Uncertain Knowledge Representation CPSC 386 Artificial Intelligence Ellen
- Slides: 26
Uncertain Knowledge Representation CPSC 386 Artificial Intelligence Ellen Walker Hiram College
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 …”
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
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
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
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)
Non. Monotonic 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”
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. Penelope is a penguin. Penelope is magical. Retract (Penelope is magical). Retract (Penelope is a penguin). (can fly) (cannot fly) (can fly)
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.
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?
JTMS Example • Each belief has justifications (+ and -) • We mark each fact as IN or OUT Suspect Abbot (IN) + Beneficiary Abbot (IN) – Alibi Abbot (OUT)
Revised Justification Suspect Abbot (OUT) – + Beneficiary Abbot (IN) Registered (IN) Alibi Abbot (IN) + + Far Away (IN) – Forged (OUT)
ATMS Example (Partial) • List all possible assumptions (e. g. A 1: register was forged, A 2: 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 A 2 but not A 1)
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)
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%
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)
Basic Probability Notation • Proposition – an assertion like “the card is red” • Random variable – Describes an event we want to know the outcome of, like “Colorof. Card. Picked” – 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
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
Conditional & unconditional Probabilities in example • Unconditional – P(Color 2 nd. Card = red) = 50% – With no other knowledge • Conditional – P(Color 2 nd. Card = red |Color 1 st. Card=red) = 25/51 – Knowing the first card was red, gives more info (lower likelihood of 2 nd card being red) – The bar is read “given”
Computing Conditional Probabilities • P(A|B) = P(A ^ B) / P(B) • The probability that the 2 nd card is red given the first card was red is (the probability that both cards are red) / (probability that 1 st card is red) • P(Car. Wont. Start |No. Gas) = P(Car. Wont. Start ^ No. Gas) / P(No. Gas) • P(No. Gas | Car. Wont. Start) = P(Car. Wont. Start ^ No. Gas) / P(Car. Wont. Start)
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
Back to Conditional Probabilities • P (Car. Wont. Start | No. Gas) – This predicts a symptom based on an underlying cause – These can be generated empirically • (Drain N gastanks, see how many cars start) • P (No. Gas | Car. Wont. Start) – This is a good example of diagnosis. We have a symptom and want to predict the cause – We can’t measure these
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!
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)?
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)
Combining Evidence ache fever flu Multiple sources of evidence leading to the same conclusion thermometer reading fever flu Chain of evidence leading to a conclusion
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