Matched generating elements in maximum entropy density reconstruction

Abstract

Moment-constrained maximum entropy (MaxEnt) reconstructs a density as an exponential family whose sufficient statistics are chosen constraint functions. We study this choice separately from the numerical coordinates used to solve the dual problem. An odd-only family on a symmetric support forces f(x)f(-x) to be constant, so parity matching is necessary. A logarithmic-rational constraint produces the Student/Cauchy family and remains population-defined when required integer moments do not exist; a parity-matched one-parameter path provides a reproducible fractional-exponent alternative containing the monomials. The numerical study uses a common damped-Newton solver and audits the six-distribution benchmark of Rajan et al. Stable parity-block divided differences, residual replay in the original constraint coordinates, and independent adaptive integration identify a raw-coordinate false convergence on D5. Under the corrected computation, the selected PATP arm is within 1.22 times the oracle on D5 but retains selected-to-oracle error ratios of 30.7, 8.88, and 3.33 on D2, D4, and D6 at n=12. These checks preserve the negative selector conclusion without treating a numerical artifact as statistical evidence. The Pearson row is reproduced, whereas the published GOPoly rows are not under the printed protocol. Matched LogRat results show family capacity; neither tested selector supports automatic choice among constraint families.

arXiv commentNumerical correction to the PATP arm: stable parity-block divided-difference coordinates with original-constraint replay replace a raw-power quadrature false convergence. D5 is resolved, while selected-to-oracle gaps remain on D2, D4, and D6, so automatic selection remains unresolved. 26 pages, three figures, eight tables. Code: https://github.com/SZabolotnii/Ku-MaxEnt-code-supplement