Belief Formation Itzhak Gilboa Tel Aviv University and HEC, Paris - - PowerPoint PPT Presentation

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Analogies and Theories in Belief Formation

Itzhak Gilboa – Tel Aviv University and HEC, Paris ISIPTA 2015 Joint works of subsets of

• A. Billot, G. Gayer, I. Gilboa, O. Lieberman, A.

Postlewaite, D. Samet, L. Samuelson, D. Schmeidler

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Background

• Classics:

– Ramsey (1926), de Finetti (1931,7) – von-Neumann-Morgenstern (1944) – Savage (1954) – Anscombe-Aumann (1963)

• Problems:

– Descriptive – Normative

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Background – cont.

• Alternative theories

– Schmeidler (1989) Choquet EU – G-Sch (1989) Maxmin EU – Klibanoff, Marinacci, Mukerji (2005) (Nau, Seo…) “Smooth Model” – Maccheroni, Marinacci, Rustichini (2006) “Variational Preferences”

• Still the “black box” paradigm

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Background – cont.

• Case-Based Decision Theory

– (w/ Schmeidler, Theory of Case Based Decisions, CUP 2001)

• Probabilities from cases

– (w/ Schmeidler and others, Case-Based Prediction, World Scientific 2012)

• Analogies and Theories

– (w/ Samuelson, Schmeidler and others, Analogies and Theories, OUP, 2015)

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Statistics and Psychology

• This project touches on both
• And we found ourselves axiomatizing

known formulae

• Surprisingly, known in both domains

– Which goes beyond this project – Sometimes, even the mistakes

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Probabilities from Cases: Similarity-weighted frequencies

The data: where and We are asked about the probability that for a new data point

  n

i i m i i

y x x



, ,...,

1

 

m p p

x x ,...,

1

 

m m i i

x x   ,...,

1

 

1 , 

i

y 1 

p

y

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Similarity-weighted frequencies – Formula (Kernel)

Choose a similarity function Given observations and a new data point estimate by

  n

i i m i i

y x x



, ,...,

1

 

 



n i p i n i i p i s p

x x s y x x s y ) , ( ) , (

 

m p p

x x ,...,

1

) 1 ( 

p

y P

 

    

m m

s :

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Similarity-weighted frequencies – Interpretation

• Special cases of

– If is constant: an estimate of the expectation (in fact, “repeated experiment” is always a matter of subjective judgment of equal similarity) – If : an estimate of the conditional expectation

• Useful when precise updating leaves us with a sparse

database

• Akin to interpolation
• But not to extrapolation!

s

 

 



n i p i n i i p i s p

x x s y x x s y ) , ( ) , (

 

 

p i

x x p i x

x s



1 ,

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Axiomatization – Setup

• bservations (case types)

A database is a multi-set of observations We will refer to a database as a sequence or a multi-set interchangeably.

1 

 

m

M



 Z M I :

     

y x x y x x

m m

, ,..., , ,...,

1 1



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Axiomatization I: Observables

• A state space
• Fix a new data point
• Databases
• A probability assignment function

 

s ,..., 1  



 Z M I :

), ( :    I I p 

 

m m p p

x x   ,...,

1

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database J 5 12 . . . 3 4 6 . . . 8 database I + J 3 2 1 Δ(Ω)

. .

9 18 . . . 11

+

The combination axiom

database I 1 2 . . . m case types M p(I) p(J) States of the world Ω = {1,2,3,…,s} p(I + J).

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The combination axiom

• Formally

for some

) ( ) 1 ( ) ( ) ( J p I p J I p      

1   

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Theorem I

• The combination axiom holds, and not all

are collinear if and only if

• For each

there are , not all collinear, and such that

– In “Probabilities as Similarity-Weighted Frequencies” w/ Billot, Samet, Schmeidler

 

I

I p ) (

M c

) (  

c

p 

c

s

 

 



M c c M c c c

s c I p s c I I p ) ( ) ( ) (

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The perspective

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For case 2 3 2 1 Δ(Ω)

. . .

p1 p2 p3

Probability

• f states

Probability = Frequency in perspective Frequency

• f cases

s1p1 s3p3 s2p2

. . .

3 2 1

.

I

.

F = (F1, F2, F3) For case 1 For case 3

.

F1s1p1 + F2s2p2 + F3s3p3

.

p(F) = p(I)

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Theorem II

Some axioms hold iff there exists a function such that ranks values by their proximity to where and The function is unique up to multiplication by

• In “Empirical Similarity” w/Lieberman and Schmeidler

 

    

m m

s :

I



 

 



n i p i n i i p i s p

x x s y x x s y ) , ( ) , (

   

n i i m i i

y x x I

1 1

, ,...,





 

m i i i

x x x ,...,

1



s

 

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Theorem III

Some additional axioms hold iff there exists a norm such that

• Satisfies “multiplicative transitivity”:
• In “Exponential Similarity” w/ Billot and Schmeidler



  m n:

 

) (

,

z x n

e z x s

 



     

z y s y x s z x s , , , 

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The Similarity – whence?

– In “Empirical Similarity” w/Lieberman and Schmeidler we propose an empirical approach: – Estimate the similarity function from the data – A parametrized approach: Consider a certain functional form – Choose a criterion to measure goodness of fit – Find the best parameters

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A functional form

• Consider a weighted Euclidean distance

and

2 1

) ( ) , (

tj ij m j j t i w

x x w x x d   



) , (

) , (

t i w

x x d t i w

e x x s





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Selection criteria

• Find weights that would minimize
• Or: round off

to get a prediction

– and then minimize

 

2

ˆ





i i i

y y

i

y ˆ

 

1 , 

p i

y

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How objective is it?

• Modeling choices that can affect the

“probability”:

– Choice of X’s and of sample – Choice of functional form – Choice of goodness of fit criterion

• As usual, objectivity may be an unattainable

ideal

• But it doesn’t mean we shouldn’t try.

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Statistical inference

– In “Empirical Similarity” w/Lieberman and Schmeidler we also develop statistical inference tools for our estimation procedure – Assume that the data were generated by a DGP of the type – Estimate the similarity function from the data – Perform statistical inference

t t i t i t i i t i t

X X s Y X X s Y P    

 

 

) , ( ) , ( ) 1 (

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Statistical inference – cont.

• Estimate the weights by maximum

likelihood

• Test hypotheses of the form
• Predict out-of-sample by the maximum

likelihood estimators (via the similarity- weighted average formula) : 

j

w H

j

w

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Failures of the combination axiom

• Integration of induction and deduction

– Learning the parameter of a coin – Linear regression Limited to case-to-case induction, generalizing empirical frequencies

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Failures of the combination axiom – cont.

• Second order induction

– Learning the similarity function In particular, doesn’t allow the similarity function to get more concentrated for large databases Combination restricted to periods of “no learning”.

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Combining Theories and Analogies

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Learning in the Model

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Modes of Reasoning

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Dynamics of Reasoning

• Under mild assumptions that mean that

– The reasoner doesn’t know the nature of the process – The reasoner is “open-minded”

• The reasoner converges away from

Bayesian reasoning

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Example

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Example – cont.