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CSCE 580
Artificial Intelligence
Ch.6 [P]: Reasoning Under Uncertainty
Sections 6.2 and 6.3: Independence and
Belief (Bayesian) Networks
Fall 2009
Marco Valtorta
[email protected]
It is remarkable that a science which began with the consideration of
games of chance should become the most important object of human
knowledge... The most important questions of life are, for the most
part, really only problems of probability. . . The theory of probabilities
is at bottom nothing but common sense reduced to calculus.
Probability does not exist.
--Pierre Simon de Laplace, 1812
--Bruno de Finetti, 1970
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Acknowledgment
• The slides are based on the textbook [P] and other sources,
including other fine textbooks
– [AIMA-2]
– David Poole, Alan Mackworth, and Randy Goebel.
Computational Intelligence: A Logical Approach. Oxford,
1998
• A second edition (by Poole and Mackworth) is under development.
Dr. Poole allowed us to use a draft of it in this course
– Ivan Bratko. Prolog Programming for Artificial Intelligence,
Third Edition. Addison-Wesley, 2001
• The fourth edition is under development
– George F. Luger. Artificial Intelligence: Structures and
Strategies for Complex Problem Solving, Sixth Edition.
Addison-Welsey, 2009
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Conditional Independence
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Conditional Independence Is Symmetric
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Example domain (diagnostic assistant)
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Department of Computer Science and Engineering
Examples of conditional independence
• The identity of the queen of Canada is independent of
whether light l1 is lit given whether there is outside power.
• Whether there is someone in a room is independent of
whether a light l2 is lit given the position of switch s3.
• Whether light l1 is lit is independent of the position of light
switch s2 given whether there is power in wire w0.
• Every other variable may be independent of whether light
l1 is lit given whether there is power in wire w0 and the
status of light l1 (if it's ok, or if not, how it's broken).
• Conditional independence is defined using numbers, but it
can often be established by qualitative arguments.
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Department of Computer Science and Engineering
Idea of belief (Bayesian) networks
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Bayesian networks
• The method of strata for constructing a Bayesian network is given above
• Usually, a Bayesian network is defined to also include probabilities, as in
the following slide.
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Components of a Bayesian network
• A belief network consists of:
– a directed acyclic graph with nodes labeled
with random variables
– a domain for each random variable
– a set of conditional probability tables for each
variable given its parents (including prior
probabilities for nodes with no parents).
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Example BN (6.10)
•
Suppose we want to use the
diagnostic assistant to
diagnose whether there is a
fire in a building based on
noisy sensor information and
possibly conflicting
explanations of what could be
going on. The agent receives a
report about whether
everyone is leaving the
building. Suppose the report
sensor is noisy: It sometimes
reports leaving when there is
no exodus, a false positive,
and sometimes does not report
when everyone is leaving, a
false negative. Suppose the
fire alarm going off can cause
the leaving, but this is not
deterministic relationship.
Either tampering or fire could
affect the alarm. Fire also
causes smoke to rise from the
building.
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Example BN (6.11)
Let’s select an ordering where the causes of a
variable are before the variable in the ordering.
For example, the variable for whether a light is
lit comes after variables for whether the light is
working and whether there is power coming into
the light.
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Department of Computer Science and Engineering
Independence Assumptions in BNs
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Example for D-Separation
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D-separation: converging connections
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D-Separation: Diverging Connections
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D-Separation: Serial Connections
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Department of Computer Science and Engineering
Using BNs: Diagnostic Assistant
The power network can be used by the diagnostic assistant in
a number of ways:
• Conditioning on the status of the switches and circuit
breakers, whether there is outside power and the position
of the switches, you can simulate the lighting.
• Given values for the switches, the outside power, and
whether the lights are lit, you can determine the posterior
probability that each switch or circuit breaker is ok or not.
• Given some switch positions and some outputs and some
intermediate values, you can determine the probability of
any other variable in the network.
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Department of Computer Science and Engineering
Updating Probabilities by Conditioning
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Department of Computer Science and Engineering
Some features and properties of BNs
• A belief network is automatically acyclic by
construction.
• A belief network is a directed acyclic graph (DAG)
where nodes are random variables.
• The parents of a node n are those variables on
which n directly depends.
• A belief network is a graphical representation of
dependence and independence:
– A variable is independent of its nondescendants given its parents.
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Constructing BNs
To represent a domain in a belief network, you need to
consider:
• What are the relevant variables?
– What will you observe?
– What would you like to find out (query)?
– What other features make the model simpler?
• What values should these variables take?
• What is the relationship between them? This should be
expressed in terms of local influence.
• How does the value of each variable depend on its
parents? This is expressed in terms of the conditional
probabilities.
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Example 5.30 and 6.14
The process of diagnosis is carried out by
conditioning on the observed symptoms and
deriving posterior probabilities of the faults or
diseases.
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Help System Example
A Naïve Bayesian Classifier:
P(H|F1,F2,…,Fn) = K x P(H) x P(F1,F2,…,Fn) =
= K x P(H) x P(F1|H) x P(F2|H) x … x P(Fn | H)
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering