Nonparametric estimation of Fisher information from real data

Area of Science:

• Statistical inference and information geometry
• Computational physics and complex systems

Background:

• The Fisher information matrix (FIM) is crucial for statistical inference, experiment design, and analyzing biological systems.
• Estimating the FIM typically involves known distributions or known parameters; this study focuses on the latter with unknown distributions.

Purpose of the Study:

• To develop and validate a novel nonparametric approach for estimating the Fisher information matrix (FIM).
• To optimize the parameter difference (Δθ) for finite-difference approximations in FIM estimation.
• To compare different nonparametric density estimation methods for FIM calculation.

Main Methods:

• Utilized nonparametric density estimation (Gaussian kernel and a novel field theory method) to compute FIM directly from data.
• Developed an optimal parameter difference (Δθ) selection based on large deviations theory.
• Compared the proposed methods against an approach using nonparametric f-divergence estimation.

Main Results:

• Validated the method using the Fisher information of the normal distribution.
• Successfully computed the FIM's temperature component in the 2D Ising model, identifying the critical temperature.
• Demonstrated the accuracy of the novel field theory density estimation method.

Conclusions:

• The developed nonparametric FIM estimation method, incorporating optimal Δθ selection, provides accurate results.
• The field theory-based density estimation offers a promising alternative for FIM computation.
• The approach is effective for analyzing complex systems like the Ising model, particularly near phase transitions.