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

Watson’s Grass Wet

Holmes’Grass Wet

Sprinkler

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- 3 Earthquake or Burglar

Alarm

Mary Calls John Calls

Burglary Earthquake

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Tour through Probability

• All probabilities are between 0 and 1

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

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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

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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)

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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

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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.

Toothache ¬Toothache

Cavity 0.04 0.06

¬Cavity 0.01 0.89

P(Toothache, Cavity)

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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)

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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

Alarm

Mary Calls John Calls

Burglary Earthquake

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Bayesian Network for Alarm Domain

Alarm

Mary Calls John Calls

P(B) P(E).001 .002

B E P(A)T T .95T F .94F T .29F F .001

A P(J)A P(M)T .70F .01

T .90F .05

Burglary Earthquake

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Retrieving Probabilities from the Conditional Distributions

• P(x1,…xn) = P(xi|Parents(xi))

• 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

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Constructing A Network

- Node Ordering and Compactness

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

Mary Calls

Alarm

Burglary

John Calls

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

Earthquake

Burglary

John Calls

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

Z

Z

Z

X YE

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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

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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.

E

Q

Q

E Q E

E

Q

E

Diagnostic Causal Intercausal Mixed

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Inference contd.

Burglary

Alarm

Mary Calls

Earthquake

John Calls

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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.

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Inference Algorithm

E

Q

Spreads out from Q to evidence nodes, root nodes and leaf nodes.

Each recursive call excludes the node fromwhich it was called.

Causal Support

Evidential SupportE

E

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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

Sprinkler & Rain

Wet Grass

C TT TF FT FF

T .08 .02 .72 .18F .40 .10 .40 .10

P(S&R)

Sprinkler Rain

Wet Grass

C P(R)

T .08 .02F .40 .10

C P(S)

T .08 .02F .40 .10

Cloudy Cloudy

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ConditioningCloudy

Sprinkler Rain

Wet Grass

Cloudy Cloudy

Sprinkler Rain

Wet Grass

Cloudy+-

+-

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Stochastic Simulation - Example

A

B C

D E

P(A=1) = 0.2

A P(B=1)0 0.21 0.8

A p(C=1)0 0.051 0.2

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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 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

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• 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

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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-Shafer theory/Bayesian Models

3) Refined Probabilities

4) Refined dependencies

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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

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Some Methods

• Bayesian (Cooper & Herskovitz 1991)

• Minimum Description Length (Lam & Bachus 1994)

• Bound and Collapse (Ramoni 1996)

Fraud

Gas Jewelry

Age Sex

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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

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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

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Dynamic Belief Networks - Forecasting Car sales

Demand Health

Supply

Price

Demand Health

Supply

Price

tt-1

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3 Other approaches to modeling Uncertainty

• Default Reasoning

• Dempster - Shafer Theory

• Fuzzy Logic ?