1 CSC 8520 Spring Paula Matuszek CS 8520: Artificial Intelligence Resolution and Bayes’ Examples Paula Matuszek Spring, 2010

2 CSC 8520 Spring Paula Matuszek To resolve a pair of clauses which are entirely disjuncts (ORs) of terms, you find a term which exists as a positive term in one and a negative term in the other, and produce a new term which contains all of the terms of both except the pair, which cancel each other. That new term is the resolvent, and showing all possible resolutions involves finding each resolvent.. The idea is that if we assume that both of the following statements are true: dog ∨ cat ¬ dog ∨ horse then (dog ∨ cat) ∧ (¬ dog ∨ horse) ⇒ (cat ∨ horse).

3 CSC 8520 Spring Paula Matuszek dog ¬ dog cathorse dog ∨ cat ¬ dog ∨ horse (dog ∨ cat ) ∧ ( ¬ dog ∨ horse) cat ∨ horse ( (dog ∨ cat ) ∧ ( ¬ dog ∨ horse)) ⇒ ( cat ∨ horse) TFTTTTTTT TFTFTFFTT TFFTTTTTT TFFFTFFFT FTTTTTTTT FTTFTTTTT FTFTFTFTT FTFFFTFFT

4 CSC 8520 Spring Paula Matuszek Midterm question: Show all the possible resolutions for the following pairs of clauses: delicious ¬ delicious ∨ anchovies The only term we can resolve on is delicious, so the resulting clause is anchovles delicious ∨ anchovies ¬ delicious ∨ ¬ anchovies We can resolve on either delicious or anchovies (but not both at once) so we can get either delicious ∨ ¬ delicious (which is trivially true) anchovies ∨ ¬ anchovies (ditto) ¬ X ∨ Y X ∨ ¬ Y ∨ Z We can resolve on x or on y. Giving us either Y ∨ ¬Y v Z or X ∨ ¬X v Z. (also both trivially true)

5 CSC 8520 Spring Paula Matuszek Bayes Theorem Bayes’ Theorem, as Eric discussed, is a way of determining the conditional probability of A given B (written A|B). In other words, if we know B has happened, how likely is A? In order to compute the conditional probability of A|B we need to know three other probabilities, called the priors: –The probability of A (without any other information) –The probability of B (without any other information) –The conditional probability of B given A

6 CSC 8520 Spring Paula Matuszek Bayes’ Theorem Figure from

7 CSC 8520 Spring Paula Matuszek Bayes Example Your child has a rash. How likely is it that he has chicken pox? In other words, what is the probability of chicken pox | rash. –Chicken pox is going around your child’s class: p(chicken pox) =.3. –Overall in your child’s class about half of the children have a rash from something: p(rash) =.5 –Almost all kids with chicken pox get a rash: –p(rash | chicken pox) =.99. So p(chicken pox | rash) = (.99 x.3)/.5 =.594 In other words, there’s about a 60% chance that if your child has a rash, it’s chicken pox.

8 CSC 8520 Spring Paula Matuszek Thinking about Bayes’ What this is basically saying is: –Two things make the probability of A|B higher: A high overall (or prior) probability of A (there is a lot of chicken pox around) A high prior probability of B|A (usually children with chicken pox get a rash) –One thing makes the probability of A|B lower: A high prior probability of B (there are a lot of rashes around) In other words, p(B) is giving us a way to normalize the other probabilities, so that we don’t decide that chicken pox is likely based on a rash if everyone has a rash. Consider what the values would be if instead of “rash” we looked at “Chicken pox | drinks milk”. Such a high proportion of children drink milk that drinking milk really has no predictive value for determining whether a child has chicken pox.