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arXiv · 2404.07311

Average entropy of Gaussian mixtures

Basheer Joudeh·Boris Škorić
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Abstract

We calculate the average differential entropy of a $q$-component Gaussian mixture in $\mathbb R^n$. For simplicity, all components have covariance matrix $σ^2 {\mathbf 1}$, while the means $\{\mathbf{W}_i\}_{i=1}^{q}$ are i.i.d. Gaussian vectors with zero mean and covariance $s^2 {\mathbf 1}$. We obtain a series expansion in $μ=s^2/σ^2$ for the average differential entropy up to order $\mathcal{O}(μ^2)$, and we provide a recipe to calculate higher order terms. Our result provides an analytic approximation with a quantifiable order of magnitude for the error, which is not achieved in previous literature.

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Basheer Joudeh, Boris Škorić. 2025-02-11. Average entropy of Gaussian mixtures. https://arxiv.org/abs/2404.07311

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