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2008

Formulas for approximating pseudo-Boolean random variables

10 years 23 days ago
Formulas for approximating pseudo-Boolean random variables
We consider {0, 1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Hammer-Holzman and Grabisch-Marichal-Roubens. We also derive a formula for the best faithful linear approximation that extends a result due to Charnes-Golany-Keane-Rousseau concerning generalized Shapley values. We show that a theorem of Hammer-Holzman that states that a pseudo-Boolean function and its best approximation of degree at most k have the same derivatives up to order k does not generalize to this setting for arbitrary probability measures, but does generalize if the probability measure is a product measure. Key words: pseudo-Boolean function, probability measure, pseudo inner product, li...
Guoli Ding, Robert F. Lax, Jianhua Chen, Peter P.
Added 26 Dec 2010
Updated 26 Dec 2010
Type Journal
Year 2008
Where DAM
Authors Guoli Ding, Robert F. Lax, Jianhua Chen, Peter P. Chen
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