Uncertainty and Reasoning

A first look

March 9, 2006

In case you missed it

…

•

Coping with uncertainty is essential

•

Modern AI has turned

probabilistic

•

Russell and

Norvig

has good coverage

Chapter 13 covers basics;

it is essential

that you

read all of it (the

Wumpus

World in Ch. 13.7 is

optional)

Chapter 14 talks about reasoning; please read this

along with and the

D

’

Ambrosio

and

Charniak

;

skip all discussions of continuous distributions

;

skip 14.5, 14.6 as well

Probability Theory Basics

•

Basic notions today

•

Get used to manipulating probabilities and

conditional probabilities

•

Basics of Bayesian Networks

Probability Basics:

Conditional Probability

Probability Basics:

Joint Probability and Marginal Probability

Variables, Events and Distributions

An Exercise in Stereotyping

Difficulties with Probabilistic Reasoning

Worked Example: Malignancy

Bayes

’

Theorem

P(A|B) =

P(B|A) P(A)

P(B)

Basics of Bayesian Reasoning:

Bayes

’

Rule: Combining Diagnostic and

Causal Reasoning

Prior = .01%

P(S)=P(S | M)P(M)+P(S | not M)P(not M)

Normalisation of posteriors

80% of people with

Meningitis have a stiff neck

Normalisation of Posteriors

Numbers not percentages

Aiding Bayesian Reasoning with Frequencies

Formal Result indicating why Probabilities

are

not

Rules of Varying Strength

Why?

Exploiting Conditional Independence 1:

Joint Distributions in terms of Conditionals

Storage: Exponential in

number of random variables

Bayesian Network Semantics

Network structure to encode (conditional)

independencies

Between variables and this

makes representation compact.

Bayesian Network Semantics - 2

Conditional Independence in a Bayesian Network

P(A,B|C)=P(A|C)P(B|C)

A is conditionally

i

ndependent of B given C

Or, P(A|B,C)=P(A|C)

No edge ~

Independence!

Basics of Bayesian Networks

•

Nodes are random variables.

•

Informally, edges represent

“

causation

”

(no directed cycles

allowed - graph is a DAG).

•

Formally, local Markov property

says: node is conditionally

independent of its non-descendants

{Z} given its parents {U}.

•

Notation:

X

⊥

{Z

1j

…

Z

nj

} | {U

1

…

U

m

}

## Comments 0

Log in to post a comment