The estimation of the mean value of copies of uncertain positions and its relation to the basic structure of quantum mechanics

In this short paper, we propose a new framework for obtaining basic aspects of quantum mechanics that originate from estimating the mean value of the position of a statistical system based on the generalized Bayes estimators. We show that while the first-order estimation leads to a classical system, the second-order estimation produces the time-independent Schrödinger equation. The Born rule describes the probabilistic nature of quantum particles, and Max Born postulated it independently from the Schrödinger equation. We show that under the proposed model, both the Schrödinger equation and the Born rule are captured organically; particularly, we show that the Born rule leads to the Schrödinger equation. Finally, we show how the proposed model deals with the transition from quantum mechanics into classical mechanics when dealing with macroscopic objects without external assumptions.