Uncertain Knowledge Representation

1. Uncertain Knowledge Representation CPSC 386 Artificial Intelligence Ellen Walker Hiram College

2. Reasoning Under Uncertainty • We have no way of getting complete information, e.g. limited sensors • We don’t have time to wait for complete information, e.g. driving • We can’t know the result of an action until after having done it, e.g. rolling a die • There are too many unlikely events to consider, e.g. “I will drive home, unless my car breaks down, or unless a natural disaster destroys the roads …”

3. But… • A decision must be made! • No intelligent system can afford to consider all eventualities, wait until all the data is in and complete, or try all possibilities to see what happens

4. So… • We must be able to reason about the likelihood of an event • We must be able to consider partial information as part of larger decisions • But • We are lazy (too many options to list, most unlikely) • We are ignorant • No complete theory to work from • All possible observations haven’t been made

5. Quick Overview of Reasoning Systems • Logic • True or false, nothing in between. No uncertainty • Non-monotonic logic • True or false, but new information can change it. • Probability • Degree of belief, but in the end it’s either true or false • Fuzzy • Degree of belief, allows overlapping of true and false states

6. Examples • Logic • Rain is precipitation • Non-monotonic • It is currently raining • Probability • It will rain tomorrow (70% chance) • Fuzzy • It is raining (.5 hard / .8 very hard / .2 a little)

7. NonMonotonic Logic • Once true (or false) does not mean always true (or false) • As information arrives, truth values can change (Penelope is a bird, penguin, magic penguin) • Implementations (you are not responsible for details) • Circumscription • Bird(x) and not abnormal(x) -> flies(x) • We can assume not abnormal(x) unless we know abnormal(x) • Default logic • “x is true given x does not conflict with anything we already know”

8. Truth Maintenance Systems • These systems allow truth values to be changed during reasoning (belief revision) • When we retract a fact, we must also retract any other fact that was derived from it • Penelope is a bird. (can fly) • Penelope is a penguin. (cannot fly) • Penelope is magical. (can fly) • Retract (Penelope is magical). (cannot fly) • Retract (Penelope is a penguin). (can fly)

9. Types of TMS • Justification based TMS • For each fact, track its justification (facts and rules from which it was derived) • When a fact is retracted, retract all facts that have justifications leading back to that fact, unless they have independent justifications. • Each sentence labeled IN or OUT • Assumption based TMS • Represent all possible states simultaneously • When a fact is retracted, change state sets • For each fact, use list of assumptions that make that fact true; each world state is a set of assumptions.

10. TMS Example (Quine & Ullman 1978) • Abbot, Babbit & Cabot are murder suspects • Abbot’s alibi: At hotel (register) • Babbit’s alibi: Visiting brother-in-law (testimony) • Cabot’s alibi: Watching ski race • Who committed the murder? • New Evidence comes in… • TV video shows Cabot at the ski race • Now, who committed the murder?

11. JTMS Example • Each belief has justifications (+ and -) • We mark each fact as IN or OUT Suspect Abbot (IN) – + Beneficiary Abbot (IN) Alibi Abbot(OUT)

12. Revised Justification Suspect Abbot (OUT) – + Beneficiary Abbot (IN) Alibi Abbot (IN) + – + Far Away(IN) Forged(OUT) Registered(IN)

13. ATMS Example (Partial) • List all possible assumptions (e.g. A1: register was forged, A2: register was not forged) • Consider all possible facts • (e.g. Abbot was at hotel.) • For each fact, determine all possible sets of assumptions that would make it valid • Eg. Abbot was at hotel (all sets that include A2 but not A1)

14. JTMS vs. ATMS • JTMS is sequential • With each new fact, update the current set of beliefs • ATMS is “pre-compiled” • Determine the correct set of beliefs for each fact in advance • When you have actual facts, find the set of beliefs that is consistent with all of them (intersection of sets for each fact)

15. Probability • The likelihood of an event occurring represented as a percentage of observed events over total observations • E.g. • I have a bag containing red & black balls • I pull 8 balls from the bag (replacing the ball each time) • 6 are red and 2 are black • Assume 75% of balls are red, 25% are black • The probability of the next ball being red is 75%

16. More examples • There are 52 cards in a deck, 4 suits (2 red, 2 black) • What is the probability of picking a red card • (26 red cards) / (52 cards) = 50% • What is the probability of picking 2 red cards? • 50% for the first card • (25 red cards) / (51 cards) for the second • Multiply for total result (26*25) / (52*51)

17. Basic Probability Notation • Proposition • an assertion like “the card is red” • Random variable • Describes an event we want to know the outcome of, like “ColorofCardPicked” • Domain is set of values such as {red, black} • Unconditional (prior) probability P(A) • In the absence of other information • Conditional probability P(A | B) • Based on specific prior knowledge

18. Some important equations • P(true) = 1; P(false) = 0 • 0 <= P(a) <= 1 • All probabilities between 0 and 1, inclusive • P(a v b) = P(a) + P(b) – P(a ^ b) • We can derive others • P(a v ~a) = 1 • P(a ^ ~a) = 0 • P(~a) + P(a) = 1

19. Conditional & unconditional Probabilities in example • Unconditional • P(Color2ndCard = red) = 50% • With no other knowledge • Conditional • P(Color2ndCard = red |Color1stCard=red) = 25/51 • Knowing the first card was red, gives more info (lower likelihood of 2nd card being red) • The bar is read “given”

20. Computing Conditional Probabilities • P(A|B) = P(A ^ B) / P(B) • The probability that the 2nd card is red given the first card was red is (the probability that both cards are red) / (probability that 1st card is red) • P(CarWontStart |NoGas) = P(CarWontStart ^ NoGas) / P(NoGas) • P(NoGas | CarWontStart) = P(CarWontStart ^ NoGas) / P(CarWontStart)

21. Product Rule and Independence • P(A^B) = P(A|B) * P(B) • (just an algebraic manipulation) • Two events are independent if P(A|B) = P(A) • E.g. 2 consecutive coin flips are independent • If events are independent, we can multiply their probabilities • P(A^B) = P(A)*P(B) when A and B are independent

22. Back to Conditional Probabilities • P (CarWontStart | NoGas) • This predicts a symptom based on an underlying cause • These can be generated empirically • (Drain N gastanks, see how many cars start) • P (NoGas | CarWontStart) • This is a good example of diagnosis. We have a symptom and want to predict the cause • We can’t measure these

23. Bayes’ Rule • P(A^B) = P(A|B) * P(B) • = P(B|A) * P(A) • So • P(A|B) = P(B|A) * P(A) / P(B) • This allows us to compute diagnostic probabilities from causal probabilities and prior probabilities!

24. Bayes’ rule for diagnosis • P(measles) = 0.1 • P(chickenpox) = 0.4 • P(allergy) = 0.6 • P(spots | measles) = 1.0 • P(spots | chickenpox) = 0.5 • P(spots | allergy) = 0.2 • (assume diseases are independent) • What is P(measles | spots)?

25. P(spots) • P(spots) was not given. • We can estimate it with the following (unlikely) assumptions • The three listed diseases are independent; no one will have two or more • There are no other causes or co-factors for spots, P.e. p(spots | none-of-the-above) = 0 • Then we can say that: • P(spots) = p(spots^measles) + p(spots^chickenpox) + p(spots^allergy) (0.42)

26. Combining Evidence ache flu fever Multiple sources of evidence leading to the same conclusion thermometer reading fever flu Chain of evidence leading to a conclusion