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Bayesian Statistics and Belief Networks

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Bayesian Statistics and Belief Networks

- Book: Ch 8.3
- Refresher on Bayesian statistics
- Bayesian classifiers
- Belief Networks / Bayesian Networks

- 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

- 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?

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).

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).

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:

ProbabilityC=VirusC=Bad Disk

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).

- 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

- 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

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:

- 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?

- 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:

- Causal Inferences
- Diagnostic Inferences
- Intercausal Inferences
- Mixed Inferences

E

Q

Q

E

Q

E

E

Q

E

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

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.

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.

Simultaneous intercausal and diagnostic inference.

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

Computing these values exactly is somewhat complicated.

- 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

...

U(1)

U(m)

X

Z(1,j)

Z(n,j)

...

Y(1)

Y(n)

Algorithm recursive, message

passing chain

- 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

- Clustering
- 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

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

- 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

- 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