Issues in Applying Probability Theory to AGI

1
Issues in Applying Probability Theory to AGI

• Pei Wang
• Temple University
• Philadelphia, USA

2
What Is Probability?

• As a mathematical theory, probability theory is
  defined by its axioms on a probability function
  P(x)
• When the theory is applied to a concrete problem,
  P(x) needs to be interpreted to measure something
  in the domain. Common choices
• Frequentist the limit of occurrence frequency
• Subjective the degree of belief of a subject
• Logical the degree of evidential support

3
Probability is Relative

• P(A) is not an intrinsic property of A under
  every interpretation, but related to a reference
• Frequentist relative to an event sequence
• Subjective relative to a given subject
• Logical relative to a body of evidence
• A probability function PR(x) cannot be legally
  used without a clear interpretation and a fixed
  reference R, though they are often implicit in
  the description

4
Frequentist Interpretation in AGI

• Given the general-purpose demand of AGI,
  probability under the frequentist interpretation
  has issues
• The reference sequence is hard to decide
• Some events' occurrence frequency has no limit
• Some events are unique and unrepeatable
• Some statements are not even events
• Frequentist interpretation is too restrict for
  AGI

5
Subjective Interpretation in AGI

• The subjective interpretation only demands the
  consistency among degrees of belief (don't
  confuse it with the logical interpretation)
• Issues when it is used in AGI
• Too weak ? the system can believe anything, as
  far as the beliefs are consistent
• Too strong ? though the consistency among beliefs
  are highly desired, it may not be achievable in
  AGI in realistic situations

6
Logical Interpretation in AGI

• Intuitively, this is the most suitable
  interpretation for AGI, since the system's degree
  of belief should measure evidential support.
• Issues
• How to define evidence? (Confirmation Paradox,
  evidential support vs. conditional probability)
• Can simplicity be used as evidence? (Occam's
  razor)
• Can all the beliefs in a system be evaluated by
  the same evidence? (Bayesian conditioning)

7
Degree of Belief in NARS

• NARS assumes insufficient knowledge and
  resources, and consequently,
• Evidence is defined on all statements in a term
  logic
• Truth-value, degree of belief, and
  evidential support are the same thing
• Each belief has its own evidential base, so the
  degrees of belief are not necessarily consistent
• Simplicity is preferred, but not factored in
  truth-value

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Conclusion

• Probability theory cannot be used as the
  foundation of AGI, because (even under a proper
  interpretation) it demands knowledge (such as a
  prior distribution that is immune to future
  revision) and resources (for global belief
  update) that AGI systems cannot afford
• When the axioms of probability theory are
  violated, the resulting models are not legal
  approximations of the theory, unless
  approximation ratio is accurately proved