Numerical Evaluation of Gaussian Mixture Entropy.

Area of Science:

• Information Theory
• Statistical Inference
• Machine Learning

Background:

• Differential entropy quantifies uncertainty in probability distributions.
• Gaussian mixture models are widely used for density estimation.
• Accurate entropy approximation is crucial for high-dimensional data analysis.

Purpose of the Study:

• To develop an approximation method for the differential entropy of q-component Gaussian mixtures in Rn.
• To introduce two specific approximation methods: h¯C,mTaylor(X) and h¯CPolyfit(X).
• To evaluate the accuracy and efficiency of the proposed methods compared to existing bounds.

Main Methods:

• Developing a novel approximation technique for differential entropy.
• Implementing Taylor series expansion for h¯C,mTaylor(X).
• Utilizing polynomial fitting for h¯CPolyfit(X).

Main Results:

• h¯C,mTaylor(X) offers a computationally simple lower bound for differential entropy.
• h¯CPolyfit(X) provides a highly accurate and efficient approximation.
• h¯CPolyfit(X) demonstrates superior accuracy and conjectured resilience in high dimensions compared to existing methods.

Conclusions:

• The proposed h¯CPolyfit(X) method offers a significant advancement in approximating differential entropy for Gaussian mixtures.
• This method is particularly valuable for applications involving high-dimensional data.
• The study provides a robust tool for information-theoretic analysis in complex statistical models.