AI 中文总结
研究未归一化统计模型中最大熵问题，在非负测度射影空间上制定最大熵。通过普遍性定理统一多种熵相关结果，确定共同优化器为$q$指数密度，规定接受区域可定变形参数，为构建有界支持统计参考分布提供方法。
AI 中文摘要
最大熵参考分布通常在归一化概率单纯形上构建。这种形式对于未归一化统计模型不太自然，其中正倍数代表相同形状，且未直接解释规定的允许区域应如何确定有界支持参考分布的变形参数。我们在非负测度的射影空间上制定最大熵并建立三个具有统计相关性的结果。首先，一个普遍性定理表明，在线性矩约束下，相同归一化幂泛函的每个允许单调变换具有完全相同的优化器。其次，共同优化器被表征为$q$指数密度。第三，规定的马氏接受区域唯一确定变形参数，得到的仿射等变参考密度是唯一的射影最大熵解，其支持与指定椭球重合。这为从稳健位置和散度估计或外部指定的允许区域构建有界支持统计参考分布提供了一种有原则的方法。
英文摘要
Maximum-entropy reference distributions are usually constructed on the normalized probability simplex. This formulation is less natural for unnormalized statistical models, in which positive multiples represent the same shape, and it does not directly explain how a prescribed admissible region should determine the deformation parameter of a bounded-support reference distribution. We formulate maximum entropy on the projective space of nonnegative measures and establish three results of statistical relevance. First, a universality theorem shows that every admissible monotone transform of the same normalized power functional has exactly the same optimizer under linear moment constraints. The result unifies the maximum-entropy implications of Tsallis and Rényi entropies, Hölder composite scores, pseudo-spherical scores, Bregman--Hölder constructions, and related homogeneous divergences without asserting a new distribution family. Second, the common optimizer is characterized as a $q$-exponential density; under mean and covariance constraints it is a compactly supported $q$-Gaussian for positive deformation and a Student-type density for negative deformation. Third, a prescribed Mahalanobis acceptance region with squared radius $R^2>d+2$ uniquely determines the deformation parameter $γ_R=2/(R^2-d-2)$. The resulting affine-equivariant reference density is the unique projective maximum-entropy solution, and its support coincides with the specified ellipsoid without an additional support constraint. This provides a principled method for constructing bounded-support statistical reference distributions from robust location and scatter estimates or from externally specified admissible regions.
