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Uncertainty ECE457 Applied Artificial Intelligence Spring 2007 Lecture #8 Outline Uncertainty Probability Bayes’ Theorem Russell & Norvig, chapter 13 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2 Limit of FOL FOL works only when facts are known to be true or false “Some purple mushrooms are poisonous” x Purple(x) Mushroom(x) Poisonous(x) In real life there is almost always uncertainty “There’s a 70% chance that a purple mushroom is poisonous” Can’t be represented as FOL sentence ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 3 Acting Under Uncertainty So far, rational decision is to pick action with “best” outcome Two actions #1 leads to great outcome #2 leads to good outcome It’s only rational to pick #1 Assumes outcome is 100% certain What if outcome is not certain? Two actions #1 has 1% probability to lead to great outcome #2 has 90% probability to lead to good outcome What is the rational decision? ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 4 Acting Under Uncertainty Maximum Expected Utility (MEU) Pick action that leads to best outcome averaged over all possible outcomes of the action Same principle as Expectiminimax, used to solve games of chance (see Game Playing, lecture #5) How do we compute the MEU? First, we need to compute the probability of each event ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 5 Types of Uncertain Variables Boolean Discrete Can take a value from a limited, countable domain Temperature {Hot, Warm, Cool, Cold} Continuous Can be true or false Warm {True, False} Can take a value from a set of real numbers Temperature [-35, 35] We’ll focus on discrete variables for now ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 6 Probability Each possible value in the domain of an uncertain variable is assigned a probability Represents how likely it is that the variable will have this value P(Temperature=Warm) Probability that the “Temperature” variable will have the value “Warm” We can simply write P(Warm) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 7 Probability Axioms P(x) [0, 1] P(x) = 1 P(x) = 0 x is necessarily false, or certain not to occur P(A B) = P(A) + P(B) – P(A B) P(A B) = 0 x is necessarily true, or certain to occur A and B are said to be mutually exclusive P(x) = 1 If all values of x are mutually exclusive ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 8 Visualizing Probabilities P(A) is the proportion of the event space in which A is true Area of green circle / Total area P(A) = 0 if the green circle doesn’t exist P(A) = 1 if the green circle covers the entire event space Event space P(A) ECE457 Applied Artificial Intelligence P(A) R. Khoury (2007) Page 9 Visualizing Probabilities P(A B) = P(A) + P(B) – P(A B) Sum of area of both circles / Total area P(A B) = 0 There is no intersection between both regions A and B can’t happen together: mutually exclusive P(A B) P(A) P(B) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 10 Prior (Unconditional) Probability Probability that A is true in the absence of any other information P(A) Example P(Temperature=Hot) = 0.2 P(Temperature=Warm) = 0.6 P(Temperature=Cool) = 0.15 P(Temperature=Cold) = 0.05 P(Temperature) = {0.2, 0.6, 0.15, 0.05} This is a probability distribution ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 11 Joint Probability Distribution Let’s add another variable Condition {Sunny, Cloudy, Raining} We can compute P(Temperature,Condition) Hot Warm Cool Cold Sunny 0.12 0.23 0.02 0.01 ECE457 Applied Artificial Intelligence Cloudy Raining 0.05 0.03 0.2 0.17 0.05 0.08 0.02 0.02 R. Khoury (2007) Page 12 Joint Probability Distribution Given a joint probability distribution P(a,b), we can compute P(a=Ai) P(Ai) = j P(Ai,Bj) Assumes all events (Ai,Bj) are mutually exclusive This is called marginalization P(Warm) = P(Warm,Sunny) + P(Warm,Cloudy) + P(Warm,Raining) P(Warm) = 0.23 + 0.2 + 0.17 P(Warm) = 0.6 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 13 Visualizing Marginalization P(A) = P(A,B) + P(A,C) + P(A,D) P(A,C) No area of A not covered by B, C or D B, C and D do not intersect inside A P(A,B) P(B) P(A) P(C) ECE457 Applied Artificial Intelligence P(D) R. Khoury (2007) P(A,D) Page 14 Posterior (Conditional) Probability Probability that A is true given that we know that B is true Can be computed using prior and joint probability P(A|B) P(A|B) = P(A,B) / P(B) P(Warm|Cloudy) = P(Warm,Cloudy) / P(Cloudy) P(Warm|Cloudy) = 0.2 / 0.32 P(Warm|Cloudy) = 0.625 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 15 Visualizing Posterior Probability P(A|B) = P(A,B) / P(B) We know that B is true We want the area of B where A is also true We don’t care about the area P(B) P(A | B) P(A) P(B) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 16 Bayes’ Theorem Start from previous conditional probability equation P(A|B)P(B) = P(A,B) P(B|A)P(A) = P(B,A) P(A|B)P(B) = P(B|A)P(A) P(A|B) = P(B|A)P(A) / P(B) (important!) P(A|B): Posterior probability P(A): Prior probability P(B|A): Likelihood P(B): Normalizing constant ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 17 Bayes’ Theorem Allows us to compute P(A|B) without knowing P(A,B) In many real-life situations, P(A|B) cannot be measured directly, but P(B|A) is available Bayes’ Theorem underlies all modern probabilistic AI systems ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 18 Visualizing Bayes’ Theorem P(A|B) = P(B|A)P(A) / P(B) We know the portion of the event space where A is true, and that where B is true We know the portion of A where B is also true We want the portion of B where A is also true P(B|A) P(A) ECE457 Applied Artificial Intelligence P(B) R. Khoury (2007) Page 19 Bayes’ Theorem Example #1 We want to design a classifier (for email spam) We know that Compute the probability that an item belongs to class C (spam) given that it exhibits feature F (the word “Viagra”) 20% of items in the world belong to class C 90% of items in class C exhibit feature F 40% of items in the world exhibit feature F P(C|F) = P(F|C) * P(C) / P(F) P(C|F) = 0.9 * 0.2 / 0.4 P(C|F) = 0.45 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 20 Bayes’ Theorem Example #2 A drug test returns “positive” if drugs are detected in an athlete’s system, but it can make mistakes If an athlete took drugs, 99% chance of + If an athlete didn’t take drugs, 10% chance of + 5% of athletes take drugs What’s the probability that an athlete who tested positive really does take drugs? P(drug|+) = P(+|drug) * P(drug) / P(+) P(+) = P(+|drug)P(drug) + P(+|nodrug)P(nodrug) P(+) = 0.99 * 0.05 + 0.1*0.95 = 0.1445 P(drug|+) = 0.99 * 0.05 / 0.1445 = 0.3426 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 21 Bayes’ Theorem We computed the normalizing constant using marginalization! P(B) = i P(B|Ai)P(Ai) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 22 Chain Rule Recall that P(A,B) = P(A|B)P(B) Can be extended to multiple variables Extend to three variables General form P(A,B,C) = P(A|B,C)P(B,C) P(A,B,C) = P(A|B,C)P(B|C)P(C) P(A1,A2,…,An) = P(A1|A2,…,An)P(A2|A3,…,An)…P(An-1|An)P(An) Compute full joint probability distribution Simple if variables conditionally independent ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 23 Visualizing Chain Rule P(A,B,C) = P(A|B,C)P(B|C)P(C) P(A,B,C) We want the proportion of the event space where A, B and C are true Proportion of B,C where A is also true * proportion of C where B is also true * proportion of the event space where C is true P(B|C) P(C) P(B) P(A) ECE457 Applied Artificial Intelligence P(A|B,C) P(B,C) R. Khoury (2007) Page 24 Independence Two variables are independent if knowledge of one does not affect the probability of the other P(A|B) = P(A) P(B|A) = P(B) P(A B) = P(A)P(B) Impact on chain rule P(A1,A2,…,An) = P(A1)P(A2)…P(An) P(A1,A2,…,An) = i=1n P(Ai) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 25 Conditional Independence Independence is hard to satisfy Two variables are conditionally independent given a third if knowledge of one does not affect the probability of the other if the value of the third is known P(A|B,C) = P(A|C) P(B|A,C) = P(B|C) Impact on chain rule P(A1,A2,…,An) = P(A1|An)P(A2|An)…P(An-1|An)P(An) P(A1,A2,…,An) = P(An) i=1n-1 P(Ai|An) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 26 Bayes’ Theorem Example #3 We want to design a classifier We know Compute the probability that an item belongs to class C given that it exhibits features F1 to Fn % of items in the world that belong to class C % of items in class C that exhibit feature Fi % of items in the world exhibit features F1 to Fn P(C|F1,…,Fn) = P(F1,…,Fn|C)*P(C)/P(F1,…,Fn) P(F1,…,Fn|C) * P(C) = P(C,F1,…,Fn) by chain rule P(C,F1,…,Fn) = P(C) i P(Fi|C) assuming features are conditionally independent given the class P(C|F1,…,Fn) = P(C) i P(Fi|C) / P(F1,…,Fn) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 27 Bayes’ Theorem Example #3 P(F1,…,Fn) Independent of class C In multi-class problems, it makes no difference! P(C|F1,…,Fn) = P(C) i P(Fi|C) This is called the Naïve Bayes Classifier “Naïve” because it assumes conditional independence of Fi given C whether it’s actually true or not Often used in practice in cases where Fi are not conditionally independent given C, with very good results ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 28