Document Type
Article
Publication Date
5-28-2008
Abstract
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. © 2007 Elsevier B.V. All rights reserved.
Publication Source (Journal or Book title)
Discrete Applied Mathematics
First Page
1581
Last Page
1597
Recommended Citation
Ding, G., Lax, R., Chen, J., & Chen, P. (2008). Formulas for approximating pseudo-Boolean random variables. Discrete Applied Mathematics, 156 (10), 1581-1597. https://doi.org/10.1016/j.dam.2007.08.034