[Paper Review] Approaching the parameter estimation quality of maximum likelihood via generalized moments

F.V. Tkachov|ArXiv.org|Jan 9, 2000

Statistical Methods and Inference1 references3 citations

TL;DR

This paper proposes a method to approximate the precision of maximum likelihood estimation using generalized moments, avoiding the computational complexity of full likelihood maximization. By deriving quasi-optimal moments through the functional derivative of the log-likelihood, it achieves near-optimal parameter estimation accuracy—approaching the Rao-Cramer lower bound—especially useful for high-dimensional or complex probability distributions in particle physics and large-scale data analysis.

ABSTRACT

A simple criterion is presented for a practical construction of generalized moments that allow one to approach the theoretical Rao-Cramer limit for parameter estimation while avoiding the complexity of the maximum likelihood method in the cases of complicated probability distributions and/or very large event samples.

Motivation & Objective

• To develop a practical alternative to maximum likelihood estimation for parameter estimation in complex or high-dimensional probability distributions.
• To address the computational infeasibility of maximum likelihood in large-scale data sets (e.g., O(10^6) events) common in high-energy physics.
• To provide a systematic way to construct moments that approach the theoretical precision limit (Rao-Cramer bound) without requiring full likelihood evaluation.
• To enable high-precision parameter estimation in scenarios where theoretical probability densities are analytically complex or derived from perturbative quantum field theory.
• To offer a framework compatible with both numerical and theoretical data processing, particularly in contexts involving unstable particles or singular generalized functions.

Proposed method

• Derives optimal moments via functional derivative of the log-likelihood function: f_opt(P) ∝ ∂/∂M [ln π(P)].
• Introduces 'quasi-optimal moments' as a practical approximation to the true optimal moments, defined by f_quasi(P) = ∂/∂M [ln π(P)] + C, where C is a constant.
• Uses the variance of the moment estimator to quantify precision, minimizing Var(M) via the condition δ/δf [Var(M)] = 0.
• Applies the method to the Breit-Wigner distribution, showing that optimal moments emphasize the slope regions where sensitivity to M is highest.
• Proposes numerical construction of f_quasi via interpolation of π(P) at multiple M values, enabling use when analytical forms are intractable.
• Suggests a hybrid approach: use χ² fitting for initial model validation and parameter estimation, then switch to quasi-optimal moments for final high-precision results.

Experimental results

Research questions

• RQ1 How can one construct moments that approach the theoretical precision limit of parameter estimation without computing the full likelihood function?
• RQ2 What is the functional form of the optimal moment that minimizes variance in parameter estimation under the Rao-Cramer inequality?
• RQ3 In what scenarios does the method of quasi-optimal moments outperform traditional χ² or histogram-based methods in precision and information retention?
• RQ4 Can quasi-optimal moments be effectively constructed when the probability density π(P) is analytically complex or derived from perturbative quantum field theory?
• RQ5 How can the method be adapted to multi-parameter estimation problems while preserving the precision of maximum likelihood?

Key findings

• The optimal moment is given by f_opt(P) = ∂/∂M [ln π(P)], which corresponds exactly to the score function used in maximum likelihood estimation.
• The variance of the parameter estimator using optimal moments achieves the Rao-Cramer lower bound, confirming theoretical optimality.
• For the Breit-Wigner distribution, the optimal moment f_opt(P) = −2(P−M)/[Γ² + (P−M)²] emphasizes the slopes of the resonance, where sensitivity to M is maximal.
• The method of quasi-optimal moments allows one to approach maximum likelihood precision with significantly reduced computational cost, especially when π(P) is complex or high-dimensional.
• In multi-parameter problems, the covariance matrix of quasi-optimal moments can be computed from data, enabling straightforward error ellipsoid mapping to parameter space.
• The method is particularly advantageous when theoretical predictions are in the form of generalized functions (e.g., singular distributions), where χ² fitting may fail or lose information.