Probabilistic Reasoning in Artificial Intelligence: Representing Uncertainty

1. Artificial Intelligence: Representation and Problem SolvingProbabilistic Reasoning (1): Probability Model 15-381 / 681 Instructors: Fei Fang (This Lecture) and Dave Touretzky feifang@cmu.edu Wean Hall 4126

2. Recap • What we have learned so far… • Search & Satisfiability in discrete space • Basic optimization in continuous and discrete space • Deterministic / Symbolic Reasoning • All focusing on deterministic settings! • Known environment • Full observability • Deterministic change Fei Fang

3. This section • Reason about uncertainty • Signature of real-world scenarios • Will it rain tomorrow? https://www.flickr.com/photos/blackplastic/4520500953 Fei Fang

4. This section • Reason about uncertainty • Signature of real-world scenarios • Partial observability • Is it raining in DC now? Fei Fang

5. This section • Reason about uncertainty • Signature of real-world scenarios • Partial observability • Sometimes, introduce uncertainty to the model to abstract how the world really works https://en.wikipedia.org/wiki/Universe https://pixabay.com/en/atom-science-research-physics-1013638/ Fei Fang

6. This section • Reason about uncertainty • Signature of real-world scenarios • Partial observability • Sometimes, introduce uncertainty to the model to abstract how the world really works • Need probabilistic models for knowledge representation & reasoning • Provide a way of summarizing the uncertainty that comes from our laziness and ignorance Fei Fang

7. Outline • Probability Model and Probabilistic Inference • Chain Rule, Independence, Bayes’ Rule Fei Fang

8. Probability Statement • The probability that it is raining in DC right now given it is not raining in Pittsburgh is • Actual situation is either raining () or not raining () • Probability statement are made w.r.t. knowledge state, not the real world • What is the knowledge state in this statement? Fei Fang

9. Quiz 1 • Antoine Gombaud (1607-1684) • What is the probability of winning? • A: • B: • C: • D: I will roll a die four times; I win if I get a 1 Fei Fang

10. From Gambling to Probability Theory • There is no math for probability at the time, how did he find out? • Gombaud invented a new scam: • However, he kept losing because • Gombaud wrote to Pascal and Fermat, who subsequently created probability theory I will roll two dice 24 times; I win if I get a double 1 Fei Fang

11. Basic Probability Notation • Sample space : set of all possible worlds (exclusive and exhaustive) • A probability model associates a probability with each possible world • Event or proposition : a set of possible worlds • Unconditional / prior probabilities: Degree of belief in the proposition in the absence of any other information • Exercise: describe Quiz 1 using these terms I will roll a die four times; I win if I get a 1 Fei Fang

12. Basic Probability Notation • Evidence: information that has been revealed • Conditional / posterior probabilities: Degree of belief in the proposition given some evidence • : Probability of given evidence • Exercise: describe the following problem using these terms • Is the statement still valid after observing first dice shows ? I roll two dice one by one, the first die shows 1. What is the probability that I get a double 1? Fei Fang

13. Basic Probability Notation • Evidence: information that has been revealed • Conditional / posterior probabilities: Degree of belief in the proposition given some evidence • : Probability of given evidence • Why? Rule out worlds where is not true I roll two dice one by one, the first die shows 1. What is the probability that I get a double 1? Fei Fang

14. Quiz 2 • Bag 1: two gold coins. Bag 2: two pennies. Bag 3: one of each. • Bag is chosen at random, and one coin from it is selected at random; the coin is gold • What is the probability that the other coin is gold given the observation? • A: 1/6 • B: 1/3 • C: 2/3 • D: 1/2 Fei Fang

15. Basic Rules • Negation relationship • Inclusion-exclusion principle Sample space Fei Fang

16. Basic Rules • Product rule • Sum rule • If ’s are mutually exclusive and exhaustive Fei Fang

17. Probability Model with Factored Representation • A random variable • Domain of : values can take • Finite or infinite; Discrete or continuous • Probability distribution • Assign probability for each value if discrete domain • Probability density function (pdf) if continuous domain I will roll a die; I win if I get a 1 Fei Fang

18. Probability Model with Factored Representation • Random variables , each has a domain • Joint probability distribution • Assign a probability for each value tuple if discrete domain • Probability density function (pdf) if continuous domain I will roll a die four times; I win if I get a 1 Fei Fang

19. Probability Model with Factored Representation • A possible world is defined to be an assignment of values to all the random variables under consideration • A probability model is completely determined by the full joint probability distribution – the joint distribution of all the random variables under consideration Fei Fang

20. Joint Probability Distribution .05 .2 0 .1 .1 0 .1 0 0 .1 .05 .1 .1 0 .1 0 Fei Fang

21. Probabilistic Inference • Probabilistic inference: Compute probability of a query variable (or variable set) taking on a value (or set of values) given some evidence on a subset of variables • No evidence: Marginal probability • With evidence: Posterior / Conditional probability May be none Fei Fang

22. Compute Marginal Probability .05 .2 0 .1 .1 0 .1 0 0 .1 .05 .1 .1 0 .1 0 Fei Fang

23. Marginal Probability Distribution • Marginal probability distribution of one variable • If there are only two random variables • If there are random variables If we compute for all , written as Fei Fang

24. Marginal Probability Distribution • Marginal distribution of a subset of variables • Called “marginalization” • Is it required that ? Fei Fang

25. Marginal Probability Distribution .05 .2 0 .1 .1 0 .1 0 0 .1 .05 .1 .1 0 .1 0 Fei Fang

26. Compute Conditional Probability • Scale the joint probability according to marginal probability .05 .2 0 .1 .1 0 .1 0 0 .1 .05 .1 .1 0 .1 0 Fei Fang

27. Conditional Probability Distribution • Conditional probability distribution: the joint distribution of given the value of the other variable Note: This is not matrix division! It represents a set of equations! Fei Fang

28. Conditional Probability Distribution • Conditional probability distribution of a subset of variables given evidence for a subset of variables • Is it required that ? • If , then read from joint probability table • If , then need to compute Fei Fang

29. Revisit Quiz 2 • Bag 1: two gold coins. Bag 2: two pennies. Bag 3: one of each. • Bag is chosen at random, and one coin from it is selected at random; the coin is gold • Define random variables • : type of selected coin • : type of the other coin in the same bag of the selected coin • Domain: A B C D E F Fei Fang

30. Revisit Quiz 2 • What is the probability that the other coin is gold given the observation? A B C D E F Fei Fang

31. Outline • Probability Model and Probabilistic Inference • Chain Rule, Independence, Bayes’ Rule Fei Fang

32. Chain Rule • Factorization using chain rule Note: This is not matrix multiplication! It represents a set of equations! Fei Fang

33. Independence • Definition: proposition and are independent iff • or • or I will roll a die tomorrow; What is the probability of getting a 1 and there is no rain? I will roll two dice one by one; What is the probability of first getting a 1 and then getting a 6? Fei Fang

34. Independence • Definition: Random variable and are independent iff • or • or • Random variables are independent iff Fei Fang

35. Quiz 1 revisited • Antoine Gombaud (1607-1684) • What is the probability of winning? I will roll a die four times; I win if I get a 1 Fei Fang

36. Conditional Independence • Definition: and are conditionally independent given iff • Once the value of is known, knowing doesn’t tell anything more about • Equivalent definition: • In a sense, the dependence between and “dissolves” once the knowledge about is made available Fei Fang

37. Bayes’ Rule • Recall product rule • Bayes’ Rule Fei Fang

38. Bayes’ Rule Fei Fang

39. Monty Hall Problem • Behind one door is a car; behind the other two are goats. Which door has the car is randomly assigned. You pick a door. Then announcer randomly opens another door that has a goat behind it and asks you whether you want to switch before he opens the door you pick. Should you stay with your choice or switch? Steve Selvin Fei Fang

40. Monty Hall Problem • Stay or switch? Random variables : which door has the car Domain Now you picked a door, say door Announcer observes your selection, and then randomly opens a door with a goat behind it Random variable : which door is opened by the announcer Domain If announcer opens door , you want to know Fei Fang

41. Monty Hall Problem : which door has the car : which door is opened Fei Fang

42. Quiz 3 • In Monty Hall problem, if there are five doors, what is the probability of winning when switching to another door? • A: • B: • C: • D: Behind one door is a car; behind the other four are goats. Which door has the car is randomly assigned. You pick a door. Then announcer randomly opens another door that has a goat behind it and asks you whether you want to switch before he opens the door you pick. Fei Fang

43. Summary • Probability Model • Random variable, Domains • Joint probability distribution • Probabilistic Inference • Use full joint probability distribution as the “knowledge base” • Compute • Marginal probability • Conditional probability • Chain Rule, Independence, Bayes’ Rule • Challenge? • Full joint distribution is hard to estimate and too big to represent explicitly Fei Fang

44. Acknowledgment • Some slides are borrowed from previous slides made by Tai Sing Lee Fei Fang

45. Backup Slides Fei Fang

46. Outline • Basics in probability theory Fei Fang

47. Concepts and Definitions • The sample space is a finite set of elements representing possible outcomes that are mutually exclusive • For each outcome , there is a probability measure • A probability distribution assigns a non-negative real probability to each element, such that (which means the sample space is exhaustive) Sample space Fei Fang

48. Concepts and Definitions • An event is a subset • If each element has equal probability, the distribution is uniform: Sample space Fei Fang

49. Concepts and Definitions • Joint probability: probability of event 𝐴 and event 𝐵 occurring together • Probability of a union of disjoint events • if and are disjoint events Sample space Fei Fang

50. The Sum Rule and Marginal Probability • The sum rule • if s are mutually exclusive and exhaustive events Sample space Fei Fang