Bayesian Statistics and Belief Networks

Overview

• Book: Ch 13,14

• Refresher on Probability

• 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?

Uncertainty

Methods for Handling Uncertainty

Probability

Making Decisions Under Uncertainty

Probability Basics

Random Variables

Prior Probability

Conditional Probability

Inference by Enumeration

Inference by Enumeration

Bayes Rule

Product Rule:

P A B P A B P B

P A B P B A P A

|

|

Equating Sides: P B A

P A B P B

P A|

( | ) ( )

( )

P Class evidenceP evidence Class P Class

P evidence|

( | ) ( )

( )i.e.

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

Inference by Enumeration

Simple Bayes Rule ExampleProbability 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)=0.8P(V)=1/1000P(C)=1/10

P V CP C V P V

P C( | )

( | ) ( )

( )

( . )( . )

( . ).

08 0 001

010 008

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

P Class evidenceP evidence Class P Class

P evidence|

( | ) ( )

( )

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:

E e e en( , ,..., )1 2

Usual simplifying assumption is conditional independence:

P e C P e Cii

n

( | ) ( | )

1

P C e

P C P e C

P e

ii

n

( | )( ) ( | )

( )

1

Bayesian Classifier ExampleProbability C=Virus C=Bad DiskP(C) 0.4 0.6P(crashes|C) 0.1 0.2P(diskfull|C) 0.6 0.1

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

P e c P e c P e c2 1 11( | ) ( | )[ ( | )] Correction factor - Difficult to compute -

n

2

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

Burglary Alarm Example

Burglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A)T T 0.95T F 0.94F T 0.29F F 0.001

A P(J)T 0.90F 0.05

A P(M)T 0.70F 0.01

Sample Bayesian Network

Using The Belief NetworkBurglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A)T T 0.95T F 0.94F T 0.29F F 0.001

A P(J)T 0.90F 0.05

A P(M)T 0.70F 0.01

P x x x P x Parents Xn i ii

n

( , ,... ) ( | ( ))1 21

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

P J A P M A P A B E P B P E( | ) ( | ) ( | ) ( ) ( ) ( . )( . )( . )( . )( . ) .0 9 0 7 0 001 0 999 0 998 0 00062

Belief Computations• Two types; both are NP-Hard• Belief Revision

– Model explanatory/diagnostic tasks– 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:

049.0)99.0)(05.0)(999.0)(998.0)(999.0(

)|()|()|()()(

AMPAJPEBAPEPBP

Belief Updating

• Causal Inferences

• Diagnostic Inferences

• Intercausal Inferences

• Mixed Inferences

Q E

Q

E

E EQ

E Q

Causal InferencesInference from cause to effect.

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

85.0

)05.0)(06.0()9.0)(94.0()|(

)05.0)(()9.0)(()|(

94.0)|(

)95.0)(002.0(1)94.0)(998.0(1)|(

)95.0)(()()94.0)(()()|(

?)|(

BJP

APAPBJP

BAP

BAP

EPBPEPBPBAP

BJP

P(M|B)=0.67 via similar calculations

Burglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A)T T 0.95T F 0.94F T 0.29F F 0.001

A P(J)T 0.90F 0.05

A P(M)T 0.70F 0.01

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

)(

)()|()|(

JP

BPBJPJBP

002517.0)(

)001.0)(999.0)(998.0()94.0)(998.0)(001.0(

)29.0)(002.0)(999.0()95.0)(002.0)(001.0()(

)001.0)(()()94.0)(()(

)29.0)(()()95.0)(()()(

AP

AP

EPBPEPBP

EPBPEPBPAP

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

052.0)(

)05.0)(9975.0()9.0)(002517.0()(

)05.0)(()9.0)(()(

JP

JP

APAPJP 016.0)052.0(

)001.0)(85.0()|( JBP

Many false positives.

Intercausal InferencesExplaining 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:

017.0)^|(

03.0)^|(

EJBP

EJAP

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)

...

...

xE

xE

zj jyzijijjiiy

i yix

u ixuiix

xx

iEzPzXyPyEPXEP

EUPuXPEXP

XEPEXPEXP

)|(),|()|()|(

)|()|()|(

)|()|()|(

\

\

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