In the literature, finite mixture models are described as linear combinations of probability distribution functions having the form $\displaystyle f(x) = \Lambda \sum_{i=1}^n w_i f_i(x)$, $x \in \mathbb{R}$, where $w_i$ are positive weights, $\Lambda$ is a suitable normalising constant and $f_i(x)$ are given probability density functions. The fact that $f(x)$ is a probability density function follows naturally in this setting. Our question is: what happens when we remove the sign condition on the coefficients $w_i$? The answer is that it is possible to determine the sign pattern of the function $f(x)$ by an algorithm based on finite sequence that we call a generalized Budan-Fourier sequence. In this paper we provide theoretical motivation for the functioning of the algorithm, and we describe with various examples its strength and possible applications.

翻译：在文献中，有限混合模型被描述为概率分布函数的线性组合，其形式为 $\displaystyle f(x) = \Lambda \sum_{i=1}^n w_i f_i(x)$，$x \in \mathbb{R}$，其中 $w_i$ 为正权重，$\Lambda$ 是适当的归一化常数，$f_i(x)$ 为给定的概率密度函数。在此设定下，$f(x)$ 作为概率密度函数这一性质自然成立。我们的问题是：当我们移除对系数 $w_i$ 的符号限制时，会发生什么？答案是，可以通过一种基于有限序列的算法来确定函数 $f(x)$ 的符号模式，我们称该序列为广义布丹-傅里叶序列。本文为该算法的运行提供了理论依据，并通过多个示例描述了其优势及潜在应用。