Seminar Outline
TDDC17
Seminar 7
Reasoning with Uncertainty
Bayesian Networks
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Propositional Logic and Models
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
•
Basic Probability Theory from a logical
perspective.
•
Bayesian Networks
•
An “efficient” means of doing probabilistic
reasoning.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
DNF Characterization of Models
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Degrees of Truth/Belief
A Language of Probability
Beliefs about Propositions
Degree of Belief
Propositions
Degree of Truth
World
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Probability Distributions
Factored representations like that used with CSPs
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Joint Probability Distributions
P Notation
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Full Joint Probability Distribution
An Example
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Conditional Probability
Some (P) Notation
From the product rule,
we may derive Bayes Rule!
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Kolmogorov’s Axioms
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Some Useful Properties
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Marginalization
Some Examples
catch
= 0.2
General
Marginalization
Rule
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Conditionalization
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Conditional Probability Distribution
Computing Conditionals
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
A General Inference Procedure
0.12
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
An Example
Comments
0.12
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Independence (1)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Independence (2)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Factoring
Absolute Independence
Independence assertions can both reduce the size
of the domain representation and make the inferencing
problem more efficient.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Baye’s Rule (Simple Case)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Intuitions
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
An Example
Generalizations: Normalized Form
P(X)
Y X
denominator of
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Many Pieces of Evidence
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
The Chain Rule
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Conditional Independence
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Naive Bayes’ Model (2)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Naive Bayes’ Model (1)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Comments
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Bayesian Networks
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Bayesian Networks
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Bayesian Network
Example (J. Pearl)
•
•
•
A person installs a new burglar alarm at home. It responds to
burglaries, but may also respond to earthquakes on occasion.
P(B)!
.001!
Burglary
P(E)!
Earthquake
.002!
The person has two neighbors, John and Mary, who promise to call
you at work when the alarm goes off.
•
John always calls when he hears the alarm, but sometimes
confuses the telephone ringing with the alarm sound.
•
Mary, who likes loud music sometimes misses the alarm
altogether
Alarm
•
Given evidence of who has or has not called, estimate the
probability of a burglary,
P(burglary | john, ~mary)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
E!
P(a)!
T!
T!
.95!
T!
F!
.94!
F!
T! € .29!
F!
.001!
F!
P(a | b ∧ e) = .95
P(a |¬b ∧ e) = .29
€
Queries
•
B!
A! P(J)!
T!
.90!
F!
.05!
JohnCalls
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
MaryCalls
A!
P(M)!
T!
.70!
F!
.01!
Semantics of Bayesian Networks
Constructing Bayesian Networks (1)
A bayesian network
is a representation of the
joint probability distribtuion
nor
m
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Constructing Bayesian Networks (2)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Exact Inference in Bayesian Networks
The main task in any probabilistic inference system is to compute the posterior
probability distribution of a set of query variables, given some observed event
(assignment of values to set of evidence variables)
Causes precede effects
other predecessors in the node ordering,
given its parents
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
We know that the terms P(X, e, y) in the joint distribution can be written as
products of conditional probabilities from the network. So, a query is answered
by computing the sums of products of conditional probabilities from the network.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
An Example
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden