Formulas for approximating pseudo-Boolean random variables

Authors:
Guoli Ding;R. F. Lax;Jianhua Chen;Peter P. Chen
Affiliations:
Department of Mathematics, LSU, Baton Rouge, LA 70803, USA;Department of Mathematics, LSU, Baton Rouge, LA 70803, USA;Department of Computer Science, 298 Coates Hall, LSU, Baton Rouge, LA 70803, USA;Department of Computer Science, 298 Coates Hall, LSU, Baton Rouge, LA 70803, USA
Venue:
Discrete Applied Mathematics
Year:
2008

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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.