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

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.

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)

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.

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?

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”

Political futures market predicted vs actual outcomes

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.

Probabilistic Methodology

0, 1, or 2 parachutes open?

type of collision

prevailing wind direction

atic U

Probabilistic assessment of dangerous climate change

from Forrest et al (2001)

from Mastrandrea and Schneider (2004)

Factoring in Risk Using Decision Theory

Step 1Step 1

P(“DAI” = 55.8%)

Step 2Step 2

P(“DAI” = 27.4%Carbon Tax 2050 = $174/Ton

Dangerous Climate Change

Uncertainty in vision: What are these?

Uncertainty in vision

Edges are not as obvious they seem

An example from Antonio Torralba

What’s this?

We constantly use other information to resolve uncertainty

Image interpretation is heavily context dependent

This phenomenon is even more prevalent in speech perception

• 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.

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.

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)

P (D = di) = 1

Rules of probability

• conditional probability

• corollary (Bayes’ rule)

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)

Discrete probability distributions

• discrete probability distribution

• joint probability distribution

• marginal probability distribution

• Bayes’ rule

• independence

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Continuous probability distributions

• probability density function (pdf)

• joint probability density

• marginal probability

• calculating probabilities using the pdf

• Bayes’ rule

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• It does not mean a probability!

• First of all, it’s not a value between 0 and 1.

• 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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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

probability theor y & uncer tainty

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