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# Bayesian Statistics and Belief Networks PowerPoint PPT Presentation

Bayesian Statistics and Belief Networks. Overview. Book: Ch 8.3 Refresher on Bayesian statistics Bayesian classifiers Belief Networks / Bayesian Networks. Why Should We Care?. Theoretical framework for machine learning, classification, knowledge representation, analysis

Bayesian Statistics and Belief Networks

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## Bayesian Statistics and Belief Networks

### Overview

• Book: Ch 8.3

• Refresher on Bayesian statistics

• Bayesian classifiers

• Belief Networks / Bayesian Networks

### Why Should We Care?

• Theoretical framework for machine learning, classification, knowledge representation, analysis

• Bayesian methods are capable of handling noisy, incomplete data sets

• Bayesian methods are commonly in use today

### Bayesian Approach To Probability and Statistics

• Classical Probability : Physical property of the world (e.g., 50% flip of a fair coin). True probability.

• Bayesian Probability : A person’s degree of belief in event X. Personal probability.

• Unlike classical probability, Bayesian probabilities benefit from but do not require repeated trials - only focus on next event; e.g. probability Seawolves win next game?

### Bayes Rule

Product Rule:

Equating Sides:

i.e.

All classification methods can be seen as estimates of Bayes’ Rule, with different techniques to estimate P(evidence|Class).

### Simple Bayes Rule Example

Probability your computer has a virus, V, = 1/1000.

If virused, the probability of a crash that day, C, = 4/5.

Probability your computer crashes in one day, C, = 1/10.

P(C|V)=0.8

P(V)=1/1000

P(C)=1/10

Even though a crash is a strong indicator of a virus, we expect only

8/1000 crashes to be caused by viruses.

Why not compute P(V|C) from direct evidence? Causal vs.

Diagnostic knowledge; (consider if P(C) suddenly drops).

### Bayesian Classifiers

If we’re selecting the single most likely class, we only

need to find the class that maximizes P(e|Class)P(Class).

Hard part is estimating P(e|Class).

Evidence e typically consists of a set of observations:

Usual simplifying assumption is conditional independence:

### Bayesian Classifier Example

P(C)0.40.6

P(crashes|C)0.10.2

P(diskfull|C)0.60.1

Given a case where the disk is full and computer crashes,

the classifier chooses Virus as most likely since

(0.4)(0.1)(0.6) > (0.6)(0.2)(0.1).

### Beyond Conditional Independence

• Include second-order dependencies; i.e. pairwise combination of variables via joint probabilities:

Linear Classifier:

C1

C2

Correction factor -

Difficult to compute -

joint probabilities to consider

### Belief Networks

• DAG that represents the dependencies between variables and specifies the joint probability distribution

• Random variables make up the nodes

• Directed links represent causal direct influences

• Each node has a conditional probability table quantifying the effects from the parents

• No directed cycles

P(B)

P(E)

Burglary

Earthquake

0.001

0.002

B EP(A)

T T0.95

Alarm

T F0.94

F T0.29

F F0.001

AP(J)

AP(M)

John Calls

Mary Calls

T0.70

T0.90

F0.01

F0.05

### Using The Belief Network

P(B)

P(E)

Burglary

Earthquake

0.002

0.001

B EP(A)

T T0.95

Alarm

T F0.94

F T0.29

F F0.001

AP(M)

JohnCalls

Mary Calls

T0.70

AP(J)

F0.01

T0.90

F0.05

Probability of alarm, no burglary or earthquake, both John and Mary call:

### Belief Computations

• Two types; both are NP-Hard

• Belief Revision

• Given evidence, what is the most likely hypothesis to explain the evidence?

• Also called abductive reasoning

• Belief Updating

• Queries

• Given evidence, what is the probability of some other random variable occurring?

### Belief Revision

• Given some evidence variables, find the state of all other variables that maximize the probability.

• E.g.: We know John Calls, but not Mary. What is the most likely state? Only consider assignments where J=T and M=F, and maximize. Best:

### Belief Updating

• Causal Inferences

• Diagnostic Inferences

• Intercausal Inferences

• Mixed Inferences

E

Q

Q

E

Q

E

E

Q

E

### Causal Inferences

P(B)

P(E)

Burglary

Earthquake

Inference from cause to effect.

E.g. Given a burglary, what is P(J|B)?

0.002

0.001

B EP(A)

T T0.95

Alarm

T F0.94

F T0.29

F F0.001

AP(M)

JohnCalls

Mary Calls

T0.70

AP(J)

F0.01

T0.90

F0.05

P(M|B)=0.67 via similar calculations

### Diagnostic Inferences

From effect to cause. E.g. Given that John calls, what is the P(burglary)?

What is P(J)? Need P(A) first:

Many false positives.

### Intercausal Inferences

Explaining Away Inferences.

Given an alarm, P(B|A)=0.37. But if we add the evidence that

earthquake is true, then P(B|A^E)=0.003.

Even though B and E are independent, the presence of

one may make the other more/less likely.

### Mixed Inferences

Simultaneous intercausal and diagnostic inference.

E.g., if John calls and Earthquake is false:

Computing these values exactly is somewhat complicated.

### Exact Computation - Polytree Algorithm

• Judea Pearl, 1982

• Only works on singly-connected networks - at most one undirected path between any two nodes.

• Backward-chaining Message-passing algorithm for computing posterior probabilities for query node X

• Compute causal support for X, evidence variables “above” X

• Compute evidential support for X, evidence variables “below” X

### Polytree Computation

...

U(1)

U(m)

X

Z(1,j)

Z(n,j)

...

Y(1)

Y(n)

Algorithm recursive, message

passing chain

### Other Query Methods

• Exact Algorithms

• Clustering

• Cluster nodes to make single cluster, message-pass along that cluster

• Symbolic Probabilistic Inference

• Uses d-separation to find expressions to combine

• Approximate Algorithms

• Select sampling distribution, conduct trials sampling from root to evidence nodes, accumulating weight for each node. Still tractable for dense networks.

• Forward Simulation

• Stochastic Simulation

### Summary

• Bayesian methods provide sound theory and framework for implementation of classifiers

• Bayesian networks a natural way to represent conditional independence information. Qualitative info in links, quantitative in tables.

• NP-complete or NP-hard to compute exact values; typical to make simplifying assumptions or approximate methods.

• Many Bayesian tools and systems exist

### References

• Russel, S. and Norvig, P. (1995). Artificial Intelligence, A Modern Approach. Prentice Hall.

• Weiss, S. and Kulikowski, C. (1991). Computer Systems That Learn. Morgan Kaufman.

• Heckerman, D. (1996). A Tutorial on Learning with Bayesian Networks. Microsoft Technical Report MSR-TR-95-06.

• Internet Resources on Bayesian Networks and Machine Learning: http://www.cs.orst.edu/~wangxi/resource.html