Bayesian Belief Networks Compound Bayesian Decision Theory

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7 Νοε 2013 (πριν από 4 χρόνια και 3 μέρες)

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Srihari: CSE 555
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Bayesian Belief Networks
Compound Bayesian Decision
Theory
Srihari: CSE 555
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Bayesian Belief Networks

In certain situations statistical properties are
not directly expressed by a parameter vector
but by causal relationships among variables
Srihari: CSE 555
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Statistically dependent and independent variables
Three-dimensional distribution which obeys p(x1, x3) = p(x1) p(x3)
Thus x1
and
x3
are statistically independent but the other feature pairs
are not
x1
x2
x3
Srihari: CSE 555
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Causal relationships

State of automobile

Temperature of engine

Pressure of brake fluid

Pressure of air in tires

Voltages in the wires

Oil pressure and air pressure are not causally
related

Engine temperature and oil temperature are
Srihari: CSE 555
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Parent-Child Relationship
Node X
has variable values (x1,x
2,….)
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Bayesian Belief Net or
Causal Network or
Belief
Net
Node A
has states {a1, a2,…} = a
Node B
has states {b1, b2,…}= b










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Conditional Probability Table
Rows sum to one
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A simple belief net
Srihari: CSE 555
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Determining a joint probability
P(a3 , b1 , x2 , c3 , d2) = P(a3) P(b1) P(x2|a3,b
1) P(c3|x2) P(d2,x2)
= 0.25 x
0.6 x
0.4 x
0.5 x 0.4
= 0.012
Only X
has 2 parents thus only the P(x
2|..) has two conditioning variables
Srihari: CSE 555
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Determining Probability of variables in a Bayes
Belief Net
Linear Chain Belief Net
To compute
proceed as above
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Determining probabilities in a net with a loop
Computing the probabilities of variables at H
in the network
Belief net with a simple loop
Differs somewhat
from linear network
because of loop
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Evidence
Given the values of some variables (evidence) determine some
configuration of other variables
Determine fish came from North Atlantic, given it is springtime
and fish is a light salmon, or P(b1|a2,x1,c1)
Query variable
Evidence
b1=North Atlantic
a2
= Spring
c1=light
x1=salmon
d = thickness unknown
Srihari: CSE 555
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Example of Evaluation (Classification)
What is the classification
when fish is light (c1),
caught in South Atlantic (b2)
and do not know time of year and thickness?
Srihari: CSE 555
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Evaluation Steps
Similarly,
P(x2|c1,b2) =
α
0.066
Normalizing,
P(x1|c1,b2)=0.63
P(x2|c1,b2)=0.37
Given the evidence
classify as a salmon
Note
Srihari: CSE 555
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Naïve Bayes
Rule
When dependency relationships among the features used by
a classfier
are unknown, assume features are conditionally independent
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Medical Diagnosis Application of
Belief Nets

Uppermost nodes (without parents)

Biological agent such as presence of bacteria or
virus

Intermediate nodes

Diseases such as emphysema or flu

Lowermost nodes

Symptoms such as high temperature or coughing

Physician enters measured values in net and
finds most likely disease or cause
Srihari: CSE 555
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Compound Bayesian Decision
Theory and Context