Maximum-entropy inference and inverse continuity of the numerical range

TL;DR

This paper investigates the conditions under which the maximum-entropy inference map remains continuous for finite-dimensional observables, linking it to the inverse numerical range map and eigenvalue functions, revealing that discontinuities are rare and independent of prior states.

Contribution

It establishes a new equivalence between the continuity of maximum-entropy inference and the strong continuity of the inverse numerical range map, providing a novel criterion based on eigenvalue functions.

Findings

Continuity of MaxEnt inference is equivalent to inverse numerical range map continuity.

Discontinuities in MaxEnt inference are very rare.

Continuity is independent of the prior state.

Abstract

We study the continuity of the maximum-entropy inference map for two observables in finite dimensions. We prove that the continuity is equivalent to the strong continuity of the set-valued inverse numerical range map. This gives a continuity condition in terms of analytic eigenvalue functions which implies that discontinuities are very rare. It shows also that the continuity of the MaxEnt inference method is independent of the prior state.