This work considers a problem of estimating a mixing probability density $f$ in the setting of discrete mixture models. The paper consists of three parts. The first part focuses on the construction of an $L_1$ consistent estimator of $f$. In particular, under the assumptions that the probability measure $\mu$ of the observation is atomic, and the map from $f$ to $\mu$ is bijective, it is shown that there exists an estimator $f_n$ such that for every density $f$ $\lim_{n\to \infty} \mathbb{E} \left[ \int |f_n -f | \right]=0$. The second part discusses the implementation details. Specifically, it is shown that the consistency for every $f$ can be attained with a computationally feasible estimator. The third part, as a study case, considers a Poisson mixture model. In particular, it is shown that in the Poisson noise setting, the bijection condition holds and, hence, estimation can be performed consistently for every $f$.

翻译：这项工作考虑了在设定离散混合物模型时估算混合概率密度(f)美元的问题。 纸张由三部分组成。 第一部分侧重于构建一个$L_ 1美元一致的估算值( f) 美元。 特别是, 假设观测的概率是原子 $\ mu 美元, 而从f美元到 $\ mu 美元的地图是双向的, 这表明存在一个估计值 $f_ n 美元, 以便每个密度( $) 美元 = mathbb}\ mathb* E} \ left [\ int \\\ f_ n - f \\\\\\\\\\\\\\ right]=0 。 第二部分讨论了执行细节。 具体地说, 这表明每美元的一致性可以用一个计算可行的估计值来达到。 第三部分作为研究案例, 考虑 Poisson 混合物模型。 特别是, 在 Poisson 噪音设置中, 双向状态, 因此每美元可以连续进行估计。