AI 中文总结
研究仅给定聚合计数变量概率分布时，哪些独立潜在计数过程与之兼容。通过发展正分解数学理论，引入正分解偏序集和分解熵，证明相关定理并进行多方面研究，为研究潜在独立结构提供框架。
AI 中文摘要
仅给定聚合计数变量的概率分布，哪些独立潜在计数过程与该观测兼容？即概率生成函数何时能分解为具有非负系数的归一化多项式？我们发展了此类正分解的数学理论。引入正分解偏序集，其元素为按细化排序的所有正分解，定义分解熵以衡量与观测分布兼容的最大潜在香农熵。证明了尖锐的熵不等式，通过潜在加法映射的单射性刻画等式情形，表明熵优化可限于最大原子化，并给出不同最大原子化具有不同熵的例子。还建立了基于支撑的正分解障碍，刻画实根和赫维茨稳定扇区，证明互质分解的局部稳定性定理，确定二次和三次的可分解区域，得到精确的四次赫维茨体积，并通过精确计算和蒙特卡罗实验研究概率单纯形内可分解区域的几何。这些结果将正分解偏序集确定为与概率生成函数相关的自然代数对象，并为研究聚合计数统计中的潜在独立结构提供了框架。
英文摘要
Given only the probability distribution of an aggregate counting variable, what independent latent counting processes are compatible with the observation? Equivalently, when does a probability-generating function admit a factorization into normalized polynomials with nonnegative coefficients? We develop a mathematical theory of such positive factorizations. We introduce the positive factorization poset, whose elements are all positive factorizations ordered by refinement, and define the factorization entropy, measuring the maximal latent Shannon entropy compatible with the observed distribution. We prove a sharp entropy inequality, characterize the equality case by injectivity of the latent addition map, show that entropy optimization may be restricted to maximal atomizations, and exhibit examples where distinct maximal atomizations have different entropy. We further establish support-based obstructions to positive factorization, characterize the real-rooted and Hurwitz-stable sectors, prove a local stability theorem for coprime factorizations, determine exactly the factorable regions in degrees two and three, obtain an exact quartic Hurwitz volume, and investigate the geometry of the factorable region inside the probability simplex through exact calculations and Monte Carlo experiments. These results identify the positive factorization poset as a natural algebraic object associated with probability-generating functions and provide a framework for studying latent independent structure in aggregate counting statistics.
