We calculate the average differential entropy of a $q$-component Gaussian mixture in $\mathbb R^n$. For simplicity, all components have covariance matrix $\sigma^2 {\mathbf 1}$, 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 $\mu=s^2/\sigma^2$ for the average differential entropy up to order $\mathcal{O}(\mu^2)$, 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.

翻译：我们计算了 $\mathbb R^n$ 中 $q$ 个分量高斯混合的平均微分熵。为简化起见，所有分量的协方差矩阵均为 $\sigma^2 {\mathbf 1}$，而均值 $\{\mathbf{W}_i\}_{i=1}^{q}$ 是独立同分布的高斯向量，均值为零且协方差为 $s^2 {\mathbf 1}$。我们得到了平均微分熵关于 $\mu=s^2/\sigma^2$ 的级数展开，最高至 $\mathcal{O}(\mu^2)$ 阶，并提供了一种计算高阶项的方法。我们的结果提供了具有可量化误差量级的解析近似，这在以往文献中尚未实现。