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$q$分量高斯混合模型的平均微分熵。为简化起见，所有分量均具有协方差矩阵$\sigma^2 {\mathbf 1}$，而均值$\{\mathbf{W}_i\}_{i=1}^{q}$是独立同分布的高斯向量，其均值为零、协方差为$s^2 {\mathbf 1}$。我们得到了平均微分熵在$\mu=s^2/\sigma^2$处直至$\mathcal{O}(\mu^2)$阶的级数展开式，并给出了计算更高阶项的方法。我们的结果提供了一个具有可量化误差量级的解析近似，这在先前文献中尚未实现。