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 𝜎2𝟏, while the means {𝐖𝑖}𝑞𝑖=1 are i.i.d. Gaussian vectors with zero mean and covariance 𝑠2𝟏. 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.