Kuznetsov independence for interval-valued expectations and sets of probability distributions: properties and algorithms

Fabio G. Cozman; Cassio P. de Campos

doi:10.1016/j.ijar.2013.09.013

Kuznetsov independence for interval-valued expectations and sets of probability distributions: properties and algorithms

Fabio G. Cozman
, Cassio P. de Campos

Research output: Contribution to journal › Article › Academic › peer-review

10 Citations (Scopus)

Abstract

Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y), E[f(X)g(Y)]=E[f(X)] E[g(Y)], where E[×] denotes interval-valued expectation and denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.

Original language	English
Pages (from-to)	666-682
Number of pages	17
Journal	International Journal of Approximate Reasoning
Volume	55
Issue number	2
DOIs	https://doi.org/10.1016/j.ijar.2013.09.013
Publication status	Published - 2014
Externally published	Yes

Funding

Thanks to Peter Walley for bringing our attention to Kuznetsovʼs work and for showing that many functions of two binary variables can be written as decomposable functions, thus suggesting Proposition 1 . Thanks to Serafín Moral for ideas concerning Kuznetsov independence and extensions; to Lev Utkin for generously translating pieces of Kuznetsovʼs book to English; to Igor Kozine for translating additional material from Kuznetsovʼs book to English; and to Enrique Miranda for clarifying results concerning many-to-one independent products. Thanks to David Avis for freely distributing the lrs package (used in the examples). We thank the reviewers for detailed reviews that led to substantial improvements to this paper. The first author has been partially supported by CNPq , and this work has been supported by FAPESP through grant 04/09568-0 . The second author has been partially supported by the Hasler Foundation grant no. 10030 .

Keywords

Graphoids
Independence concepts
Lower expectations
Probability and expectation intervals
Sets of probability distributions

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10.1016/j.ijar.2013.09.013

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abstract = "Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y), E[f(X)g(Y)]=E[f(X)] E[g(Y)], where E[×] denotes interval-valued expectation and denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.",

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T1 - Kuznetsov independence for interval-valued expectations and sets of probability distributions

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AU - Cozman, Fabio G.

AU - de Campos, Cassio P.

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N2 - Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y), E[f(X)g(Y)]=E[f(X)] E[g(Y)], where E[×] denotes interval-valued expectation and denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.

AB - Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y), E[f(X)g(Y)]=E[f(X)] E[g(Y)], where E[×] denotes interval-valued expectation and denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.

KW - Graphoids

KW - Independence concepts

KW - Lower expectations

KW - Probability and expectation intervals

KW - Sets of probability distributions

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DO - 10.1016/j.ijar.2013.09.013

M3 - Article

SN - 0888-613X

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JO - International Journal of Approximate Reasoning

JF - International Journal of Approximate Reasoning

IS - 2

ER -

Kuznetsov independence for interval-valued expectations and sets of probability distributions: properties and algorithms

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Keywords

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