Uncertainty & Bayesian Belief Networks

Uncertainty & Bayesian Belief Networks

2

Data
-
Mining with Bayesian Networks on
the Internet

•
Internet can be seen as a massive repository
of Data

•
Data is often updated

•
Once meaningful data has been collected
from the Internet, some model is needed
which is able to:

–
be learnt from the vast amount of available data

–
enable the user to reason about the data.

–
Be easily updated given new data

3

Section 1
-

Bayesian Networks


An Introduction

•
Brief Summary of Expert Systems

•
Causal Reasoning

•
Probability Theory

•
Bayesian Networks
-

Definition, inference

•
Current issues in Bayesian Networks

•
Other Approaches to Uncertainty



4

Expert Systems

1 Rule Based Systems

•
1960s
-

Rule Based Systems

•

Model human Expertise using IF .. THEN rules or
Production Rules.

•
Combines the rules (or
Knowledge Base
) with an
inference engine
to reason about the world.

•
Given certain observations, produces conclusions.

•
Relatively successful but limited.

5

2 Uncertainty

•
Rule based systems failed to handle uncertainty

•
Only dealt with true or false facts

•
Partly overcome using
Certainty factors

•
However, other problems: no differentiation
between causal rules and diagnostic rules.


6

3 Normative Expert Systems

•
Model Domain rather than Expert

•
Classical probability used rather than ad
-
hoc
calculus

•
Expert support rather than Expert Model

•
1980s
-

More Powerful Computers make complex
probability calculations feasible

•
Bayesian Networks introduced (Pearl 1986) e.g.
MUNIN.

7

Causality
-

1

Icy Roads

Icy Roads

Holmes Crashes

Watson Crashes

8

-

2
Wet Grass

Rain

Watson’s

Grass Wet

Holmes’

Grass Wet

Sprinkler

9

-

3
Earthquake or Burglar

Alarm

Mary Calls

John Calls

Burglary

Earthquake

10

Tour through Probability


•
All probabilities are between 0 and 1


•
Necessarily true propositions have probability=1
and necessarily false propositions have
probability=0


11

Conjunctions and Disjunctions

•
P(A & B) =
P(A) x P(B)



•
P(A v B) =
P(A) + P(B)

(mutually exclusive)


•
P(A v B)

=P(A)+P(B)
-

P(A & B)

(not mutually exclusive)


A

B

A

B

Venn Diagrams

A

B

12

Conditional probability & independence

•
Probability of B “given” A:




•
Independence:



P(B|A)=P(A&B)



P(A)

E.g. P(Hearts|Heart last time)

E.g. P(Heads|Even) = P(Heads)

P(B|A)=P(B)

13

Probability Distributions

•
Probability Distribution:





–
p(Weather=Sunny) = 0.5

–
p(Weather=Rain)= 0.2

–
p(Weather=Cloud)= 0.2

–
p(Weather=Snow)= 0.1


•
NB Distribution sums to 1.


0.5

0.2

0.1

S R C S

14

Joint Probability

•
Completely specifies all beliefs in a problem
domain.

•
Joint prob Distribution is an n
-
dimensional table
with a probability in each cell of that state
occurring.

•
Written as P(X
1
, X
2
, X
3

…, X
n
)





•
When instantiated as P(x
1
,x
2

…, x
n
)

15

Joint Distribution Example

•
Domain with 2
variables each of
which can take on 2
states.


Toothache

¬Toothache


Cavity

0.04


0.06

¬Cavity

0.01


0.89

P(Toothache, Cavity)

16

Bayes’ Theorem

Simple:



P(Y|X) = P(X|Y)P(Y)







P(X)

General:



P(Y|X,E) = P(X|Y,E)P(Y|E)





P(X|E)

17

Bayesian Probability

•
No need for
repeated Trials

•
Appear to follow
rules of Classical
Probability

•
How well do we
assign
probabilities?

The Probability Wheel:

A Tool for Assessing Probabilities

18

Bayesian Network
-

Definition


•
Causal Structure

•
Interconnected Nodes

•
Directed Acyclic Links

•
Joint Distribution formed
from conditional
distributions at each node.

19

Earthquake or Burglar

Alarm

Mary Calls

John Calls

Burglary

Earthquake

20

Bayesian Network for Alarm Domain

Alarm

Mary Calls

John Calls

P(B)

P(E)

.001

.002

B E P(A)

T T .95

T F .94

F T .29

F F .001

A P(J)

A P(M)

T .70

F .01

T .90

F .05

Burglary

Earthquake

21

Retrieving Probabilities from the
Conditional Distributions

•
P(x
1
,…x
n
) =
P

P(x
i
|Parents(x
i
))

•
E.g.



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 0.001 x 0.999 x 0.998


=

0.00062

i=1

n

22

Constructing A Network

-

Node Ordering and Compactness

•
Mary Calls

•
John Calls

•
Alarm

•
Burglary

•
Earthquake

Mary Calls

Alarm

Burglary

John Calls

Earthquake

23

Node Ordering and Compactness contd.

•
Mary Calls

•
Johns Calls

•
Earthquake

•
Burglary

•
Alarm


24

Node Ordering and Compactness contd.

•
Mary Calls

•
Johns Calls

•
Earthquake

•
Burglary

•
Alarm


Mary Calls

Earthquake

Burglary

John Calls

Alarm

25

Conditional Independence

revisited
-

D
-
Separation


•
To do inference in a Belief Network we have to
know if two sets of variables are conditionally
independent given a set of evidence.


•
Method to do this is called Direction
-
Dependent
Separation or D
-
Separation.

26

D
-
Separation contd.

•
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.


•
X is a set of variables with unknown values

•
Y is a set of variables with unknown values

•
E is a set of variables with known values.

27

D
-
Separation contd.

•
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:


1) Z is in E and Z has one arrow leading in and one leading
out.


2) Z is in E and has both arrows leading out.


3) Neither Z nor any descendant of Z is in E and both path
arrows lead in to Z.


28

Blocking

Z

Z

Z

X

Y

E

29

D
-
Separation
-

Example

•
Moves and Battery are
independent given it is
known about Ignition

•
Moves and Radio are
independent if it is known
that Battery works

•
Petrol and Radio are
independent given no
evidence. But are
dependent given evidence
of Starts

Radio

Ignition

Petrol

Starts

Moves

Battery

30

Inference

•
Diagnostic Inferences (effects to causes)

•
Causal Inferences (causes to effects)

•
Intercausal Inferences
-

or ‘Explaining
Away’ (between causes of common effect)

•
Mixed Inferences (combination of two or
more of the above)


31

Inference contd.

E

Q

Q

E

Q

E

E

Q

E


Diagnostic Causal Intercausal Mixed

32

Inference contd.

Burglary

Alarm

Mary Calls

Earthquake

John Calls

33

Inference in Singly Connected Networks

•
E.g. P(X|E):



Involves computing two values:

–
Causal Support (evidence variables above X
connected through it’s parents)

–
Evidential Support (evidence variables below X
connected through it’s children


•
Algorithm can perform in Linear Time.



34

Inference Algorithm

E

Q

Spreads out from Q to


evidence nodes, root

nodes and leaf nodes.


Each recursive call

excludes the node from

which it was called.

Causal Support

Evidential Support

E

E

35

Inference in Multiply Connected
Networks

•
Exact Inference is known to be NP
-
Hard

•
Approaches include:

–
Clustering

–
Conditioning

–
Stochastic Simulation

•
Stochastic Simulation is most often used,
particularly on large networks.


36

Clustering

Sprinkler & Rain

Wet Grass

C TT TF FT FF


T .08 .02 .72 .18

F .40 .10 .40 .10

P(S&R)

Sprinkler

Rain

Wet Grass

C P(R)


T .08 .02

F .40 .10

C P(S)


T .08 .02

F .40 .10

Cloudy

Cloudy

37

Conditioning

Cloudy

Sprinkler

Rain

Wet Grass

Cloudy

Cloudy

Sprinkler

Rain

Wet Grass

Cloudy

+

-

+

-

38

Stochastic Simulation
-

Example

A

B

C

D

E

P(A=1) = 0.2

A P(B=1)

0

0.2

1

0.8

A p(C=1)

0

0.05

1

0.2

39


Stochastic

Simulation

Run repeated simulations to estimate the probability
distribution


•
Let Wx = the states of all other variables except x.

•
Let the Markov Blanket of a node be all of its parents,
children and parents of children.

•
Distribution of each node, x, conditioned upon Wx can be
computed locally from their own probability with their
children’s :

•
P(a|Wa) =


.
P(a) . P(b|a) . P(c|a)

•
P(b|Wb) =


.
P(b|a) . P(d|b,c)

•
P(c|Wc) =


.
P(c|a) . P(d|b,c) . P(e|c)

•
Therefore, only the Markov blanket of a node is required
to compute the distribution

40

•
Set all observed nodes to their values

•
Set all other nodes to random values


•
STEP 1

•
Select a node randomly from the network

•
According to the states of the node’s markov blanket,
compute P(x=state, Wx) for all states

•
STEP 2

•
Use a random number generator that is biased according to
the distribution computed in step 1 to select the next value
of the node

•
Repeat

The Algorithm

41

Algorithm contd.

•
The final probability distribution of each unobserved node
is calculated from either:


1) the number of times each node took a particular state

2) the average conditional probability of each node taking a
particular state given the other variables states.

42

Case Study
-

Pathfinder

•
Diagnostic Expert System for Lymph
-
Node Diseases



4 Versions of Pathfinder :

1) Rule Based

2) Experimented with Certainty Factors/Dempster
-
Shafer
theory/Bayesian Models

3) Refined Probabilities

4) Refined dependencies

43

Section 2
-

Research Issues in
Uncertainty


•
Assume no
Knowledge of
Probabilities
Distributions or
Causal Structure.

•
Is it possible to
infer both of these
from data?


Case Fraud Gas Jewellery Age Sex


1 No No No 30
-
50 F


2 No No No 30
-
50 M


3 Yes Yes Yes >50 M


4 No No No 30
-
50 M


5 No Yes No <30 F


6 No No No <30 F


7 No No No >50 M


8 No No Yes 30
-
50 F


9 No Yes No <30 M


10 No No No <30 F

1 Learning Belief Networks from Data

44

Some Methods

•
Bayesian (Cooper & Herskovitz 1991)

•
Minimum Description Length (Lam &
Bachus 1994)

•
Bound and Collapse (Ramoni 1996)

Fraud

Gas

Jewelry

Age

Sex

45

2 Dynamics
-

Markov Models

State t
-
2

State t
-
1

State t

State t+1

State t+2

Percept t
-
2

Percept t
-
1

Percept t

Percept t+1

Percept t+2

Sensor Model

State Transition Model

46

Updating over time

State t

Percept t

State t

State t+1

Percept t

Percept t+1

State t
-
1

State t

Percept t
-
1

Percept t

47


Dynamic Belief Networks
-


Forecasting Car sales



Demand

Health

Supply

Price

Demand

Health

Supply

Price

t

t
-
1

48

3 Other approaches to modeling
Uncertainty


•
Default Reasoning


•
Dempster
-

Shafer
Theory


•
Fuzzy Logic