Bayesian%20Networks

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Uncertainty Bayesian Belief Networks
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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

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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

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Expert Systems1 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.

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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.

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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.

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Causality - 1 Icy Roads
Icy Roads
Holmes Crashes
Watson Crashes
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- 2 Wet Grass
Rain
Sprinkler
Watsons Grass Wet
Holmes Grass Wet
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- 3 Earthquake or Burglar
Burglary
Earthquake
Alarm
Mary Calls
John Calls
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Tour through Probability

• All probabilities are between 0 and 1
• Necessarily true propositions have probability1
  and necessarily false propositions have
  probability0

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Conjunctions and Disjunctions
Venn Diagrams

• 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
A
B
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Conditional probability independence

• Probability of B given A
• Independence

P(BA)P(AB) P(A)
E.g. P(HeartsHeart last time)
P(BA)P(B)
E.g. P(HeadsEven) P(Heads)
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Probability Distributions

• Probability Distribution
• p(WeatherSunny) 0.5
• p(WeatherRain) 0.2
• p(WeatherCloud) 0.2
• p(WeatherSnow) 0.1
• NB Distribution sums to 1.

0.5
0.2
0.1
S R C S
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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(X1, X2, X3 , Xn)
• When instantiated as P(x1,x2 , xn)

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Joint Distribution Example

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

P(Toothache, Cavity)
Toothache Toothache
Cavity 0.04 0.06
Cavity 0.01 0.89
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Bayes Theorem

• Simple
• P(YX) P(XY)P(Y)
  P(X)

• General
• P(YX,E) P(XY,E)P(YE)
  P(XE)

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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
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Bayesian Network - Definition

• Causal Structure
• Interconnected Nodes
• Directed Acyclic Links
• Joint Distribution formed from conditional
  distributions at each node.

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Earthquake or Burglar
Burglary
Earthquake
Alarm
Mary Calls
John Calls
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Bayesian Network for Alarm Domain
Burglary
Earthquake
P(B)
P(E)
.001
.002
B E P(A)
T T .95
T F .94
Alarm
F T .29
F F .001
A P(J)
A P(M)
T .70
T .90
F .01
F .05
Mary Calls
John Calls
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Retrieving Probabilities from the Conditional
Distributions
n

• P(x1,xn) P P(xiParents(xi))
• E.g.
• P(J M A B E)
• P(JA)P(MA)P(AB,E)P(B)P(E)
• 0.9 x 0.7 x 0.001 x 0.999 x 0.998
• 0.00062

i1
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Constructing A Network- Node Ordering and
Compactness

• Mary Calls
• John Calls
• Alarm
• Burglary
• Earthquake

Mary Calls
John Calls
Alarm
Burglary
Earthquake
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Node Ordering and Compactness contd.

• Mary Calls
• Johns Calls
• Earthquake
• Burglary
• Alarm

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Node Ordering and Compactness contd.

• Mary Calls
• Johns Calls
• Earthquake
• Burglary
• Alarm

Mary Calls
John Calls
Earthquake
Burglary
Alarm
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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.

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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.

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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.

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Blocking
X
Y
E
Z
Z
Z
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D-Separation - Example
Battery

• 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
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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)

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Inference contd.
Q
E
Q
E
E
Q
E
Q
E
Diagnostic Causal Intercausal
Mixed
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Inference contd.
Burglary
Earthquake
Alarm
Mary Calls
John Calls
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Inference in Singly Connected Networks

• E.g. P(XE)
• Involves computing two values
• Causal Support (evidence variables above X
  connected through its parents)
• Evidential Support (evidence variables below X
  connected through its children
• Algorithm can perform in Linear Time.

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Inference Algorithm
E
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
Q
E
E
Evidential Support
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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.

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Clustering
Cloudy
Cloudy
Sprinkler Rain
Sprinkler
Rain
P(SR)
C P(R) T .08 .02 F .40 .10
C TT TF FT FF T .08 .02 .72
.18 F .40 .10 .40 .10
C P(S) T .08 .02 F .40 .10
Wet Grass
Wet Grass
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Conditioning
-
-


Cloudy
Cloudy
Cloudy
Cloudy
Sprinkler
Rain
Sprinkler
Rain
Wet Grass
Wet Grass
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Stochastic Simulation - Example
P(A1) 0.2
A
A P(B1) 0 0.2 1 0.8
A p(C1) 0 0.05 1 0.2
B
C
E
D
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Stochastic SimulationRun 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 childrens
• P(aWa) ? . P(a) . P(ba) . P(ca)
• P(bWb) ? . P(ba) . P(db,c)
• P(cWc) ? . P(ca) . P(db,c) . P(ec)
• Therefore, only the Markov blanket of a node is
  required to compute the distribution

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The Algorithm

• 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 nodes markov
  blanket, compute P(xstate, 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

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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.

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Case Study - Pathfinder

• Diagnostic Expert System for Lymph-Node Diseases
• 4 Versions of Pathfinder
• 1) Rule Based
• 2) Experimented with Certainty Factors/Dempster-Sh
  afer theory/Bayesian Models
• 3) Refined Probabilities
• 4) Refined dependencies

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Section 2 - Research Issues in Uncertainty
1 Learning Belief Networks from Data

• 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 gt50 M 4 No
No No 30-50 M 5
No Yes No lt30 F
6 No No No lt30
F 7 No No No
gt50 M 8 No No
Yes 30-50 F 9 No Yes
No lt30 M 10 No
No No lt30 F
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Some Methods

• Bayesian (Cooper Herskovitz 1991)
• Minimum Description Length (Lam Bachus 1994)
• Bound and Collapse (Ramoni 1996)

Fraud
Age
Sex
Gas
Jewelry
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2 Dynamics - Markov Models
State Transition Model
State t-2
State t-1
State t
State t1
State t2
Percept t-2
Percept t-1
Percept t
Percept t1
Percept t2
Sensor Model
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Updating over time
State t-1
State t
Percept t-1
Percept t
State t
Percept t
State t
State t1
Percept t
Percept t1
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Dynamic Belief Networks - Forecasting Car sales
t
t-1
Price
Price
Demand
Health
Demand
Health
Supply
Supply
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3 Other approaches to modeling Uncertainty

• Default Reasoning
• Dempster - Shafer Theory
• Fuzzy Logic

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