Probabilistic Reasoning

---

The Problem

• We normally deal with assertions and their causal connections:
  • John has fever
  • John has the flu
  • If somebody has the flu then that person has fever.

• We are not certain that such assertions are true. We believe/disbelieve them to some degree. [Though "belief" and "evidence" are not the same thing, for our purposes they will be treated synonymously.]

• Our problem is how to associate a degree of belief or of disbelief with assertions
  • How do we associate beliefs with elementary assertions
  • How do we combine beliefs in composite assertions from the beliefs of the component assertions
  • What is the relation between the beliefs of causally connected assertions.

• Estimates for elementary assertions are obtained
  • From Experts (subjective probability)
  • From frequencies (if given enough data)

• It is very hard to come up with good estimates for beliefs. Always consider the question "What if the guess is bad".

• Estimates are needed, given the belief in assertions A and B, for the assertions ~A, A & B, A v B

• Evidence must be combined in cases such as:
  • We have a causal connection from assertion A to assertion B, what can we say about B if A is true, or, viceversa, about A if B is true

  • We have a causal connection from assertion A to assertions B1 and B2, what can we say about A if both B1 and B2 are true

  • We have a causal connection from assertion A1 to B and a causal connection from A2 to B, what can we say about B when both A1 and A2 are true.

Proposed Solutions

                   Probabilistic Reasoning



        Single Number Estimates             Belief Interval



   Fuzzy Logic    Bayesian  
                             Certainty Factors
                                        Dempster-Shafer Theory

      Odds Propagation  Bayesian Networks

Probabilistic Reasoning in AI Textbooks

        Giarratano & Riley:   149 pages out of 630 pages
        Ginsberg:              23 pages out of 405 pages
        Luger & Stubblefield:   8 pages out of 704 pages
        Nillson:                0 pages out of 427 pages
        Patterson:             25 pages out of 431 pages
        Rich & Knight:         19 pages out of 582 pages
        Shoham:                34 pages out of 315 pages
        Tanimoto:              39 pages out of 562 pages
        Winston:                0 pages out of 691 pages

        Pearl,J.:       Probabilistic Reasoning in Intelligent Systems
                        Morgan-Kauffman, 1988
        Shafer,Pearl:   Readings in Uncertain Reasoning
                        Morgan-Kauffman, 1990
        Neapolitan,R.E.:Probabilistic Reasoning in Expert Systems
                        J.Wiley, 1990