A Bayesian Approach to Inverse Quantum Statistics

TL;DR

This paper introduces a nonparametric Bayesian method for reconstructing quantum potentials from experimental data, effectively handling heterogeneous data and incorporating explicit prior information, demonstrated through a 1D numerical example.

Contribution

It presents a novel Bayesian framework combining quantum likelihood models with stochastic process priors for inverse quantum problems.

Findings

Successful numerical implementation in one dimension

Explicit prior information improves potential reconstruction

Method handles heterogeneous data effectively

Abstract

A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information over potentials implemented in form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly, i.e., directly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation was feasible for one--dimensional problems. Using correct a priori information turned out to be essential.