ECE457 Applied Artificial Intelligence Spring 2008 Lecture #8 Uncertainty ECE457 Applied Artificial Intelligence Spring 2008 Lecture #8

Outline Uncertainty Probability Bayes’ Theorem Russell & Norvig, chapter 13 ECE457 Applied Artificial Intelligence R. Khoury (2008) 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 (2008) 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? #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 (2008) 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 (2008) Page 5

Types of Uncertain Variables Boolean Can be true or false Warm  {True, False} Discrete Can take a value from a limited, countable domain Temperature  {Hot, Warm, Cool, Cold} Continuous 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 (2008) 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 (2008) Page 7

Probability Axioms P(x)  [0, 1] P(x) = 1 P(x) = 0 x is necessarily true, or certain to occur 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 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 (2008) Page 8

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 (2008) Page 9

Joint Probability Distribution Let’s add another variable Condition  {Sunny, Cloudy, Raining} We can compute P(Temperature,Condition) Sunny Cloudy Raining Hot 0.12 0.05 0.03 Warm 0.23 0.2 0.17 Cool 0.02 0.08 Cold 0.01 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 10

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 (2008) Page 11

Posterior (Conditional) Probability Probability that A is true given that we know that B is true P(A|B) Can be computed using prior and joint probability 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 (2008) Page 12

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 (2008) Page 13

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 (2008) Page 14

Bayes’ Theorem Example #1 We want to design a classifier (for email spam) Compute the probability that an item belongs to class C (spam) given that it exhibits feature F (the word “Viagra”) We know that 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 (2008) Page 15

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 (2008) Page 16

Bayes’ Theorem We computed the normalizing constant using marginalization! P(B) = i P(B|Ai)P(Ai) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 17

Chain Rule Recall that P(A,B) = P(A|B)P(B) Can be extended to multiple variables Extend to three variables P(A,B,C) = P(A|B,C)P(B,C) P(A,B,C) = P(A|B,C)P(B|C)P(C) General form 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 (2008) Page 18

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 (2008) Page 19

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 (2008) Page 20

Bayes’ Theorem Example #3 We want to design a classifier Compute the probability that an item belongs to class C given that it exhibits features F1 to Fn We know % 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 (2008) Page 21

Naïve Bayes Classifier 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 (2008) Page 22