Lecture 4A: Probability Theory Review

Advanced Artificial Intelligence

Outline

• Axioms of Probability• Product and chain rules • Bayes Theorem• Random variables• PDFs and CDFs• Expected value and variance

Introduction

• Sample space - set of all possible outcomes of a random experiment– Dice roll: {1, 2, 3, 4, 5, 6}– Coin toss: {Tails, Heads}

• Event space - subsets of elements in a sample space– Dice roll: {1, 2, 3} or {2, 4, 6}– Coin toss: {Tails}

• Axioms of Probability– for all

examples

• Coin flip– P(H)– P(T)– P(H,H,H)– P(x1=x2=x3=x4)– P({x1,x2,x3,x4} contains more than 3 heads)

Set operations

• Let and • and

• Properties:

• If then

Conditional Probability

• A and B are independent if

• A and B are conditionally independent given C if

Conditional Probability

• Joint probability:

• Product rule:

• Chain rule of probability:

examples

• Coin flip– P(x1=H)=1/2– P(x2=H|x1=H)=0.9– P(x2=T|x1=T)=0.8– P(x2=H)=?

Conditional Probability

• probability of A given B• Example:– probability that school is closed – probability that it snows– probability of school closing if it snows– probability of snowing if school is closed

Conditional Probability

• probability of A given B• Example:– probability that school is closed – probability that it snows

P(A, B) 0.005

P(B) 0.02

P(A|B) 0.25

𝑃 ( 𝐴∨𝐵 )= 𝑃 ( 𝐴∩𝐵 )𝑃 (𝐵 )

Quiz

• P(D1=sunny)=0.9• P(D2=sunny|D1=sunny)=0.8• P(D2=rainy|D1=sunny)=?• P(D2=sunny|D1=rainy)=0.6• P(D2=rainy|D1=rainy)=?• P(D2=sunny)=?• P(D3=sunny)=?

Joint Probability

• Multiple events: cancer, test result

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Has cancer? Test positive? P(C,TP)

yes yes 0.018

yes no 0.002

no yes 0.196

no no 0.784

Joint Probability

• The problem with joint distributions

It takes 2D-1 numbers to specify them!

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Conditional Probability

• Describes the cancer test:

• Put this together with: Prior probability

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Has cancer? Test positive?

P(TP, C)

yes yes

yes no

no yes

no no

Has cancer? Test positive? P(TP, C)

yes yes 0.018

yes no 0.002

no yes 0.196

no no 0.784

Conditional Probability

• We have:

• We can now calculate joint probabilities

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Conditional Probability

• “Diagnostic” question: How likely do is cancer given a positive test?

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Has cancer? Test positive? P(TP, C)

yes yes 0.018

yes no 0.002

no yes 0.196

no no 0.784

Bayes Theorem

• We can relate P(A|B) and P(B|A) through Bayes’ rule:

Bayes Theorem

• We can relate P(A|B) and P(B|A) through Bayes’ rule:

Posterior Probability Likelihood

Normalizing Constant

Prior Probability

Bayes Theorem

• We can relate P(A|B) and P(B|A) through Bayes’ rule:

• can be eliminated with law of total probability:

=

Random Variables

• Random variable X is a function s.t.

• Examples:– Russian roulette: X = 1 if gun fires and X = 0 otherwise and – X = # of heads in 10 coin tosses

Cumulative Distribution Functions

• Defined as such that • Properties:

Probability Density Functions

• For discrete random variables, defined as:

• Relates to CDFs:

• =

Probability Density Functions

• For continuous random variables, defined as:

• Relates to CDFs:

Probability Density Functions

• Example:• Let X be the angular position of freely spinning pointer on

a circle

Probability Density Functions

• Example:• Let X be the angular position of freely spinning pointer on

a circle f(X)

1/2

2

X

Probability Density Functions

• Example:• Let X be the angular position of freely spinning pointer on

a circle f(X)

1/2

2

X

Probability Density Functions

• Example:• Let X be the angular position of freely spinning pointer on

a circle

• for

f(x)

1/2

2

x

F(x)

1

2

x

Probability Density Functions

• Example:• Let X be the angular position of freely spinning pointer on

a circle

• for • =

f(x)

1/2

2

x

F(x)

1

2

x

Expectation

• Given a discrete r.v. X and a function

• For a continuous r.v. X and a function

Expectation

• Expectation of a r.v. X is known as the first moment, or the mean value, of that variable ->

• Properties:• for any constant • for any constant • +

Variance

• For a random variable X, define:

• Another common form:

• Properties:• for any constant • for any constant

Gaussian Distributions

• Let , then

where represents the mean and the variance• Occurs naturally in many phenomena– Noise/error– Central limit theorem