🤖 AI Summary
This study addresses the limitations of traditional maximum entropy methods, which are constrained to monomial moment constraints and thus struggle to accurately model non-Gaussian distributions exhibiting algebraic tails or symmetric multimodality, while also suffering from ill-conditioned dual problems. Building upon Kunchenko’s decomposition space, the work generalizes constraint basis functions to designable generators, introduces a tail-class-matching strategy for generator selection, and develops three novel generator families—fractional-power, trigonometric, and logarithmic-rational functions—supported by parity-admissibility theorems and a design atlas. Integrating a unified dual solver, an analytical product-moment evaluator, and a variance-optimal element selection rule (oPMM-alpha), the proposed framework reconstructs Student’s and Cauchy distributions under a single constraint and automatically selects fractional moments. Experiments demonstrate an 8.5-fold reduction in MSE for bimodal Gaussian mixtures; for heavy-tailed distributions, fractional-power generators achieve feasible recovery in 19 out of 20 trials with Kolmogorov–Smirnov statistic KS = 0.068, while logarithmic-rational generators precisely recover the Cauchy tail exponent using only one constraint.
📝 Abstract
Moment-constrained maximum entropy (MaxEnt) reconstructs probability densities from a few moments in uncertainty evaluation (GUM) and reliability analysis. The classical method uses monomial constraints x^i. We show that monomials are merely one choice of generating element of the underlying Kunchenko decomposition space, and that this choice -- more than the solver -- governs which densities are representable and how well-conditioned the dual problem is. We study three elements under one dual solver: a fractional-power element (PATP) that reduces fractional-moment exponent selection to a one-dimensional scan on signed supports; a trigonometric (characteristic-function) element whose constraints exist for every distribution and keep the dual Hessian bounded; and a logarithmic-rational element log(1+(x/s)^2) whose single constraint yields the Student/Cauchy family (1+(x/s)^2)^lambda, representing algebraic tails the first two do not produce. A parity-admissibility theorem shows that an element of odd functions cannot represent any non-uniform symmetric density; the unifying lesson is a design map matching the element to the target's tail class. Empirically, on a bimodal Gaussian mixture the scan-selected fractional member cuts reconstruction MSE by 8.5x over the six-moment monomial baseline (all 20 seeds), while the trigonometric element is best-conditioned. On heavy tails the fractional element restores feasibility where monomial MaxEnt is infeasible (19/20 seeds) and reconstructs the body (KS 0.068) but not the tail, whereas the matched logarithmic element recovers the Cauchy tail index from one constraint. A variance-optimal rule (oPMM-alpha) selects the element for the reported functional. An analytical product-moment evaluator makes a measurement-and-verification optimization fitness exactly deterministic and faster than Monte Carlo, removing its noise-induced violations.