Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Descision theory with Bayesian Networks

Margus Niitsoo,Aivo J¨urgenson

April 22,2009

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Outline

1

Model of descision theory

2

Descision trees and computing them

3

Inﬂuence diagram for one descision

4

Inﬂuence diagrams in general

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Utility

We associate numbers with diﬀerent states of the world

Better states get higher numbers

That number is called the Utility

The theory is that maximal expected utility is most desirable.

Some situations are hard to model

For instance - Avoiding a disaster

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Where do we get utilities

Out of thin air

Something objective (money,grades)

The gamblers way!

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Axioms of two true Gamblers

The axioms of rational choice by Neumann and Morgenstern:

Let A,B,C,D be diﬀerent distributions of world outcomes

(lotteries).Assume we have a partial ordering ≥ on all possible

outcome distributions that satisﬁes

Reﬂexivity,Completeness,Transitivity

Preference increasing:pA+(1 −p)B ≥ qA+(1 −q)B iﬀ

p ≥ q

Continuity:A ≥ B ≥ C means there exists a q such that

B ≡ pA+(1 −p)B.

Independence:C = pA+(1 −p)B and C ≡ D imply

D ≡ pA+(1 −p)B

If the set of atoms is ﬁnite,this guarantees a utility function exists.

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Descision trees

Parts

Descision nodes

Chance nodes

Utility leaves

History is known!

Exponential growth

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Descision trees

Parts

Descision nodes

Chance nodes

Utility leaves

History is known!

Exponential growth

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Descision trees

Parts

Descision nodes

Chance nodes

Utility leaves

History is known!

Exponential growth

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Descision trees

Using Bayesian Networks

Solution concept - Strategy

Calculating the tree

Coalescing

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Inﬂuence diagrams

Get rid of the tree

Introduce extra nodes to BN

Calculate!

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

In general

Descisions aﬀect one another

Descision sequence ﬁxed!

No Forgetting!

Partitions whole BN into sets

The concept of a solution - Policy

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

In general

Descisions aﬀect one another

Descision sequence ﬁxed!

No Forgetting!

Partitions whole BN into sets

The concept of a solution - Policy

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

In general

Descisions aﬀect one another

Descision sequence ﬁxed!

No Forgetting!

Partitions whole BN into sets

The concept of a solution - Policy

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

In general

Descisions aﬀect one another

Descision sequence ﬁxed!

No Forgetting!

Partitions whole BN into sets

The concept of a solution - Policy

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

Model of descision theory

Descision trees and computing them

Inﬂuence diagram for one descision

Inﬂuence diagrams in general

Questions anyone?

Aivo will continue next week with generalizations

Margus Niitsoo,Aivo J¨urgenson

Descision theory with Bayesian Networks

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