We calculate the smoothest mixture density under a variety of prescribed specifications. This includes constraints on certain moments, specifications on density values and/or its derivatives, and prescribed probability masses in certain regions. As a roughness measure, we use Fisher Information (FI) in the space of mixtures $\cal M$. For mixtures, FI cannot be calculated in closed form. We define the space $\cal R$ of root mixtures (RMs) living on the Hilbert sphere. A transformation of FI to $\cal R$ admits a closed-form solution and yields the desired result in $\cal M$. This naturally leads to a tandem processing with two density representations maintained simultaneously in $\cal R$ and $\cal M$. FI is calculated in RM space $\cal R$ while the constraints are evaluated in mixture space $\cal M$.

翻译：我们计算了在多种指定条件下最平滑的混合密度。这些条件包括对特定矩的约束、对密度值及其导数的指定，以及对特定区域内概率质量的预设。作为粗糙度度量，我们使用混合空间 $\cal M$ 中的 Fisher 信息 (FI)。对于混合模型，FI 无法通过闭式求解。我们定义了位于希尔伯特球面上的根混合 (RM) 空间 $\cal R$。将 FI 变换到 $\cal R$ 后可获得闭式解，并在 $\cal M$ 中得出所需结果。这自然形成了双处理流程，即同时在 $\cal R$ 和 $\cal M$ 中维护两种密度表示：FI 在 RM 空间 $\cal R$ 中计算，而约束条件在混合空间 $\cal M$ 中评估。