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(AB) =
P(BA) 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,BC)=P(AC)P(BC)
A is conditionally
i
ndependent of B given C
Or, P(AB,C)=P(AC)
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 nondescendants
{Z} given its parents {U}.
•
Notation:
X
⊥
{Z
1j
…
Z
nj
}  {U
1
…
U
m
}
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