axioms

PMR 2728 / 5228Probability Theory in AI and

Robotics

Subjectivist AxiomsFabio G. Cozman - Ofce MS08

PMR 2728 / 5228Probability Theory in AI and Robotics Subjectivist Axioms p.1/36

Foundations: decision making

First step is to define acts.1. An act is a function X :

Utility theory

1. Preferences amongst consequences2. In fact, preferences amongst lotteries3. Basic lottery: (r1, , r2) yields combination of r1 and r2

with weight 4. Obvious interpretation:

(r1, , r2) = r1 + (1 )r2.

5. That is, lotteries are convex combinations


von Neumann-Morgenstern theorem

Theorem 1. SupposeR is a set of lotteries, and is a binary relationamong lotteries such that:

1. the relation is complete and transitive;2. if l1 l2 and (0, 1), then (l1, , l3) (l2, , l3);3. if l1 l2 l3, then there exists , (0, 1) such that

(l1, , l3) l2 (l1, , l3);then there is a function u : R < such that

l1 l2 u(l1) u(l2),

u((l1, , l2)) = u(l1) + (1 )u(l2)

and u is unique up to a linear transformation au + b with a > 0.


What does it show??1. A rational agent has a utility function u2. Utility is unique up to linearity3. Decision making without uncertainty:

X1 10X2 0X3 5

X1 dominates the others, should be selected


Preferences amongst acts

Consider several acts:

1 . . . j . . . n

X1 c11 c1j c1n...

Xi ci1 cij cin...

Xm c1m cmj cmn

How to compare them?


Example

Take a ball from an urn.Urn with 10 red balls, 20 yellow balls, 30 black balls.

Red Yellow BlackX1 100 0 0X2 100 100 100X3 0 0 100X4 0 100 100


Ellsbergs example

Take a ball from an urn.Urn with 30 red balls and 60 yellow/black balls.

Red Black YellowX1 100 0 0X2 0 100 0X3

X4


Ellsbergs example


Red Black YellowX1 100 0 0X2 0 100 0X3 100 0 100X4 0 100 100


Rules of the game

1. There is a relation amongst acts2. Axioms on :

Partial Order is reflexive and transitive.Independence/Sure-thing X Y iff Z, (0, 1),

X + (1 )Z Y + (1 )Z.

Dominance If X() > Y () for all , then

X Y and not Y X.

Continuity If Xi X and Y Xi Z then Y X Z.


Transitivity


Red Yellow BlackX1 100 0 0X2 0 100 0X3 0 0 100


Independence/Sure-thing




Dominance




Back to Ellsbergs example


Red Black YellowX1 100 0 0X2 0 100 0X3 100 0 100X4 0 100 100


Direct consequences of axioms:

1. X Y iff X Y 0.2. X 0 and Y 0 imply X + Y 0.3. X 0 then X 0 for > 0.4. X Y then X Y for > 0.

That is, there is a cone of desirable acts.Note that if X 0, then 0 X.


Cones1. There is a cone of desirable acts.2. There is a cone of undesirable acts.3. There is a region of indecision (if X is there, X is

there).

How do we represent such cones?


Main theoremTheorem 2 (Giron and Rios 1980). If satisfies the axioms, then thereis a credal set K such that:

X Y P K : EP [X] EP [Y ]

Moreover, the maximal credal set is convex.


Example

1. Binary possibility space2. X = [1, 2]3. Y = [3, 0]4. X Y5. Then E[X] E[Y ] implies

1p1 + 2p2 3p1

p2 p1,

and also p1 + p2 = 1.


Continuing...

1. X = [1, 2], Y = [3, 0]2. X Y ; then p2 p1.3. 5Y/3 X4. Then 5E[Y ] /3 E[X] implies

5p1 1p1 + 2p2

p2 2p1,

and also p1 + p2 = 1.


Geometry

1. A general assessment on the expectation for a variablerepresents a hyperplane on desirable acts.

2. Each expectation produces a constraint on probabilitymeasures.

3. A convex set of probability measures represents a setof desirable acts.


Completeness

1. Additional axiom: is complete (either X Y or Y Xor both.

2. Then:(a) Suppose we have a credal set with P1 6= P2

representing .(b) Suppose EP1 [X] > EP2 [X].(c) Take = (EP1 [X] + EP2 [X])/2.(d) Then EP1 [X ] = (EP1[X] EP2[X])/2 > 0(e) Then

EP1[X ] > 0 > EP2 [X ]

and this contradicts completeness (neither X nor X).


Bayesian theory: Basic theorem

Theorem 3. Preferences satisfying partial order, independence,dominance, continuity and completenessare represented by a single probability measure.


Other axioms for Bayesian theory

1. Savages theory: obtain utility and probability fromaxioms.

2. Anscombe/Aumanns: obtain utility and probability,assuming chances but single ordering .

3. Dutch book arguments (Ramsey, de Finetti)

(Also, some more general theories; SSK95, etc.)


Dutch book: basics

1. Theorem of the alternative: basic result in linearalgebra, with many versions.

2. Consequence of the theorem of the alternative:Theorem 4. A is a matrix, p a vector, a vector. Then exactly oneof the following systems has a solution:

Ap = 0, p 0,

i

pi = 1.

A < 0.

3. Suppose each row of A is (Xi i).4. Now we interpret: there is a probability measure p such

that E[Xi i] = 0 iff A < 0 has no solution.


Dutch book

1. So, when the i fail expectation axioms, there is no p,then

[1 . . . n]

X1 1...

Xn n

< 0

has a solution.2. It means that it is possible to form an act that will be

always against you...3. If one can create such an act, one can make a Dutch

book you always lose the game!


Dutch book: ConclusionEither:you have a probability measure representing yourpreferencesorsomeone will create a Dutch book against you (and you willbe miserable)


Conditional preferences

1. For event B define B as

X B Y XB +Z(1B) Y B +Z(1B), any Z.

2. Given independence axiom, Z can be suppressed:XB Y B is enough.

3. Thus, if B(X Y ) 0 then X Y is desirable given B.4. The relation B satisfies all axioms ONLY IF P(B) > 0!!


Conditional expectations

1. If P(B) > 0, then

X B Y B(X Y ) 0 E[B(X Y )] 0.

2. Then

E[XB] E[Y B] E[X|B] P(B) E[Y |B] P(B)

and then

X B Y E[X|B] E[Y |B] .

That is, conditional expectations represent conditional pref-erences!


A few advanced issues

Conditioning can be defined even on events ofprobability zero.Binary preferences lead to convexity; more generalcredal sets can be produced with general choice.

We will not look into these issues any further in this course.


Problem with zero probabilities

1. Suppose conditioning event B with probability zero.2. Then E[XB] E[Y B] for any X, Y !3. Now suppose (XB)() > (Y B)() for every B; this

violates dominance for B!!

There are theories that allow conditioning even if P(B) = 0.


Universal conditioning

1. Instead of dening

P(A|B) =P(A B)

P(B),

we adoptP(A B) = P(A|B)P(B)

as an additional axiom.2. Now B can have probability zero.3. de Finettis theory, also known as Popper measure

theory; popular in some specialized fields, but notwidely used.


Convexity?

1. So far, only binary choices.2. Maximal credal sets are convex.3. But convexity is not really required.4. Are there decisions that move away from convexity?


Sets of acts1. Consider 4 acts in a binary possibility space:

W = [2/5, 2/5],X = [0, 1],Y = [1, 0],Z = [0, 0].

2. Assume P(1) [1/4, 3/4].3. Note that Z is never the best.4. So, Z is dominated and discarded.5. How about the others?6. All are undominated (named maximal).7. Only X and Y can be optimal (E-admissible).


Walleys theorem

Theorem 5. Suppose set of acts is convex. Then all maximal acts areE-admissible.


Seidenfelds axiom1. Now suppose we require that decisions should not

change if the set of acts is convexified.2. Then, every decision must be an E-admissible

decisions!


General credal sets1. With Seidenfelds axiom, we can use sets of acts to

differentiate between any two credal sets!2. That is, any credal set represents a unique pattern of

preferences...3. This is the most general theory that has a behavioral

interpretation (that is, a theory that really makes anysense).


Foundations: decision makingUtility theoryvon Neumann-Morgenstern theoremWhat does it show??Preferences amongst actsExampleEllsberg's exampleEllsberg's exampleRules of the gameTransitivityIndependence/Sure-thingDominanceBack to Ellsberg's exampleDirect consequences of axioms:ConesMain theoremExampleContinuing...GeometryCompletenessBayesian theory: Basic theoremOther axioms for Bayesian theoryDutch book: basicsDutch bookDutch book: ConclusionConditional preferencesConditional expectationsA few advanced issuesProblem with zero probabilitiesUniversal conditioningConvexity?Sets of actsWalley's theoremSeidenfeld's axiomGeneral credal sets

axioms

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