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Reasoning with Uncertainty

Reasoning with Uncertainty. We have accepted that (A -> B) is TRUE followed by A is TRUE, would result in B is TRUE (that is, modus ponens rule is true) e.g. A := it rains B := umbrella opens

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Reasoning with Uncertainty

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  1. Reasoning with Uncertainty • We have accepted that (A -> B) is TRUE followed by A is TRUE, would result in B is TRUE • (that is, modus ponens rule is true) • e.g. • A := it rains • B := umbrella opens • 1) A -> B 1) A -> B • 2) A 2) B • -------------------- ----------------- • 3) B 3) A ? • What about the reverse? : A -> B, followed by B is true • (Does that mean that “it must be raining?”)

  2. Not Necessarily (that it is raining) A B A -> B A -> B is true and B is true does that mean A is true? TTT T F F FTT Not necessarily because the umbrella may be open to protect us from bright sun also ! F F T

  3. Uncertainty with Non-Independent Eventsand “Probabilistic” Logic Consider the following (weather and behavior) events: - if it rains then umbrella opens : A -> B is TRUE; - there is a 30% chance of (raining Λ umbrella opens); - there is a 5% chance of (sunny Λ and umbrella opens) .30 .70 .35 .65 If the umbrella is open does that mean that it is raining? Answer is “may be” because: P (raining | umbrella is open) = P (raining Λ umbrella open) / P (umbrella open) = .30 / .35 = 6/7 chance that it is raining when umbrella is open

  4. How About the Original Case? If it is raining, does that mean the umbrella is open? Answer: “definitely” P (umbrella is open | raining) = P (raining Λ umbrella open) / P (raining) = .30 / .30 = 1/1 = 100% chance that umbrella is open if it rains Thus : A -> B A ----------------- B is 100% and thus modus ponens works as a syllogism

  5. What about other conditions? If the umbrella is open, we said that there is a 6/7 or 86% chance that it is raining. However, if the umbrella is open there is also a 1/7 or 14% chance that it is not raining! - P( not raining| umbrella open) = P (not raining Λ umbrella open)/P (umbrella open) = .05 / .35 = 1/7 = 14% What about modus tollens? A -> B ~B ----------------- ~ A P(~A | ~B) = P (~ A Λ ~B) / P ( ~B) = .65 /.65 = 100%

  6. Interesting Scenarios From the sample - “ -if it is raining the umbrella will be open; -it is raining; therefore, the umbrella is open” we actually have some of the following interesting cases: 1) P( umbrella is open | raining) = .30/.30 = 100% (modus ponens) 2) P( raining | umbrella is open) = .30/.35 = 6/7 = 86% 3) P( not raining | umbrella is open) = .05/.35 = 14% 4) P( not raining | umbrella is not open) = .65/.65 = 100% (modus tollens)

  7. Look at “simple pseudo-code” of non-independent events Pseudo-code segment: Let A : k = true and B: y = 80 & our code segment shows A -> B . . if (k is true) { z := 50 ; y := z +30; } else { y := z + 50; } print y . . Question: if “print y” shows y = 80, is k = true? Answer: - most likely but NOT necessarily: P(A | B) = P( A Λ B) / P(B) - if P(A Λ B) = P(B), then yes. - but we must check the code “above and below” the shown code segment (e.g. is Z set to 30 before? ) Question: print y shows y ≠ 80, is k = true? Answer: - NO (via modus tollens)

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