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

2. Outline Uncertainty Probability Bayes’ Theorem
Russell & Norvig, chapter 13
ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 2

3. 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

4. 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?
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5. 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
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6. 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
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7. 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)
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8. 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

9. 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
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10. 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
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11. 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) = P(Warm) = 0.6
ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 11

12. 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
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13. 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
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14. 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
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15. Bayes’ Theorem Example #1
We want to design a classifier (for 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
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16. 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 * *0.95 =
P(drug|+) = 0.99 * 0.05 / =
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17. Bayes’ Theorem
We computed the normalizing constant using marginalization!
P(B) = i P(B|Ai)P(Ai)
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18. 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

19. 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)
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20. 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)
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21. 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)
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22. 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
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