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Comonotone lower probabilities for bivariate Introduction and - - PowerPoint PPT Presentation

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Comonotone lower probabilities for bivariate Introduction and discrete structures Comonotonicity for lower probabilities Building Ignacio Montes and Sebastien Destercke comonotone lower probabilities Conclusions University of Oviedo,

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone lower probabilities for bivariate and discrete structures

Ignacio Montes and Sebastien Destercke

University of Oviedo, Spain Technologic University of Compiegne, France

ISIPTA 2015, July 24

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

About the authors... S. Destercke

https://www.hds.utc.fr/ sdesterc/dokuwiki/fr/accueil

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

About the authors... I. Montes

http://unimode.uniovi.es

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Overview

Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Overview

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Motivation

Multivariate distributions take into account the dependence between the variables. Possible dependence between the variables: independence, comonotonicity, countermonotonicity, . . . What happens when we have imprecise information? Independence for imprecise probabilities has already been studied. And what about comonotonicity?

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Overview

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone random variables

Let (X, Y ) be a random vector defined on X × Y, and denote by PX,Y its probability distribution. PX,Y is called comonotone when the following equivalent conditions hold: For any (x, y) ∈ X × Y, either P(X ≤ x, Y > y) = 0 or P(X > x, Y ≤ y) = 0. The support of PX,Y is an increasing subset of R2. FX,Y(x, y) = min(FX(x), FY(y)) for any (x, y).

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone random variables

Let (X, Y ) be a random vector defined on X × Y, and denote by PX,Y its probability distribution. PX,Y is called comonotone when the following equivalent conditions hold: For any (x, y) ∈ X × Y, either P(X ≤ x, Y > y) = 0 or P(X > x, Y ≤ y) = 0. The support of PX,Y is an increasing subset of R2. FX,Y(x, y) = min(FX(x), FY(y)) for any (x, y). ω1 ω2 ω3 ω4 X 2 2 3 1 Y 1 2 2 1

1 2 1 2 3

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone lower probabilities

Assume that P : P(X × Y) → [0, 1] is a lower probability mod- elling the imprecise knowledge about PX,Y. We say that P is comonotone when any P ∈ M(P) is comonotone.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone lower probabilities

Assume that P : P(X × Y) → [0, 1] is a lower probability mod- elling the imprecise knowledge about PX,Y. We say that P is comonotone when any P ∈ M(P) is comonotone. Example: Consider X = {x1, x2} and Y = {y1, y2} (x1 < x2 and y1 < y2). If P({(x1, y2)}) = 0, any P ∈ M(P) is comonotone:

x1 x2 y1 y2

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Characterizations

Comonotone lower probabilities supp(P) is an increasing set ∀(x, y), either P({(u, v) : u > x, v ≤ y}) = 0 or P({(u, v) : u ≤ x, v > y}) = 0. F(x, y) = min(F X(x), F Y(y)) F(x, y) = min(F X(x), F Y(y)). Any P ∈ M(P) is comonotone.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Overview

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Building comonotone lower probabilities

Theorem

Let BelX and BelY be two belief function with nested focal el- ements. Then, it is always possible to build a joint coherent comonotone lower probability with these fixed marginals, and in fact, it is also a belief function with nested focal elements.

Theorem

Let BelX and BelY be two belief functions with focal sets A1, . . . , An and B1, . . . , Bn such that Ai = [ai, ai] and Bi = [bi, bi] are in- tervals for i = 1, . . . , n and mX(Ai) = mY(Bi). Then, there exists a joint coherent comonotone lower probability with these fixed marginals. Furthermore, it is a belief function.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Example

y1 y2 y3 x1 x2 x3 mX({x2}) = 0.6, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.8.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Example

y1 y2 y3 x1 x2 x3 mX({x2}) = 0.6, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.8. mX({x2}) = 0.2, mX({x2}) = 0.4, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.4, mY({y1, y2, y3}) = 0.4.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Example

y1 y2 y3 x1 x2 x3 mX({x2}) = 0.6, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.8. mX({x2}) = 0.2, mX({x2}) = 0.4, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.4, mY({y1, y2, y3}) = 0.4.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Example

y1 y2 y3 x1 x2 x3 mX({x2}) = 0.6, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.8. mX({x2}) = 0.2, mX({x2}) = 0.4, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.4, mY({y1, y2, y3}) = 0.4.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Example

y1 y2 y3 x1 x2 x3 mX({x2}) = 0.6, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.8. mX({x2}) = 0.2, mX({x2}) = 0.4, mX({x1, x2, x3}) = 0.4. mY({y1, y2}) = 0.2, mY({y1, y2, y3}) = 0.4, mY({y1, y2, y3}) = 0.4.

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Non-uniqueness

The joint comonotone belief function may not be unique!!

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Non-uniqueness

The joint comonotone belief function may not be unique!!

y1 y2 x1 x2

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Non-uniqueness

The joint comonotone belief function may not be unique!!

y1 y2 x1 x2

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Non-uniqueness

The joint comonotone belief function may not be unique!!

y1 y2 x1 x2

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Overview

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Highlights

This work investigates the notion of comonotonicity for lower probabilities. Definition, characterizations and properties. Not all marginals allow to define a joint comonotone coher- ent lower probability. It does for particular cases of belief functions. Comonotonicity for bivariate p-boxes: too much restrictive. All our results can be extended to countermonotonicity.

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Introduction Comonotonicity for lower probabilities Building comonotone lower probabilities Conclusions

Comonotone lower probabilities for bivariate and discrete structures

Ignacio Montes and Sebastien Destercke

University of Oviedo, Spain Technologic University of Compiegne, France

ISIPTA 2015, July 24

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