Chapter 15: Probabilistic Reasoning Systems

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Chapter 15 Probabilistic Reasoning Systems

• Overview
• Belief networks
• Reasoning about independent events
• Inferences in belief networks
• Other approaches to uncertain reasoning
• Fuzzy sets

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Belief Networks - 1

• Structure
• Directed acyclic graph (DAG)
• Nodes
• Random variables
• Directed link
• From random variable (node) X to random variable
  (node) Y
• Variable X has direct influence on variable Y
• Give conditional relations
• Conditional probabilities
• based on parents for node

Burglary
Earthquake
Alarm
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Belief Network Example

• Events
• Burglary House is robbed
• Earthquake
• Alarm Sounds in house
• Might be because of burglary
• Might be due to earthquake
• JohnCalls
• Neighbor John calls you at work
• Might be because he hears alarm, or because he
  hears telephone ringing
• MaryCalls
• Neighbor Mary calls you at work
• Calls when she hears alarm, but misses hearing
  alarm sometime

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Network Construction

• Add root causes first
• Then add the variables they influence
• Etc until we reach the leaves (symptoms or
  effects)
• No direct causal influence on other variables
• Or we are stopping our analysis at that point
• Temporal ordering from root causes to effects
• Causality works forward in time usually

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Example p. 468, 15.2(a)

• Nuclear power station
• Alarm checks gauge temperature sensor
• exceed threshold?
• Gauge is a sensor that measures core temperature
• Categorical events
• Boolean valued
• A alarm sounds
• Fa alarm is faulty
• Fg gauge is faulty
• Multivalued
• G gauge reading
• T actual core temperature
• Draw a belief network for this domain, given that
  the gauge is more likely to fail when the core
  temperature is too high

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Power plant belief network
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Inference (Probability Computation)In Belief
Networks

• Problem
• Given
• Query variables
• Evidence variables
• Compute
• posterior (conditional) probability
• P(Query Evidence) ?
• E.g.,
• Burglary query variable
• JohnCalls evidence variable
• P(Burglary JohnCalls) ?

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Kinds of Inference UsingBelief Networks

• Causal
• Using a cause to infer an effect
• From top to bottom of network
• E.g., P(JohnCalls Burglary)
• Diagnostic
• Using an effect to infer a cause
• From bottom of network to top
• E.g., P(Burglary JohnCalls)
• Convert to causal P(B J) P(J B) P(B) /
  P(J)
• Intercausal
• Between causes of a common effect
• P(Burglary Alarm ? Earthquake)
• Mixed
• Combinations of these

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Causal Inference Method

• Show this
• diagnostic inference can be converted to causal
  inference by Bayes rule
• end up with two equations with one unknown solve

Causal inference P(C A) ? P(C A) P(C ??
B A) P(C ?? ?B A) P(C A) P(C A ? B)
P(B A) P(C ?B ? A) P(?B A) Given
independent causes P(BA) P(B) and P(?B A)
P(?B) P(C A) P(C A ? B) P(B) P(C ?B ?
A) P(?B)
causes
A
B
C
effects
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Why does P(C ?? B A) P(C A ? B) P(B A)?

• Need Joint Probability Equation (p. 440 in text)
• P(A?B?C??Z) P(AB?C??Z) P(BC??) P(Z)
• 1) P(C ?? B A) P(C ? B ? A) / P(A) Bayes
  rule
• 2) P(C ? B ? A) P(C B ? A) P(BA) P(A) JPE
• 3) P(C ?? B A) P(C B ? A) P(BA) P(A) /
  P(A)
• substituted 2 into 1
• 4) P(C ?? B A) P(C B ? A) P(BA)
  cancellation

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Larger Networks

• Compute internal node, C, probability
• With equation just developed
• Use as P(C) below
• Compute lower (effect) probability
• P(E) P(E ? C ) P(E ? ?C )
• P(E C) P(C ) P(E ?C ) P(?C )
• Use have P(C ) and from the Belief net we P(E
  C) and P(E ?C)

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Example

• Given our two equations
• P(C A) P(C A ? B) P(B) P(C ?B ? A)
  P(?B)
• P(E) P(E C) P(C ) P(E ?C ) P(?C )
• And our belief network for the burglary scenario,
  compute the causal probability P(JohnCalls
  Burglary)

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Solution

• Let A Alarm, B Burglary, J JohnCalls, E
  Earthquake
• Calculate causal probability P(J B)
• P(AB) P(AB?E)P(E) P(AB ??E)P(?E)
• .95 .002 .95 .998
• .0019 .9481 0.95
• P(J) P(JA) P(A) P(J ?A) P(?A)
• .9 .95 0.05(1 0.95)
• .855 .0025 0.8575
• So, P(J B) 0.8575 (given independence
  assumptions)

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Diagnostic Reasoning

• Want to know P(Top-event Bottom-event)
• Reasoning using an effect to infer probability of
  a cause
• Use Bayes rule to convert to causal reasoning
• Use causal reasoning method just described
• E.g., P(Burglary JohnCalls)
• P(B J) P(J B) P(B) / P (J)
• Also compute
• P(?B J) P(J ? B) P(? B) / P (J)
• Get P(J B) and P(J ? B) from causal
  reasoning
• Have prior probabilities P(B), P(? B)
• Get P(J) from P(B J) P(?B J) 1
• P(J B) P(B) / P (J) P(J ? B) P(? B) / P
  (J) 1
• P(J) P(J B) P(B) P(J ? B) P(? B)

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Example Diagnostic Reasoning

• Given our two causal reasoning equations
• P(C A) P(C A ? B) P(B) P(C ?B ? A)
  P(?B)
• P(E) P(E C) P(C ) P(E ?C ) P(?C )
• And given our diagnostic reasoning equation
• P(J B) P(B)
• P(B J) ____________________
• P(J B) P(B) P(J ? B) P(? B)
• And our belief network for the burglary scenario,
  compute the diagnostic probability
• P(Burglary JohnCalls)

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Solution

• Calculate other causal probability P(J ?B)
• P(A?B) P(A?B?E)P(E) P(A?B ??E)P(?E)
• .29 .002 .001 (1 - .002)
• .00058 .000998 0.001578
• P(J) P(JA) P(A) P(J ?A) P(?A)
• .9 .001578 0.05(1 0. 001578 )
• .0014202 .0499211 0.0513413
• P(J B) P(B)
• P(B J) ____________________
• P(J B) P(B) P(J ? B) P(? B)
• 0.8575 .001 /(0.8575 .001 0.0513413
  .999)
• 0.0008575 / (0.0008575 0.051289958)
• 0.0008575 / .052147458
• 0.01644 (given independence assumptions)

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Comparision of Probabilitistic vs. Logical
Reasoning
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Fuzzy Sets

• Systems of functions giving possibilities
• Fuzzy set membership
• Math sets an element is either a member or not a
  member of a set
• Fuzzy sets an element can be a partial member of
  multiple sets
• Values of functions between 0 and 1
• F(x) y where F is the fuzzy function y is the
  possiblity value
• Standard math sets
• crisp sets yes/no answers for membership
• Useful for representing vagueness
• E.g., in English language terms such as tall or
  small

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Fuzzy Set Membership

• In the male-height example (Luger Stubblefield)
• Say Fred 4 10 for a particular man, Fred
• He is a member of two fuzzy sets
• Small and Medium
• Sets(Fuzzy-Variable) set of all fuzzy sets to
  which the Fuzzy-Variable has membership
• (This is a crisp set).
• Let X 4 10
• Sets(X) Small, Medium
• X.Fuzzy-Set possibility value of X for the
  given fuzzy set defined if Fuzzy-Set ? Sets(X)
• X.Small 0.2
• X.Medium 0.2

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Fuzzy Variable Distribution

• In the male-height example (Luger Stubblefield)
• Fuzzy distribution for height variable
• Composed of three separate functions
• Small is defined over region from x0 to x5
• Medium is defined over region from x46 to x6
• Tall is defined over region from x 56 to x
  66

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Fuzzy Function Combination

• We often have Sets(X) gt 1
• Depending on the application
• We may want to combine the fuzzy functions for
  Sets(X)
• I.e., get a single result value for the input
  value of X
• Conventions for and, or and not
• f1(x) and f2(x) min(f1(x), f2(x))
• f1(x) or f2(x) max(f1(x), f2(x))
• not f1(x) 1 f1(x)

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Fuzzy Rules

• Rules will be of the form
• If ltconditiongt then ltcompute fuzzy resultgt
• ltconditiongt is written in terms of specific
  values for fuzzy variables, X1, X2, XN
• ltcompute fuzzy resultgt will specify a fuzzy
  output variable and give it both a function and a
  value
• For example
• If Fuzzy-function1 ? Sets(X1) and Fuzzy-function2
  ? Sets(X1), then
• Let O1.Fuzzy-output-function
• min(X1.Fuzzy-function1, X1.Fuzzy-function2)
• Fuzzy output variable will have another fuzzy
  distribution that needs to be specified
• Fuzzy output function needs to be defined on that
  fuzzy output distribution

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One Application Method - 1

• Input set of crisp input variables, say, X1, X2,
  XN
• In general, each input variable will have a
  different fuzzy distribution
• Output Set of crisp output variables Y1, Y2,
  .., YM

Classify, generate fuzzy Outputs, combine fuzzy
outputs, Generate crisp result
X1, X2, XN
Y1, Y2, YN
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One Application Method - 2

• Classify variables by fuzzy membership
• Sets(X1), Sets(X2), , Sets(XN)
• Use a set of fuzzy rules
• Each rule will compute an output fuzzy variable
• Will have one or more output variables that have
  fuzzy distributions
• For multiple possibility values for one variable
• Different possibilities for separate fuzzy
  functions
• Union areas of possibilities for variable
• Use a method to combine and generate crisp
  output
• E.g., centroid method

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Pendulum Balancing Controller

• Problem
• Sensor values
• Motor output