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A Not-So-Quick Overview of Probability

A Not-So-Quick Overview of Probability. William W. Cohen Machine Learning 10-605. Warmup : Zeno’s paradox. 0. 1. 1+0.1+0.01+0.001+0.0001+… = ?. Lance Armstrong and the tortoise have a race Lance is 10x faster Tortoise has a 1m head start at time 0.

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A Not-So-Quick Overview of Probability

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  1. A Not-So-Quick Overview of Probability William W. Cohen Machine Learning 10-605

  2. Warmup: Zeno’s paradox 0 1 1+0.1+0.01+0.001+0.0001+… = ? • Lance Armstrong and the tortoise have a race • Lance is 10x faster • Tortoise has a 1m head start at time 0 • 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 -? unresolved until calculus was invented

  3. The prosecution calls Gottfried Leibniz.

  4. The Problem of Induction • David Hume (1711-1776): pointed out • Empirically, induction seems to work • Statement (1) is an application of induction. • This stumped people for about 200 years • Of the Different Species of Philosophy. • Of the Origin of Ideas • Of the Association of Ideas • Sceptical Doubts Concerning the Operations of the Understanding • Sceptical Solution of These Doubts • Of Probability9 • Of the Idea of Necessary Connexion • Of Liberty and Necessity • Of the Reason of Animals • Of Miracles • Of A Particular Providence and of A Future State • Of the Academical Or Sceptical Philosophy

  5. 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”.

  6. 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

  7. 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 a possible outcome of an “experiment” the experiment is not deterministic

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

  9. 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 = I have Ebola • A = The US president in 2023 will be male • A = You wake up tomorrow with a headache • A = the 1,000,000,000,000th digit of π is 7 a possible outcome of an “experiment” the experiment is not deterministic

  10. 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) Events, random variables, …., probabilities “Dice” “Experiments”

  11. (This is Andrew’s joke) The Axioms Of Probability

  12. 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”)

  13. 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

  14. 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

  15. A B 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)

  16. A B 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) P(A or B) B P(A and B) Simple addition and subtraction

  17. 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) = P(A) + P(~A) - P(A and ~A) 1 = P(A) + P(~A) - 0 P(A or ~A) = 1 P(A and ~A) = 0

  18. Elementary Probability in Pictures • P(~A) + P(A) = 1 A ~A

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

  20. 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)

  21. Elementary Probability in Pictures • P(A) = P(A ^ B) + P(A ^ ~B) A ^ B B A ^ ~B ~B

  22. The LAWSOfProbability Laws of probability: Axioms … Monty Hall Problem proviso

  23. The Monty Hall Problem 3 • 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! • You now can either • stick with your guess • always change doors • flip a coin and pick a new door randomly according to the coin

  24. 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. The Monty Hall Problem Moral: ?

  25. 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? …

  26. 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

  27. 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 frequently need to assume the independence of something to solve any learning problem.

  28. 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

  29. 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?

  30. 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 {v1,v2, .. vk} • Example: V={aaliyah, aardvark, …., zymurge, zynga} • Example: V={aaliyah_aardvark, …, zynga_zymgurgy} • Thus…

  31. Terms: Binomials and Multinomials • 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 {v1,v2, .. vk} • Example: V={aaliyah, aardvark, …., zymurge, zynga} • Example: V={aaliyah_aardvark, …, zynga_zymgurgy} • The distribution Pr(A) is a multinomial • For k=2 the distribution is a binomial

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

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

  35. Elementary Probability in Pictures A=aaliyah … A=… A=zynga A=…. A=aardvark

  36. 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 … and what is z?

  37. 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? Moral: ?

  38. 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

  39. 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)

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

  41. 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 “mixture model” λ = (0.5 - 0.137)/0.4 = 0.9075

  42. Another picture for this problem • 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) A (fair die) ~A (loaded) P(B|A) P(B|~A) ~A and B A and B

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

  44. Some practical problems “marginalizing out” A • 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

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

  46. 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) ?

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

  48. P(B|A) * P(A) P(A|B) * P(B) P(A|B) = P(B|A) = P(A) P(B) posterior prior Bayes’ rule 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…

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