Average Entropy of Gaussian Mixtures.

Area of Science:

• Information Theory
• Probability Theory
• Statistical Inference

Background:

• Gaussian mixture models are widely used for density estimation.
• Calculating differential entropy for mixtures is computationally challenging.
• Existing methods lack precise error quantification.

Purpose of the Study:

• To derive an analytic approximation for the average differential entropy of Gaussian mixtures.
• To provide a method for calculating higher-order terms in the approximation.
• To establish quantifiable error bounds for the approximation.

Main Methods:

• Derivation of a series expansion for differential entropy.
• Analysis of a q-component Gaussian mixture in Rn.
• Expansion in terms of the ratio of covariance matrices (μ=s²/σ²).

Main Results:

• An analytic approximation for average differential entropy up to O(μ²) was obtained.
• A systematic method for computing higher-order terms was developed.
• The approximation includes a quantifiable error magnitude.

Conclusions:

• The derived series expansion offers a significant improvement over previous literature.
• This work provides a valuable tool for analyzing information-theoretic properties of Gaussian mixtures.
• The method allows for accurate estimation of differential entropy with controlled error.