Quantum Statistical Inference

Summary

ABSTRACT. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity.

Introduction

Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic. That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.