1. Slide 1 A Quick Overview of Probability William W. Cohen Machine Learning 10-601 Jan 2008

2. Slide 2 Probabilistic and Bayesian Analytics Andrew W. Moore Professor School of Computer Science Carnegie Mellon University www.cs.cmu.edu/~awm awm@cs.cmu.edu 412-268-7599 Note to other teachers and users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrew’s tutorials: http://www.cs.cmu.edu/~awm/tutorials. Comments and corrections gratefully received. http://www.cs.cmu.edu/~awm/tutorials Copyright © Andrew W. Moore [Some material pilfered from http://www.cs.cmu.edu/~awm/tutorials]http://www.cs.cmu.edu/~awm/tutorials

3. Slide 3 Announcements Recitation this week: Matlab tutorial Probability q/a and tutorial HW1 is due on Monday

4. Slide 4 Zeno’s paradox Lance Armstrong and the tortoise have a race Lance is 10x faster Tortoise has a 1m head start at time 0 01 So, when Lance gets to 1m the tortoise is at 1.1m So, when Lance gets to 1.1m the tortoise is at 1.11m … So, when Lance gets to 1.11m the tortoise is at 1.111m … and Lance will never catch up -? 1+0.1+0.01+0.001+0.0001+… = ? unresolved until calculus was invented

5. Slide 5 The Problem of Induction David Hume (1711- 1776): pointed out 1.Empirically, induction seems to work 2.Statement (1) is an application of induction. This stumped people for about 200 years 1.Of the Different Species of Philosophy.Of the Different Species of Philosophy. 2.Of the Origin of IdeasOf the Origin of Ideas 3.Of the Association of IdeasOf the Association of Ideas 4.Sceptical Doubts Concerning the Operations of the UnderstandingSceptical Doubts Concerning the Operations of the Understanding 5.Sceptical Solution of These DoubtsSceptical Solution of These Doubts 6.Of Probability 9Of Probability 9 7.Of the Idea of Necessary ConnexionOf the Idea of Necessary Connexion 8.Of Liberty and NecessityOf Liberty and Necessity 9.Of the Reason of AnimalsOf the Reason of Animals 10.Of MiraclesOf Miracles 11.Of A Particular Providence and of A Future StateOf A Particular Providence and of A Future State 12.Of the Academical Or Sceptical PhilosophyOf the Academical Or Sceptical Philosophy

6. Slide 6 A Second Problem of Induction A black crow seems to support the hypothesis “all crows are black”. A pink highlighter supports the hypothesis “all non-black things are non-crows” Thus, a pink highlighter supports the hypothesis “all crows are black”.

7. Slide 7 A Third Problem of Induction You have much less than 200 years to figure it all out.

8. Slide 8 Probability Theory Events discrete random variables, boolean random variables, compound events Axioms of probability What defines a reasonable theory of uncertainty Compound events Independent events Conditional probabilities Bayes rule and beliefs Joint probability distribution

9. Slide 9 Discrete Random Variables A is a Boolean-valued random variable if A denotes an event, there is uncertainty as to whether A occurs. Examples A = The US president in 2023 will be male A = You wake up tomorrow with a headache A = You have Ebola A = the 1,000,000,000,000 th digit of π is 7 Define P(A) as “the fraction of possible worlds in which A is true” We’re assuming all possible worlds are equally probable

10. Slide 10 Discrete Random Variables A is a Boolean-valued random variable if A denotes an event, there is uncertainty as to whether A occurs. Define P(A) as “the fraction of experiments in which A is true” We’re assuming all possible outcomes are equiprobable Examples You roll two 6-sided die (the experiment) and get doubles (A=doubles, the outcome) I pick two students in the class (the experiment) and they have the same birthday (A=same birthday, the outcome) a possible outcome of an “experiment ” the experiment is not deterministic

11. Slide 11 Visualizing A Event space of all possible worlds Its area is 1 Worlds in which A is False Worlds in which A is true P(A) = Area of reddish oval

12. Slide 12 The Axioms of Probability 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B)

13. Slide 13 The Axioms Of Probabi lity (This is Andrew’s joke)

14. Slide 14 These Axioms are Not to be Trifled With There have been many many other approaches to understanding “uncertainty”: Fuzzy Logic, three-valued logic, Dempster-Shafer, non- monotonic reasoning, … 25 years ago people in AI argued about these; now they mostly don’t Any scheme for combining uncertain information, uncertain “beliefs”, etc,… really should obey these axioms If you gamble based on “uncertain beliefs”, then [you can be exploited by an opponent]  [your uncertainty formalism violates the axioms] - di Finetti 1931 (the “Dutch book argument”)

15. Slide 15 Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) The area of A can’t get any smaller than 0 And a zero area would mean no world could ever have A true

16. Slide 16 Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) The area of A can’t get any bigger than 1 And an area of 1 would mean all worlds will have A true

17. Slide 17 Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) A B

18. Slide 18 Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) A B P(A or B) B P(A and B) Simple addition and subtraction

19. Slide 19 Theorems from the Axioms 0 <= P(A) <= 1, P(True) = 1, P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B)  P(not A) = P(~A) = 1-P(A) P(A or ~A) = 1 P(A and ~A) = 0 P(A or ~A) = P(A) + P(~A) - P(A and ~A) 1 = P(A) + P(~A) - 0

20. Slide 20 Elementary Probability in Pictures P(~A) + P(A) = 1 A ~A

21. Slide 21 Side Note I am inflicting these proofs on you for two reasons: 1.These kind of manipulations will need to be second nature to you if you use probabilistic analytics in depth 2.Suffering is good for you (This is also Andrew’s joke)

22. Slide 22 Another important theorem 0 <= P(A) <= 1, P(True) = 1, P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B)  P(A) = P(A ^ B) + P(A ^ ~B) A = A and (B or ~B) = (A and B) or (A and ~B) P(A) = P(A and B) + P(A and ~B) – P((A and B) and (A and ~B)) P(A) = P(A and B) + P(A and ~B) – P(A and A and B and ~B)

23. Slide 23 Elementary Probability in Pictures P(A) = P(A ^ B) + P(A ^ ~B) B ~B A ^ ~B A ^ B

24. Slide 24 The Monty Hall Problem You’re in a game show. Behind one door is a prize. Behind the others, goats. You pick one of three doors, say #1 The host, Monty Hall, opens one door, revealing…a goat! 3 You now can either stick with your guess always change doors flip a coin and pick a new door randomly according to the coin

25. Slide 25 The Monty Hall Problem Case 1: you don’t swap. W = you win. Pre-goat: P(W)=1/3 Post-goat: P(W)=1/3 Case 2: you swap W1=you picked the cash initially. W2=you win. Pre-goat: P(W1)=1/3. Post-goat: W2 = ~W1 Pr(W2) = 1-P(W1)=2/3. Moral: ?

26. Slide 26 The Extreme Monty Hall/Survivor Problem You’re in a game show. There are 10,000 doors. Only one of them has a prize. You pick a door. Over the remaining 13 weeks, the host eliminates 9,998 of the remaining doors. For the season finale: Do you switch, or not? …

27. Slide 27 Some practical problems You’re the DM in a D&D game. Joe brings his own d20 and throws 4 critical hits in a row to start off DM=dungeon master D20 = 20-sided die “Critical hit” = 19 or 20 Is Joe cheating? What is P(A), A=four critical hits? A is a compound event: A = C1 and C2 and C3 and C4

28. Slide 28 Independent Events Definition: two events A and B are independent if Pr(A and B)=Pr(A)*Pr(B). Intuition: outcome of A has no effect on the outcome of B (and vice versa). We need to assume the different rolls are independent to solve the problem. You almost always need to assume independence of something to solve any learning problem.

29. Slide 29 Some practical problems You’re the DM in a D&D game. Joe brings his own d20 and throws 4 critical hits in a row to start off DM=dungeon master D20 = 20-sided die “Critical hit” = 19 or 20 What are the odds of that happening with a fair die? Ci=critical hit on trial i, i=1,2,3,4 P(C1 and C2 … and C4) = P(C1)*…*P(C4) = (1/10)^4 Followup: D=pick an ace or king out of deck three times in a row: D=D1 ^ D2 ^ D3

30. Slide 30 Some practical problems The specs for the loaded d20 say that it has 20 outcomes, X where P(X=20) = 0.25 P(X=19) = 0.25 for i=1,…,18, P(X=i)= Z * 1/18 What is Z?

31. Slide 31 Multivalued Discrete Random Variables Suppose A can take on more than 2 values A is a random variable with arity k if it can take on exactly one value out of {v 1,v 2,.. v k } Thus…

32. Slide 32 More about Multivalued Random Variables Using the axioms of probability… 0 <= P(A) <= 1, P(True) = 1, P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) And assuming that A obeys… It’s easy to prove that

33. Slide 33 More about Multivalued Random Variables Using the axioms of probabilityand assuming that A obeys… It’s easy to prove that And thus we can prove

34. Slide 34 Elementary Probability in Pictures A=1 A=2 A=3 A=4 A=5

35. Slide 35 Some practical problems The specs for the loaded d20 say that it has 20 outcomes, X P(X=20) = P(X=19) = 0.25 for i=1,…,18, P(X=i)= Z * 1/18 and what is Z?

36. Slide 36 Some practical problems You (probably) have 8 neighbors and 5 close neighbors. What is Pr(A), A=one or more of your neighbors has the same sign as you? What’s the experiment? What is Pr(B), B=you and your close neighbors all have different signs? What about neighbors? ncn c*c ncn Moral: ?

37. Slide 37 Some practical problems I bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves? P(X=20) = P(X=19) = 0.25 for i=1,…,18, P(X=i)= 0.5 * 1/18

38. Slide 38 Some practical problems I have 3 standard d20 dice, 1 loaded die. Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die. What is P(B)? P(B) = P(B and A) + P(B and ~A)= 0.1*0.75 + 0.5*0.25 = 0.2 using Andrew’s “important theorem” P(A) = P(A ^ B) + P(A ^ ~B)

39. Slide 39 Elementary Probability in Pictures P(A) = P(A ^ B) + P(A ^ ~B) B ~B A ^ ~B A ^ B Followup: What if I change the ratio of fair to loaded die in the experiment?

40. Slide 40 Some practical problems I have lots of standard d20 die, lots of loaded die, all identical. Experiment is the same: (1) pick a d20 uniformly at random then (2) roll it. Can I mix the dice together so that P(B)=0.137 ? P(B) = P(B and A) + P(B and ~A)= 0.1*λ + 0.5*(1- λ) = 0.137 λ = (0.5 - 0.137)/0.4 = 0.9075“mixture model”

41. Slide 41 Another picture for this problem A (fair die)~A (loaded) A and B ~A and B It’s more convenient to say “if you’ve picked a fair die then …” i.e. Pr(critical hit|fair die)=0.1 “if you’ve picked the loaded die then….” Pr(critical hit|loaded die)=0.5 Conditional probability: Pr(B|A) = P(B^A)/P(A) P(B|A)P(B|~A)

42. Slide 42 Definition of Conditional Probability P(A ^ B) P(A|B) = ----------- P(B) Corollary: The Chain Rule P(A ^ B) = P(A|B) P(B)

43. Slide 43 Some practical problems I have 3 standard d20 dice, 1 loaded die. Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die. What is P(B)? P(B) = P(B|A) P(A) + P(B|~A) P(~A) = 0.1*0.75 + 0.5*0.25 = 0.2 “marginalizing out” A

44. Slide 44 A (fair die)~A (loaded) A and B ~A and B P(B|A)P(B|~A) P(A) P(~A) P(B) = P(B|A)P(A) + P(B|~A)P(~A)

45. Slide 45 Some practical problems I have 3 standard d20 dice, 1 loaded die. Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die. Suppose B happens (e.g., I roll a 20). What is the chance the die I rolled is fair? i.e. what is P(A|B) ?

46. Slide 46 A (fair die)~A (loaded) A and B ~A and B P(B|A)P(B|~A) P(A) P(~A) P(A and B) = P(B|A) * P(A) P(A and B) = P(A|B) * P(B) P(A|B) * P(B) = P(B|A) * P(A) P(B|A) * P(A) P(B) P(A|B) = P(B) P(A|B) = ?

47. Slide 47 A (fair die)~A (loaded) A and B ~A and B P(B|A)P(B|~A) P(A) P(~A) P(A and B) = P(B|A) * P(A) P(A and B) = P(A|B) * P(B) P(A|B) * P(B) = P(B|A) * P(A) P(B|A) * P(A) P(B) P(A|B) = P(B)

48. Slide 48 P(B|A) * P(A) P(B) P(A|B) = P(A|B) * P(B) P(A) P(B|A) = Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418 …by no means merely a curious speculation in the doctrine of chances, but necessary to be solved in order to a sure foundation for all our reasonings concerning past facts, and what is likely to be hereafter…. necessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning… Bayes’ rule prior posterior

49. Slide 49 More General Forms of Bayes Rule

50. Slide 50 More General Forms of Bayes Rule

51. Slide 51 Useful Easy-to-prove facts

52. Slide 52 More about Bayes rule An Intuitive Explanation of Bayesian Reasoning: Bayes' Theorem for the curious and bewildered; an excruciatingly gentle introduction - Eliezer YudkowskyAn Intuitive Explanation of Bayesian Reasoning Problem: Suppose that a barrel contains many small plastic eggs. Some eggs are painted red and some are painted blue. 40% of the eggs in the bin contain pearls, and 60% contain nothing. 30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue. What is the probability that a blue egg contains a pearl?

53. Slide 53 Some practical problems Joe throws 4 critical hits in a row, is Joe cheating? A = Joe using cheater’s die C = roll 19 or 20; P(C|A)=0.5, P(C|~A)=0.1 B = C1 and C2 and C3 and C4 Pr(B|A) = 0.0625 P(B|~A)=0.0001

54. Slide 54 What’s the experiment and outcome here? Outcome A: Joe is cheating Experiment: Joe picked a die uniformly at random from a bag containing 10,000 fair die and one bad one. Joe is a D&D player picked uniformly at random from set of 1,000,000 people and n of them cheat with probability p>0. I have no idea, but I don’t like his looks. Call it P(A)=0.1

55. Slide 55 Remember: Don’t Mess with The Axioms A subjective belief can be treated, mathematically, like a probability Use those axioms! There have been many many other approaches to understanding “uncertainty”: Fuzzy Logic, three-valued logic, Dempster-Shafer, non- monotonic reasoning, … 25 years ago people in AI argued about these; now they mostly don’t Any scheme for combining uncertain information, uncertain “beliefs”, etc,… really should obey these axioms If you gamble based on “uncertain beliefs”, then [you can be exploited by an opponent]  [your uncertainty formalism violates the axioms] - di Finetti 1931 (the “Dutch book argument”)

56. Slide 56 Some practical problems Joe throws 4 critical hits in a row, is Joe cheating? A = Joe using cheater’s die C = roll 19 or 20; P(C|A)=0.5, P(C|~A)=0.1 B = C1 and C2 and C3 and C4 Pr(B|A) = 0.0625 P(B|~A)=0.0001 Moral: with enough evidence the prior P(A) doesn’t really matter.

57. Slide 57 Some practical problems I bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves?

58. Slide 58 One solution I bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves? P(1)=0 P(2)=0 P(3)=0 P(4)=0.1 … P(19)=0.25 P(20)=0.2 MLE = maximum likelihood estimate

59. Slide 59 A better solution I bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves? P(A=i|B) = P(B|A=i)*P(A=i) / P(B) B = observed counts P(A=i) = prior probability (subjective) … and we’ll need some tricks to make this simple to compute MAP = maximum a posteriori estimate

60. Slide 60 Some practical problems I have 1 standard d6 die, 2 loaded d6 die. Loaded high: P(X=6)=0.50 Loaded low: P(X=1)=0.50 Experiment: pick one d6 uniformly at random (A) and roll it. What is more likely – rolling a seven or rolling doubles? Three combinations: HL, HF, FL P(D) = P(D ^ A=HL) + P(D ^ A=HF) + P(D ^ A=FL) = P(D | A=HL)*P(A=HL) + P(D|A=HF)*P(A=HF) + P(A|A=FL)*P(A=FL)

61. Slide 61 Elementary Probability in Pictures A=HL A=HF A=FL B ^ A=HL B ^ A=HF B ^ A=LF

62. Slide 62 Elementary Probability in Pictures A=1 A=2 A=3 A=4 A=5 B ^ A=1 B ^ A=2 B ^ A=3 B ^ A=4 B ^ A=5

63. Slide 63 Elementary Probability in Pictures A=2 A=3 A=4 A=5 A=1 P(A=1) Think of multiplying by P(A=1) as “squeezing” an area by some fraction

64. Slide 64 Elementary Probability in Pictures A=2 A=3 A=4 A=5 A=1 P(A=1) P(B|A=1) P(A and B)

65. Slide 65 Some practical problems I have 1 standard d6 die, 2 loaded d6 die. Loaded high: P(X=6)=0.50 Loaded low: P(X=1)=0.50 Experiment: pick one d6 uniformly at random (A) and roll it. What is more likely – rolling a seven or rolling doubles? Three combinations: HL, HF, FL P(D) = P(D ^ A=HL) + P(D ^ A=HF) + P(D ^ A=FL) = P(D | A=HL)*P(A=HL) + P(D|A=HF)*P(A=HF) + P(A|A=FL)*P(A=FL)

66. Slide 66 Some practical problems I have 1 standard d6 die, 2 loaded d6 die. Loaded high: P(X=6)=0.50 Loaded low: P(X=1)=0.50 Experiment: pick one d6 uniformly at random (A) and roll it. What is more likely – rolling a seven or rolling doubles? 123456 1D7 2D7 3D7 47D 57D 67D Three combinations: HL, HF, FL Roll 1 Roll 2

67. Slide 67 A brute-force solution ARoll 1Roll 2P FL111/3 * 1/6 * ½ FL121/3 * 1/6 * 1/10 FL1…… …16 21 2… ……… 66 HL11 12 ……… HF11 … Comment doubles seven doubles A joint probability table shows P(X1=x1 and … and Xk=xk) for every possible combination of values x1,x2,…., xk With this you can compute any P(A) where A is any boolean combination of the primitive events (Xi=Xk), e.g. P(doubles) P(seven or eleven) P(total is higher than 5) ….

68. Slide 68 The Joint Distribution Recipe for making a joint distribution of M variables: Example: Boolean variables A, B, C

69. Slide 69 The Joint Distribution Recipe for making a joint distribution of M variables: 1.Make a truth table listing all combinations of values of your variables (if there are M Boolean variables then the table will have 2 M rows). Example: Boolean variables A, B, C ABC 000 001 010 011 100 101 110 111

70. Slide 70 The Joint Distribution Recipe for making a joint distribution of M variables: 1.Make a truth table listing all combinations of values of your variables (if there are M Boolean variables then the table will have 2 M rows). 2.For each combination of values, say how probable it is. Example: Boolean variables A, B, C ABCProb 0000.30 0010.05 0100.10 0110.05 100 1010.10 1100.25 1110.10

71. Slide 71 The Joint Distribution Recipe for making a joint distribution of M variables: 1.Make a truth table listing all combinations of values of your variables (if there are M Boolean variables then the table will have 2 M rows). 2.For each combination of values, say how probable it is. 3.If you subscribe to the axioms of probability, those numbers must sum to 1. Example: Boolean variables A, B, C ABCProb 0000.30 0010.05 0100.10 0110.05 100 1010.10 1100.25 1110.10 A B C 0.05 0.25 0.100.05 0.10 0.30

72. Slide 72 Using the Joint One you have the JD you can ask for the probability of any logical expression involving your attribute

73. Slide 73 Using the Joint P(Poor Male) = 0.4654

74. Slide 74 Using the Joint P(Poor) = 0.7604

75. Slide 75 Inference with the Joint

76. Slide 76 Inference with the Joint P(Male | Poor) = 0.4654 / 0.7604 = 0.612

77. Slide 77 Inference is a big deal I’ve got this evidence. What’s the chance that this conclusion is true? I’ve got a sore neck: how likely am I to have meningitis? I see my lights are out and it’s 9pm. What’s the chance my spouse is already asleep?