Average entropy of Gaussian mixtures

We calculate the average differential entropy of a q-component Gaussian mixture in Rⁿ. For simplicity, all components have covariance matrix σ² 1, while the means ᵢ\ᵢ₌₁q are i.i.d. Gaussian vectors with zero mean and covariance s² 1. We obtain a series expansion in μ=s²/σ² for the average differential entropy up to order O(μ²), 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.