Transforms of pseudo-Boolean random variables

Abstract

a b s t r a c tAs in earlier works, we consider {0, 1} n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.

Set Functions, Games and Capacities in Decision Making

“…We mention here the result found by Ding et al, without proof[91]. We mention here the result found by Ding et al, without proof[91].…”

“…(ii) Some authors have proposed to use a more general inner product (Ding et al [90,91] and Marichal and Mathonet [232]), starting from probabilities p 1 ; : : : ; p n , where p i indicates the probability that x 2 f0; 1g n has coordinate x i D 1. Considering that coordinates are statistically independent, this induces a probability distribution over f0; 1g n given by…”

“…It is shown in Ding et al [91] that there still exists an orthonormal basis of pseudo-Boolean functions, given by: It is shown in Ding et al [91] that there still exists an orthonormal basis of pseudo-Boolean functions, given by: …”

“…(see Ding et al [91], and Marichal and Mathonet [232]). An important fact is that the best approximation f k has the same Banzhaf interaction transform as the original function (up to truncation, of course).…”

Least Square Approximations and Linear Values of Cooperative Games

“…Hammer and Holzman ([8]) 2 studied both the above version and the unconstrained version with equal weights (α S = 1 ∀S), and proved that the optimal solutions of the unconstrained version yield the Banzhaf value [1] (see also Section 5 below). More general versions of the unconstrained problem were solved by Grabisch et al [7] with the approximation being relative to the space of k-additive games (i.e., games whose Möbius transform vanishes for subsets of size greater than k) 3 .…”