Estimation of expected value of function of i.i.d. Bernoulli random variables

TL;DR

This paper develops methods to estimate the expected value of functions of i.i.d. Bernoulli variables, demonstrating the approach with a specific example involving graph Laplacians and random matrices.

Contribution

The paper introduces a novel estimation technique for expected values of functions of Bernoulli variables, supported by computational evidence on complex matrix functions.

Findings

Expected value of the normalized trace is between 0.2006 and 0.2030

Method applies to functions involving graph Laplacians and random matrices

Computational approach effectively estimates complex matrix function expectations

Abstract

We estimate the expected value of certain function $f:\{-1,1\}^{n}\to\mathbb{R}$. For example, with computer assistance, we show that if $\Delta$ is the Laplacian of the Cayley graph of $(\mathbb{Z}/15\mathbb{Z})\times(\mathbb{Z}/15\mathbb{Z})$ and $D$ is a diagonal $225\times 225$ matrix with entries chosen independently and uniformly from $\{-1,1\}$, then the expected value of the normalized trace of $(2I+D-\Delta)^{-1}$ is between $0.2006$ and $0.2030$.