Probability Theor y & Uncer tainty
Transcript of Probability Theor y & Uncer tainty
AI: 15-780 / 16-731Mar 1, 2007
Probability Theory & UncertaintyRead Chapter 13 of textbook
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
What you will learn today
• 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
- discrete and continuous probability
- joint and marginal probability
- calculating probability
• next probability lecture: the process of probabilistic inference
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 3
What is the role of probability and inference in AI?
• Many algorithms are designed as if knowledge is perfect, but it rarely is.
• There are almost always things that are unknown, or not precisely known.
• Examples:
- bus schedule
- quickest way to the airport
- sensors
- joint positions
- finding an H-bomb
• An agent making optimal decisions must take into account uncertainty.
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Probability as frequency: k out of n possibilities
• Suppose we’re drawing cards from a standard deck:
- P(card is the Jack ♥ | standard deck) = 1/52
- P(card is a ♣ | standard deck) = 13/52 = 1/4
• What’s the probability of a drawing a pair in 5-card poker?
- 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)
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 5
Making rational decisions when faced with uncertainty
• Probability
the precise representation of knowledge and uncertainty
• Probability theory
how to optimally update your knowledge based on new information
• Decision theory: probability theory + utility theory
how to use this information to achieve maximum expected utility
• Consider again the bus schedule. What’s the utility function?
- 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.
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Probability of uncountable events
• How do we calculate probability that it will rain tomorrow?
- 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?
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
The Iowa Electronic Markets: placing probabilities on single events
• http://www.biz.uiowa.edu/iem/
• “The Iowa Electronic Markets are real-money futures markets in which contract payoffs depend on economic and political events such as elections.”
• Typical bet: predict vote share of candidate X - “a vote share market”
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Political futures market predicted vs actual outcomes
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
John Craven and the missing H-Bomb
• In Jan. 1966, used Bayesian probability and subjective odds to locate H-bomb missing in the Mediterranean ocean.
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Probabilistic Methodology
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0, 1, or 2 parachutes open?
type of collision
prevailing wind direction
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
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Probabilistic assessment of dangerous climate change
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from Forrest et al (2001)
from Mastrandrea and Schneider (2004)
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Factoring in Risk Using Decision Theory
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Step 1Step 1
P(“DAI” = 55.8%)
Step 2Step 2
P(“DAI” = 27.4%Carbon Tax 2050 = $174/Ton
Dangerous Climate Change
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Uncertainty in vision: What are these?
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Uncertainty in vision
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Edges are not as obvious they seem
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
An example from Antonio Torralba
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What’s this?
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
We constantly use other information to resolve uncertainty
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Image interpretation is heavily context dependent
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
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.
• 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.
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
De Finetti’s definition of probability
• 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
- 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.
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 21
Axioms of probability
• 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:
- A single random variable must sum to 1:
- 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)
n∑
i=1
P (D = di) = 1
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Rules of probability
• conditional probability
• corollary (Bayes’ rule)
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Pr(A|B) =Pr(A andB)
Pr(B), P r(B) > 0
Pr(B|A)Pr(A) = Pr(A andB) = Pr(A|B)Pr(B)
⇒ Pr(B|A) =Pr(A|B)Pr(B)
Pr(A)
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Discrete probability distributions
• discrete probability distribution
• joint probability distribution
• marginal probability distribution
• Bayes’ rule
• independence
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 24
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All the nice looking slides like this one from now on are from Andrew Moore.
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007
Continuous probability distributions
• probability density function (pdf)
• joint probability density
• marginal probability
• calculating probabilities using the pdf
• Bayes’ rule
31
Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 32
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 34
What does p(x) mean?
• It does not mean a probability!
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• It’s just a value, and an arbitrary one at that.
• The likelihood of p(a) can only be compared relatively to other values p(b)
• It indicates the relative probability of the integrated density over a small delta:
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Michael S. Lewicki ! Carnegie MellonAI: Probability Theory ! Mar 1, 2007 63
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