AI: 15-780 / 16-731
Mar 1, 2007
Probability Theory & Uncertainty
Read Chapter 13 of textbook
What you will learn today
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fundamental role of uncertainty in AI
probability theory can be applied to many of these problems
probability as uncertainty
probability theory is the calculus of reasoning with uncertainty
probability and uncertainty in different contexts
review of basis probabilistic concepts
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discrete and continuous probability
joint and marginal probability
calculating probability
next probability lecture: the process of probabilistic inference
AI: Probability Theory ! Mar 1, 2007
2
Michael S. Lewicki ! Carnegie Mellon
What is the role of probability and inference in AI?
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Many algorithms are designed as if knowledge is perfect, but it rarely is.
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Examples:
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There are almost always things that are unknown, or not precisely known.
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bus schedule
quickest way to the airport
sensors
joint positions
finding an H-bomb
An agent making optimal decisions must take into account uncertainty.
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
Probability as frequency: k out of n possibilities
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Suppose we’re drawing cards from a standard deck:
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P(card is a ♣ | standard deck) = 13/52 = 1/4
What’s the probability of a drawing a pair in 5-card poker?
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P(card is the Jack ♥ | standard deck) = 1/52
P(hand contains pair | standard deck) =
# of hands with pairs
_______________
total # of hands
Counting can be tricky (take a course in combinatorics)
Other ways to solve the problem?
General probability of event given some conditions:
P(event | conditions)
AI: Probability Theory ! Mar 1, 2007
4
Michael S. Lewicki ! Carnegie Mellon
Making rational decisions when faced with uncertainty
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Probability
the precise representation of knowledge and uncertainty
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Probability theory
how to optimally update your knowledge based on new information
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Decision theory: probability theory + utility theory
how to use this information to achieve maximum expected utility
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Consider again the bus schedule. What’s the utility function?
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Suppose the schedule says the bus comes at 8:05.
Situation A: You have a class at 8:30.
Situation B: You have a class at 8:30, and it’s cold and raining.
Situation C: You have a final exam at 8:30.
AI: Probability Theory ! Mar 1, 2007
5
Michael S. Lewicki ! Carnegie Mellon
Probability of uncountable events
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How do we calculate probability that it will rain tomorrow?
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Look at historical trends?
Assume it generalizes?
What’s the probability that there was life on Mars?
What was the probability the sea level will rise 1 meter within the century?
What’s the probability that candidate X will win the election?
AI: Probability Theory ! Mar 1, 2007
6
Michael S. Lewicki ! Carnegie Mellon
The Iowa Electronic Markets: placing probabilities on single events
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http://www.biz.uiowa.edu/iem/
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Typical bet: predict vote share of candidate X - “a vote share market”
“The Iowa Electronic Markets are real-money futures markets in which contract
payoffs depend on economic and political events such as elections.”
AI: Probability Theory ! Mar 1, 2007
7
Michael S. Lewicki ! Carnegie Mellon
Political futures market predicted vs actual outcomes
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
John Craven and the missing H-Bomb
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In Jan. 1966, used Bayesian probability and subjective odds to
locate H-bomb missing in the Mediterranean ocean.
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
10
Michael S. Lewicki ! Carnegie Mellon
Probabilistic Methodology
type of collision
prevailing wind direction
0, 1, or 2 parachutes open?
AI: Probability Theory ! Mar 1, 2007
Probabilistic assessment of dangerous climate change
Forest et al., 2001
Climatic Uncertainty
from Mastrandrea and Schneider (2004)
from Forrest et al (2001)
AI: Probability Theory ! Mar 1, 2007
Michael S. Lewicki ! Carnegie Mellon
11
Step 1
Factoring in Risk Using Decision Theory
P(“DAI” = 55.8%)
Dangerous Climate Change
Step 2
P(“DAI” = 27.4%
Carbon Tax 2050
= $174/Ton
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
Uncertainty in vision: What are these?
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
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Michael S. Lewicki ! Carnegie Mellon
Uncertainty in vision
AI: Probability Theory ! Mar 1, 2007
Edges are not as obvious they seem
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
An example from Antonio Torralba
What’s this?
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
We constantly use other information to resolve uncertainty
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
Image interpretation is heavily context dependent
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
This phenomenon is even more prevalent in speech perception
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It is very difficult to recognize phonemes from naturally spoken speech when they
are presented in isolation.
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All modern speech recognition systems rely heavily on context (as do we).
HMMs model this contextual dependence explicitly.
This allows the recognition of words, even if there is a great deal of uncertainty in
each of the individual parts.
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
De Finetti’s definition of probability
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Was there life on Mars?
You promise to pay $1 if there is, and $0 if there is not.
Suppose NASA will give us the answer tomorrow.
Suppose you have an oppenent
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You set the odds (or the “subjective probability”) of the outcome
But your oppenent decides which side of the bet will be yours
de Finetti showed that the price you set has to obey the axioms of probability or
you face certain loss, i.e. you’ll lose every time.
AI: Probability Theory ! Mar 1, 2007
20
Michael S. Lewicki ! Carnegie Mellon
Axioms of probability
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Axioms (Kolmogorov):
0 ! P(A) ! 1
P(true) = 1
P(false) = 0
P(A or B) = P(A) + P(B) " P(A and B)
•
Corollaries:
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A single random variable must sum to 1:
n
!
P (D = di ) = 1
i=1
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The joint probability of a set of variables must also sum to 1.
If A and B are mutually exclusive:
P(A or B) = P(A) + P(B)
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
Rules of probability
•
conditional probability
P r(A|B) =
•
P r(A and B)
,
P r(B)
P r(B) > 0
corollary (Bayes’ rule)
P r(B|A)P r(A)
⇒ P r(B|A)
AI: Probability Theory ! Mar 1, 2007
= P r(A and B) = P r(A|B)P r(B)
P r(A|B)P r(B)
=
P r(A)
22
Michael S. Lewicki ! Carnegie Mellon
Discrete probability distributions
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•
•
•
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discrete probability distribution
joint probability distribution
marginal probability distribution
Bayes’ rule
independence
AI: Probability Theory ! Mar 1, 2007
Michael S. Lewicki ! Carnegie Mellon
23
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Michael S. Lewicki
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Continuous probability distributions
•
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•
probability density function (pdf)
joint probability density
marginal probability
calculating probabilities using the pdf
Bayes’ rule
AI: Probability Theory ! Mar 1, 2007
31
Michael S. Lewicki ! Carnegie Mellon
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AI: Probability Theory ! Mar 1, 2007
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AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon
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Michael S. Lewicki ! Carnegie Mellon
Next time: The process of probabilistic inference
1. define model of problem
2. derive posterior distributions and estimators
3. estimate parameters from data
4. evaluate model accuracy
AI: Probability Theory ! Mar 1, 2007
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Michael S. Lewicki ! Carnegie Mellon