AI Lecture 7 (uncertainty)
- 1. Topic – 7: UncertaintyTopic – 7: Uncertainty Fall 2018Fall 2018 Tajim Md. Niamat Ullah Akhund Lecturer Department of Computer Science and Engineering Daffodil International University Email: tajim.cse@diu.edu.bd
- 2. Acting under Uncertainty Basic Probability Notation The Axioms of Probability Inference Using Full Joint Distributions Independence Bayes’ Rule and Its Use Topic ContentsTopic Contents
- 3. Autonomous AgentsAutonomous Agents 3 Real World Driving Agent Sensors camera tachometer engine status temperature Effectors accelerator brakes steering Reasoning & Decisions Making Model of vehicle location & status road status Actions change speed change steering Prior Knowledge physics of movement rules of road Goals drive home
- 4. Uncertainty in the World Model The agent can never be completely certain about the state of the external world since there is ambiguity and uncertainty. Why? sensors have limited precision e.g. camera has only so many pixels to capture an image sensors have limited accuracy e.g. tachometer’s estimate of velocity is approximate there are hidden variables that sensors can’t “see” e.g. vehicle behind large truck or storm clouds approaching the future is unknown, uncertain, i.e. cannot foresee all possible future events which may happen 4 Majidur RahmanMajidur Rahman
- 5. Rules and Uncertainty Say we have a rule: if toothache then problem is cavity But not all patients have toothaches due to cavities so we could set up rules like: if toothache and not(gum disease) and not(filling) and ... then problem = cavity This gets complicated, a better method would be: if toothache then problem is cavity with 0.8 probability or P(cavity|toothache) = 0.8 the probability of cavity is 0.8 given toothache is all that is known 5 Majidur RahmanMajidur Rahman
- 6. Uncertainty in the World Model True uncertainty: rules are probabilistic in nature rolling dice, flipping a coin? Laziness: too hard to determine exceptionless rules takes too much work to determine all of the relevant factors too hard to use the enormous rules that result Theoretical ignorance: don't know all the rules problem domain has no complete theory (medical diagnosis) Practical ignorance: do know all the rules BUT haven't collected all relevant information for a particular case 6 Majidur RahmanMajidur Rahman
- 7. LogicsLogics Logics are characterized by what they commit to as "primitives". 7 Logic What Exists in World Knowledge States Propositional facts true/false/unknown First-Order facts, objects, relations true/false/unknown Temporal facts, objects, relations, times true/false/unknown Probability Theory facts degree of belief 0..1 Fuzzy degree of truth degree of belief 0..1 Majidur RahmanMajidur Rahman
- 8. Probability Theory Probability theory serves as a formal means for representing and reasoning with uncertain knowledge of manipulating degrees of belief in a proposition (event, conclusion, diagnosis, etc.) 8 Majidur RahmanMajidur Rahman
- 9. Utility Theory Every state has a degree of usefulness or utility and the agent will prefer states with higher utility. 9 Majidur RahmanMajidur Rahman
- 10. Decision Theory An agent is rational if and only if it chooses the action that yields the highest expected utility, averaged over all the possible outcomes of the action. Decision theory = probability theory + utility theory 10 Majidur RahmanMajidur Rahman
- 11. Kolmogorov's Axioms of Probability 1. 0 ≤ P(a) ≤ 1 probabilities are between 0 and 1 inclusive 1. P(true) = 1, P(false) = 0 probability of 1 for propositions believed to be absolutely true probability of 0 for propositions believed to be absolutely false 1. P(a ∨ b) = P(a) + P(b) - P(a ∧ b) probability of two states is their sum minus their “intersection” 11
- 12. Inference Using Full Joint Distribution Start with the joint probability distribution: 12 Majidur RahmanMajidur Rahman
- 13. Inference by Enumeration Start with a full joint probability distribution for the Toothache, Cavity, Catch world: P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2 13 Majidur RahmanMajidur Rahman
- 14. Inference by Enumeration 14 Start with the joint probability distribution: Can also compute conditional probabilities: P(¬cavity | toothache) = P(¬cavity ∧ toothache) P(toothache) = 0.016+0.064 0.108 + 0.012 + 0.016 + 0.064 = 0.4 Majidur RahmanMajidur Rahman
- 15. IndependenceIndependence 15 Majidur RahmanMajidur Rahman Independence between propositions a and b can be written as: P(a | b) = P(a) or P(b | a) = P(b) or P(a ∧ b) = P(a) P(b) Independence assertions are usually based on knowledge of the domain. As we have seen, they can dramatically reduce the amount of information necessary to specify the full joint distribution. If the complete set of variables can be divided into independent subsets, then the full joint can be factored into separate joint distributions on those subsets. For example, the joint distribution on the outcome of n independent coin flips, P(C1, . . . , Cn), can be represented as the product of n single-variable distributions P(Ci).
- 16. 16 Majidur RahmanMajidur Rahman Bayes’ TheoremBayes’ Theorem
- 17. P(A∧B) = P(A|B)P(B) P(B∧A) = P(B|A)P(A) P(B|A)P(A) = P(A|B)P(B) P(B|A) = P(A|B)P(B) P(A) Bayes’ law (also Bayes’ law or Bayes’ rule) is fundamental to probabilistic reasoning in AI!! 17 Majidur RahmanMajidur Rahman Bayes’ TheoremBayes’ Theorem
- 18. Bayes’ in Action Bayes’ rule requires three terms - a conditional probability and two unconditional probabilities - just to compute one conditional probability. Bayes’ rule is useful in practice because there are many cases where we do have good probability estimates for these three numbers and need to compute the fourth. In a task such as medical diagnosis, we often have conditional probabilities on causal relationships and want to derive a diagnosis. 18 Majidur RahmanMajidur Rahman
- 19. Bayes’ in Action Example:Example: A doctor knows that the disease meningitis causes the patient to have a stiff neck, say, 50% of the time. The doctor also knows some unconditional facts: the prior probability that a patient has meningitis is 1/50,000, and the prior probability that any patient has a stiff neck is 1/20. Let s be the proposition that the patient has a stiff neck and m be the proposition that the patient has meningitis. Solution:Solution: This question can be answered by using the well-known Bayes’ theorem. 19 Majidur RahmanMajidur Rahman
- 20. Bayes’ in ActionBayes’ in Action Solution:Solution: P(s|m) = 0.5 P(m) = 1/50000 P(s) = 1/20 20 Majidur RahmanMajidur Rahman
- 21. 21 Bayes’ Theorem ExampleExample Consider a football game between two rival teams: Team 0 and Team 1. Suppose Team 0 wins 65% of the time and Team 1 wins the remaining matches. Among the games won by Team 0, only 30% of them come from playing on Team 1 ’s football field. On the other hand, 75% of the victories for Team 1 are obtained while playing at home. If Team 1 is to host the next match between the two teams, which team will most likely emerge as the winner? Solution:Solution: This question can be answered by using the well-known Bayes’ theorem. Majidur RahmanMajidur Rahman
- 22. 22 Bayes’ TheoremBayes’ Theorem Majidur RahmanMajidur Rahman
- 23. 23 Bayes’ TheoremBayes’ Theorem Majidur RahmanMajidur Rahman
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- 27. Using Bayes’ Theorem More Realistically Bayes’ian updating P(Cavity | Toothache ∧ Catch) = P(Toothache ∧ Catch | Cavity) P(Cavity) 27 Majidur RahmanMajidur Rahman
- 29. COURTESY: Md. Tarek Habib Assistant Professor Daffodil International University Majidur RahmanMajidur Rahman
- 30. ********** ********** If You Need Me ********** ********** Mail: tajim.cse@diu.edu.bd Website: https://www.tajimiitju.blogspot.com ORCID: https://orcid.org/0000-0002-2834-1507 LinkedIn: https://www.linkedin.com/in/tajimiitju ResearchGate: https://www.researchgate.net/profile/Tajim_Md_Niamat_Ullah_Akhund YouTube: https://www.youtube.com/tajimiitju?sub_confirmation=1 SlideShare: https://www.slideshare.net/TajimMdNiamatUllahAk Facebook: https://www.facebook.com/tajim.mohammad GitHub: https://github.com/tajimiitju Google+: https://plus.google.com/+tajimiitju Gmail: tajim.mohammad.3@gmail.com Twitter: https://twitter.com/Tajim53 Thank you Tajim Md. Niamat Ullah AkhundTajim Md. Niamat Ullah Akhund