Dr. Matthew Ikl

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Imprecise Probabilities and Their Role in General
Intelligence A Pragmatic Approach to Calculating
Weight of Evidence Combining Imprecise
Probabilities and Confidence Intervals
Dr. Matthew Iklé Department of Mathematics and
Computer Science Adams State College
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Probability Theory

• A Principled Foundation for Artificial General
  Intelligence
• BUT Constraints are placed by
• The need to operate within realistic
  computational resources.
• The current, incomplete state of probabilistic
  mathematics.
• THUS Probability theory requires augmentation
  with heuristic approaches to be pragmatic for
  general intelligence.

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Probabilistic Logic Networks (PLN)

• A Logical Inference System
• Combines rigorous probabilistic formulas with
  heuristic rules.
• Reasoning based on uncertain knowledge and/or
  reasoning leading to uncertain conclusions.
• Ability to encompass within logic things such as
  induction, abduction, analogy and speculation,
  and reasoning about time and causality.
• Effectively propagates uncertainties through
  complex inferences involving quantifiers,
  higher-order functions, etc.
• Designed for integration with a general-purpose
  cognition process (in the Novamente AI system)

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Probabilistic Logic Networks (PLN)

• A Rich Set of Inference Rules
• Deduction, Bayes Rule, Unification,
  Intensional/Extensional Inference, Belief
  Revision,
• Each rule comes with uncertain truth value
  formulas, calculating the truth value of the
  conclusion from the truth values of the premises
• Inference is controlled by highly flexible
  forward and backward chaining processes able to
  take feedback from external processes and thus
  behave adaptively

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Belief Revision

• One simple, but critical rule within PLN and
  other uncertain inference systems
• Allows the combination of two different estimates
  of the truth value of the same proposition, to
  form a composite estimate
• Is awkwardly handled within standard
  probabilistic approaches
• Different estimates may come from different
  external sources, OR from different internal
  inference trails

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Belief Revision A Heuristic Rule

• lts,dgt -- ltstrength, weight of evidencegt
• count n k d/(1-d) d n/(nk), assume k10
• eat (cat, mouse) lt.8,.7gt // data source 1
• eat(cat, mouse) lt.2,.4gt // data source 2
• -
• eat(cat, mouse) lt.58, .75gt // sources 1 and 2
• (.8 .7 .2.4)/(.7.4) .58
• d1.7 --gt N1 23
• d2.4 --gt N2 7
• N N1 N2 30 (assuming no dependence)
• d .75

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Weight of Evidence

• What is it?
• Why is it important?
• E.g. belief revision
• One approach weight of ev. interval width
• .2,.8 means less evidence than .4,.6
• Pei Wangs NARS system
• Imprecise probabilities
• Heuristic approaches (Izabela Freire Goertzel)

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Imprecise Probabilities

• The foundation of one approach to weight of
  evidence calculations within PLN
• Peter Walleys Imprecise Beta-Binomial (IBB)
  theory, developed in his seminal work,
  Statistical Inference with Imprecise
  Probabilities, 1991.
• Uses a parametrized envelope of
  (Beta-disribution) priors rather than assuming a
  single prior.

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Imprecise Probabilities

• Advantages of Imprecise Probabilities in General
• Weakness of the traditional approach to
  statistics with its reliance on often unmotivated
  assumptions regarding the functional forms of
  probability distributions.
• More natural and consistent with uncertain and
  incomplete information.
• Standard Bayesian methods offer no generally
  viable way to assess or reason about
  second-order uncertainties or
  weight-of-evidence (eloquently pointed out by
  Pei Wang).

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Imprecise Probabilities are not (quite) the
entire answer

• Disadvantages of Imprecise Probabilities
• Overly conservative.
• Professing ignorance rather than giving guidance
  for practical decision-making.
• Even with significant information to the
  contrary, imprecise probability intervals rapidly
  expand to 0,1.

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The PLN Approach

• A Hybrid of Imprecise Probabilities and
  Traditional Confidence Intervals
• Walleys key ideas provide a solid foundation.
• Natural generalization of Walleys parametrized
  distributions.
• All distributions replaced by envelopes of
  distributions.

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Three Basic Stages

• To calculate the weight of evidence associated
  with the conclusion of an uncertain inference
  rule (e.g. deduction, Bayes rule,)
• Translate premise strength (probability) values
  s, count (weight-of-evidence) values n, and
  standard confidence levels b, into initial
  intervals L,U.
• Calculate final L,U interval using the
  inference rule and Monte-Carlo methods (see next
  slide)
• Translate final L,U interval back to get final
  strength, count, and confidence level values.

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Monte-Carlo methods
final probability interval
initial probability interval
Imprecise Probability level
Translation layer
strength, count, confidence level
strength, count, confidence level
AI engine level
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An Example

• Suppose we have
• 100 gerbils of unknown color
• 10 gerbils of known color, 5 of which are blue
• and 100 rats of known color, 10 of which are
  blue.
• We wish to estimate the probability of a randomly
  chosen blue rodent being a gerbil, using Bayes
  rule
• P(gerbil blue) ?

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Experimental Results
PLN Approach
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Experimental Results
Bayesian (Standard Confidence Interval) Approach
Walleys Approach
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The PLN Approach

• Advantages of the PLN Hybrid Method
• Introduction of traditional Bayesian confidence
  intervals at each stage provides an easily
  configurable way to control the expansion of the
  probability intervals.
• Both Walleys IBB theory and standard Bayesian
  inference follow from the PLN approach as special
  cases.
• Allows for the modeling of all probabilities by
  any family of distributions.
• Allows for considerably more flexibility in
  accounting for known and unknown quantities.

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Conclusions

• The PLN Hybrid Method
• Combines the solid philosophical underpinnings of
  imprecise probability theory with the
  practicality of standard Bayesian methods.
• Provides the ability to adjust interval widths
  based on confidence levels.
• Interoperates smoothly with non-probabilistic
  heuristic methods.

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QUESTIONS?