decision Networks
Dynamic Bayesian Networks (DBNs)

5. Definition of Bayesian Networks
A data structure that represents the dependence between variables
Gives a concise specification of the joint probability distribution
A Bayesian Network is a directed acyclic graph (DAG) in which the following holds:
A set of random variables makes up the nodes in the network
A set of directed links connects pairs of nodes
Each node has a probability distribution that quantifies the effects of its parents

6. Conditional Independence
The relationship between :
conditional independence
and BN structure
is important for understanding how BNs work

7. Conditional Independence – Causal Chains
Causal chains give rise to conditional independence
Example: “Smoking causes cancer, which causes dyspnoea” A B C A B C
smoking
cancer
dyspnoea

8. Conditional Independence – Common Causes
Common Causes (or ancestors) also give rise to conditional independence Example: “Cancer is a common cause of the two symptoms: a positive Xray and dyspnoea”
cancer B
dyspnoea
Xray A C
 (A indep C) | B ( )
I have dyspnoea (C ) because of cancer (B) so I do not need an Xray test

9. Conditional Dependence – Common Effects
Common effects (or their descendants) give rise to conditional dependence
Example: “Cancer is a common effect of pollution and smoking” Given cancer, smoking “explains away” pollution
pollution A C
???
smoking B
cancer
pollution
cancer ( ) 
We know that you smoke and have cancer, we do not need to assume that your cancer was caused by pollution

10. Joint Distributions for describing uncertain worlds
• Researchers found already numerous and dramatic benefits of Joint Distributions for describing uncertain worlds • Students in robotics and Artificial Intelligence have to understand problems with using Joint Distributions • You should discover how Bayes Net methodology allows us to build Joint Distributions in manageable chunks

11. Bayes Net methodology Why Bayesian methods matter?
Bayesian Methods are one of the most important conceptual advances in the Machine Learning / AI field to have emerged since 1995.
A clean, clear, manageable language and methodology for expressing what the robot designer is certain and uncertain about
Already, many practical applications in medicine, factories, helpdesks: for instance
P(this problem | these symptoms) // we will use P as probability
anomalousness of this observation
choosing next diagnostic test | these observations

13. Problem 1: Creating Joint Distribution Table
Joint Distribution Table is an important concept

14. Truth table of all combinations of Boolean Variables
Probabilistic truth table
You can guess this table, you can take data from some statistics,
You can build this table based on some partial tables
Truth table of all combinations of Boolean Variables

15. Idea – use decision diagrams to represent these data.

16. Use of independence while creating the tables

17. Wet – Sprinkler – Rain Example

18. Wet-Sprinkler-Rain Example

19. Problem 1: Creating the Joint Table

20. Our Goal is to derive this table
Let us observe that if I know 7 of these, the eight is obviously unique , as their sum = 1
So I need to guess or calculate or find 2n-1 = 7 values
But the same data can be stored explicitely or implicitely, not necessarily in the form of a table!!
What extra assumptions can help to create this table?

21. Wet-Sprinkler-Rain Example

22. Wet-Sprinkler-Rain Example
Sprinkler on under condition that it rained
You need to understand causation when you create the table
Wet-Sprinkler-Rain Example
Understanding of causation

23. Independence simplifies probabilities
We use independence of variables S and R
P(S|R) = Sprinkler on under condition that it rained
We can use these probabilities to create the table
S and R are independent
Wet-Sprinkler-Rain Example

24. Wet-Sprinkler-Rain Example Conditional Probability Table (CPT)
We create the CPT for S and R based on our knowledge of the problem
Conditional Probability Table (CPT)
Sprinkler was on
It rained
Grass is wet
What about children playing or a dog pissing?
It is still possible by this value 0.1
This first step shows the collected data

25. Full joint for only S and R
Independence of S and R is used
0.95
0.90
0.90
0.01
Use chain rule for probabilities
Wet-Sprinkler-Rain Example

26. Chain Rule for Probabilities
Random variables
0.95
0.90
0.90
0.01

27. Full joint probability
You have a table
You want to calculate some probability
P(~W)
Wet-Sprinkler-Rain Example

28. Wet-Sprinkler-Rain Example
Independence of S and R implies calculating fewer numbers to create the complete Joint Table for W, S and R
Six numbers
We reduced only from seven to six numbers
Wet-Sprinkler-Rain Example

29. Explanation of Diagrammatic Notations
such as Bayes Networks
You do not need to build the complete table!!

30. You can build a graph of tables or nodes which correspond to certain types of tables

31. Wet-Sprinkler-Rain Example

32. Wet-Sprinkler-Rain Example Conditional Probability Table (CPT)
Sprinkler was on
It rained
Grass is wet
This first step shows the collected data
Conditional Probability Table (CPT)

33. Full joint probability
You have a table
You want to calculate some probability
When you have this table you can modify it,
you can also calculate everything!!
P(~W)

34. Problem 2: Calculating conditional probabilities from the Joint Distribution Table

35. Wet-Sprinkler-Rain Example
Probability that S=T and W=T
= Probability that grass is wet under assumption that sprinkler was on
Probability that S=T

36. Wet-Sprinkler-Rain Example

37. Wet-Sprinkler-Rain Example
We showed examples of both causal inference and diagnostic inference
We will use this in next slide
Wet-Sprinkler-Rain Example

38. “Explaining Away” the facts from the table
Calculated earlier from this table
<
Wet-Sprinkler-Rain Example

39. Conclusions on this problem
Table can be used for Explaining Away
Table can be used to calculate conditional independence.
Table can be used to calculate conditional probabilities
Table can be used to determine causality

40. Problem 3: What if S and R are dependent
Problem 3: What if S and R are dependent? Calculating conditional independence

41. Conditional Independence of S and R
Wet-Sprinkler-Rain Example

42. Diagrammatic notation for conditional Independence of two variables
Wet-Sprinkler-Rain Example extended

43. Conditional Independence formalized for sets of variables

44. Now we will explain conditional independence
CLOUDY - Wet-Sprinkler-Rain Example

45. Example – Lung Cancer Diagnosis

46. Example – Lung Cancer Diagnosis
A patient has been suffering from shortness of breath (called dyspnoea) and visits the doctor, worried that he has lung cancer.
The doctor knows that other diseases, such as tuberculosis and bronchitis are possible causes, as well as lung cancer.
She also knows that other relevant information includes whether or not the patient is a smoker (increasing the chances of cancer and bronchitis) and what sort of air pollution he has been exposed to.
A positive Xray would indicate either TB or lung cancer.

47. Nodes and Values in Bayesian Networks
Q: What are the nodes to represent and what values can they take?
A: Nodes can be discrete or continuous
Boolean nodes – represent propositions taking binary values Example: Cancer node represents proposition “the patient has cancer”
Ordered values Example: Pollution node with values low, medium, high
Integral values Example: Age with possible values 1-120
Lung Cancer

48. Lung Cancer Example: Nodes and Values
Node name
Type
Values
Pollution
Binary
{low,high}
Smoker
Boolean
{T,F}
Cancer
Dyspnoea
Xray
{pos,neg}
Shortness of breath
Example of variables as nodes in BN

49. Lung Cancer Example: Bayesian Network Structure
Pollution
Smoker
Cancer
Xray
Dyspnoea
Lung Cancer

50. Conditional Probability Tables (CPTs) in Bayesian Networks

51. Conditional Probability Tables (CPTs) in Bayesian Networks
After specifying topology, we must specify
the CPT for each discrete node
Each row of CPT contains the conditional probability of each node value for each possible combination of values in its parent nodes
Each row of CPT must sum to 1
A CPT for a Boolean variable with n Boolean parents contains 2n+1 probabilities
A node with no parents has one row (its prior probabilities)

52. Lung Cancer Example: example of CPT
Smoking is true
Probability of cancer given state of variables P and S
Pollution low
C= cancer
P = pollution
S = smoking
X = Xray
D = Dyspnoea
Bayesian Network for cancer
Lung Cancer

53. Several small CPTs are used to create larger JDTs.

54. The Markov Property for Bayesian Networks
Modelling with BNs requires assuming the Markov Property:
There are no direct dependencies in the system being modelled which are not already explicitly shown via arcs
Example: smoking can influence dyspnoea only through causing cancer

55. Software NETICA for Bayesian Networks and joint probabilities

56. Reasoning with Numbers – Using Netica software
Here are the collected data
Lung Cancer

57. Representing the Joint Probability Distribution: Example
We want to calculate this
P = pollution
S = smoking
X = Xray
D = Dyspnoea
This graph shows how we can calculate the joint probability from other probabilities in the network
Lung Cancer

58. Problem 4: Determining Causality and Bayes Nets: Advertisement example

59. Causality and Bayes Nets: Advertisement example
Bayes nets allow one to learn about causal relationships
One more Example:
Marketing analysts want to know whether to increase, decrease or leave unchanged the exposure of some advertisement in order to maximize profit from the sale of some product
Advertised (A) and Buy (B) will be variables for someone having seen the advertisement or purchased the product
Advertised-Buy Example

60. Causality Example
So we want to know the probability that B=true given that we force A=true, or A=false
We could do this by finding two similar populations and observing B based on A=true for one and A=false for the other
But this may be difficult or expensive to find such populations
Advertised (A) seen the advertisement
Buy (B) will be variables for someone purchased the product
Advertised-Buy Example

61. How causality can be represented in a graph?

62. Markov condition and Causal Markov Condition
But how do we learn whether or not A causes B at all?
The Markov Condition states:
Any node in a Bayes net is conditionally independent of its non-descendants given its parents
The CAUSAL Markov Condition (CMC) states:
Any phenomenon in a causal net is independent of its non-effects given its direct causes
Advertised (A) and Buy (B)
Advertised-Buy Example

63. Acyclic Causal Graph versus Bayes Net
Thus, if we have a directed acyclic causal graph C for variables in X, then, by the Causal Markov Condition, C is also a Bayes net for the joint probability distribution of X
The reverse is not necessarily true—a network may satisfy the Markov condition without depicting causality
Advertised-Buy Example

64. Causality Example: when we learn that p(b|a) and p(b|a’) are not equal
Given the Causal Markov Condition CMC, we can infer causal relationships from conditional (in)dependence relationships learned from the data
Suppose we learn with high Bayesian probability that p(b|a) and p(b|a’) are not equal
Given the CMC, there are four simple causal explanations for this: (more complex ones too)

65. Causality Example: four causal explanations
A causes B
If they advertise more you buy more
B causes A
If you buy more, they have more money to advertise

66. Causality Example: four causal explanations
selection bias
Hidden common cause of A and B (e.g. income)
A and B are causes for data selection
(a.k.a. selection bias,
perhaps if database didn’t record false instances of A and B)
In rich country they advertise more and they buy more
If you increase information about Ad in database, then you increase also information about Buy in the database

67. Causality Example continued
But we still don’t know if A causes B
Suppose
We learn about the Income (I) and geographic Location (L) of the purchaser
And we learn with high Bayesian probability the network on the right
Advertised (A=Ad) and Buy (B)
Advertised-Buy Example

68. Causality Example - using CMC
Given the Causal Markov Condition CMC, the ONLY causal explanation for the conditional (in)dependence relationships encoded in the Bayes net is that Ad is a cause for Buy
That is, none of the other relationships or combinations thereof produce the probabilistic relationships encoded here
Advertised (Ad) and Buy (B)
Advertised-Buy Example

69. Causality in Bayes Networks
Thus, Bayes Nets allow inference of causal relationships by the Causal Markov Condition (CMC)

70. Problem 5: Determine D-separation in Bayesian Networks

71. D-separation in Bayesian Networks
We will formulate a Graphical Criterion of conditional independence
We can determine whether a set of nodes X is independent of another set Y, given a set of evidence nodes E, via the Markov property:
If every undirected path from a node in X to a node in Y is d-separated by E, then X and Y are conditionally independent given E

72. Determining D-separation (cont)
A set of nodes E d-separates two sets of nodes X and Y, if every undirected path from a node in X to a node in Y is blocked given E
A path is blocked given a set of nodes E, if there is a node Z on the path for which one of three conditions holds:
Z is in E and Z has one arrow on the path leading in and one arrow out (chain)
Z is in E and Z has both path arrows leading out (common cause)
Neither Z nor any descendant of Z is in E, and both path arrows lead into Z (common effect)
Chain
Common cause
Common effect

73. Another Example of Bayesian Networks: Alarm
Alarm Example
Let us draw BN from these data

74. Bayes Net Corresponding to Alarm-Burglar problem
Alarm Example

75. Compactness, Global Semantics, Local Semantics and Markov Blanket
Compactness of Bayes Net
Burglar
Earthquake
John calls
Mary calls
Alarm Example

76. Global Semantics, Local Semantics and Markov Blanket for BNs
Useful concepts

77. Alarm Example

79. Markov’s blanket are:
Parents
Children
Children’s parent

80. Problem 6 How to systematically Build a Bayes Network -- Example

82. Alarm Example

83. Alarm Example

84. Alarm Example

85. So we add arrow
Alarm Example

86. Alarm Example

87. Alarm Example

88. Bayes Net for the car that does not want to start
Such networks can be used for robot diagnostics or diagnostic of a human done by robot

89. Inference in Bayes Nets and how to simplify it
Alarm Example

90. First method of simplification: Enumeration
Alarm Example

91. Alarm Example

92. Second method: Variable Elimination
Alarm Example
Variable A was eliminated
Variable E was eliminated

93. 3SAT Example
Polytrees are better

94. IDEA:
Convert DAG to polytrees

95. Clustering is used to convert non-polytree BNs

96. EXAMPLE: Clustering is used to convert non-polytree BNs
Not a polytree
Is a polytree
Alarm Example

97. Approximate Inference
Direct sampling methods
Rejection sampling
Likelihood weighting
Markov chain Monte Carlo

98. 1. Direct Sampling Methods

99. Direct Sampling
Direct Sampling generates minterms with their probabilities

100. We start from top
W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

101. W=wet
C= cloudy
R = rain
S = sprinkler
Cloudy = yes
Wet Sprinkler Rain Example

102. W=wet
C= cloudy
R = rain
S = sprinkler
Cloudy = yes
Wet Sprinkler Rain Example

103. W=wet
C= cloudy
R = rain
S = sprinkler
sprinkler = no
Wet Sprinkler Rain Example

104. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

105. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

106. W=wet
C= cloudy
R = rain
S = sprinkler
We generated a sample minterm
C S’ R W
Wet Sprinkler Rain Example

107. 2. Rejection Sampling Methods

108. Rejection Sampling Reject inconsistent samples
Wet Sprinkler Rain Example

109. 3. Likelihood weighting methods

111. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

112. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

113. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

114. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

115. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

116. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

117. W=wet
C= cloudy
R = rain
S = sprinkler
Wet Sprinkler Rain Example

118. Likelihood weighting vs. rejection sampling
Both generate consistent estimates of the joint distribution conditioned on the values of the evidence variables
Likelihood weighting converges faster to the correct probabilities
But even likelihood weighting degrades with many evidence variables because a few samples will have nearly all the total weight

120. Sources Prof. David Page Matthew G. Lee Nuria Oliver, Barbara Rosario
Alex Pentland
Ehrlich Av
Ronald J. Williams
Andrew Moore tutorial with the same title
Russell &Norvig’s AIMA site
Alpaydin’s Introduction to Machine Learning site.