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Uncertainty

Uncertainty. Logical approach problem: we do not always know complete truth about the environment Example: Leave(t) = leave for airport t minutes before flight Query: ?. Problems. Why can’t we determine t exactly? Partial observability road state, other drivers’ plans

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Uncertainty

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  1. Uncertainty Logical approach problem: we do not always know complete truth about the environment Example: Leave(t) = leave for airport t minutes before flight Query: ?

  2. Problems • Why can’t we determine t exactly? • Partial observability • road state, other drivers’ plans • Uncertainty in action outcomes • flat tire • Immense complexity of modelling and predicting traffic

  3. Problems • Three specific issues: • Laziness • Too much work to list all antecedents or consequents • Theoretical ignorance • Not enough information on how the world works • Practical ignorance • If if we know all the “physics”, may not have all the facts

  4. What happens with a purely logical approach? • Either risks falsehood: • “Leave(45) will get me there on time” • Leads to conclusions to weak to do anything with: • “Leave(45) will get me there on time if there’s no snow and there’s no train crossing Route 19 and my tires remain intact and...” • Leave(1440) might work fine, but then I’d have to spend the night in the airport

  5. Solution: Probability • Given the available evidence, Leave(35) will get me there on time with probability 0.04 • Probability address uncertainty, not degree of truth • Degree of truth handled by fuzzy logic • IsSnowing is true to degree 0.2 • Probabilities summarize effects of laziness and ignorance • We will use combination of probabilities and utilities to make decisions

  6. Subjective or Bayesian probability • We will make probability estimates based on knowledge about the world • P(Leave(45) | No Snow) = 0.55 • Not assertions about the world • Probability assessment if the world were a certain way • Probabilities change with new information • P(Leave(45) | No Snow, 5 AM) = 0.75 • Analagous to entailment, not truth

  7. Making decision under uncertainty • Suppose I believe the following: • P(Leave(35) gets me there on time | ...) = 0.04 • P(Leave(45) gets me there on time | ...) = 0.55 • P(Leave(60) gets me there on time | ...) = 0.95 • P(Leave(1440) gets me there on time | ...) = 0.9999 • Which action do I choose? • Depends on my preferences for missing flight vs. eating in airport, etc. • Utility theory used to represent preferences • Decision theory takes into account utility and probabilities

  8. Axioms of Probability • For any propositions A and B: • Example: • A = computer science major • B = born in Minnesota

  9. Notation and Concepts • Unconditional probability or prior probability: • P(Cavity) = 0.1 • P(Weather = Sunny) = 0.55 • corresponds to belief prior to arrival of any new evidence • Weather is a multivalued random variable • Could be one of <Sunny, Rain, Cloudy, Snow> • P(Cavity) shorthand for P(Cavity=true)

  10. Probability Distributions • Probability Distribution gives probability values for all values • P(Weather) = <0.55, 0.05, 0.2, 0.2> • must be normalized: sum to 1 • Joint Probability Distribution gives probability values for combinations of random variables • P(Weather, Cavity) = 4 x 2 matrix

  11. Earthquake=false Earthquake=true Posterior Probabilities • Conditional or Posterior probability: • P(Cavity | Toothache) = 0.8 • For conditional distributions: • P(Weather | Earthquake) =

  12. Posterior Probabilities • More knowledge does not change previous knowledge, but may render old knowledge unnecessary • P(Cavity | Toothache, Cavity) = 1 • New evidence may be irrelevant • P(Cavity | Toothache, Schiller in Mexico) = 0.8

  13. Definition of Conditional Probability • Two ways to think about it

  14. Definition of Conditional Probablity • Another way to think about it • Sanity check: Why isn’t it just: • General version holds for probability distributions: • This is a 4 x 2 set of equations

  15. Bayes’ Rule • Product rule given by • Bayes’ Rule: • Bayes’ rule is extremely useful in trying to infer probability of a diagnosis, when the probability of cause is known.

  16. Bayes’ Rule example • Does my car need a new drive axle? • If a car needs a new drive axle, with 30% probability this car jerks around • P(jerks | needs axle) = 0.3 • Unconditional probabilites: • P(car jerks) = 1/1000 • P(needs axle) = 1/10,000 • Then: • P(needs axle | jerks) = P(jerks | needs axle) P(needs axle) ------------------------------------------ P(jerks) • = (0.3 x 1/10,000) / (1/1000) = .03 • Conclusion: 3 of every 100 cars that jerk need an axle

  17. Not dumb question • Question: • Why should I be able to provide an estimate of P(B|A) to get P(A|B)? • Why not just estimate P(A|B) and be done with the whole thing?

  18. Not dumb question • Answer: • Diagnostic knowledge is often more tenuous than causal knowledge • Suppose drive axles start to go bad in an “epidemic” • e.g. poor construction in a major drive axle brand two years ago is now haunting us • P(needs axle) goes way up, easy to measure • P(needs axle | jerks) should (and does) go up accordingly – but how to estimate? • P(jerks | needs axle) is based on causal information, doesn’t change

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