无限混合模型中可识别性的偏微分方程障碍
Partial Differential Equation Barriers to Identifiability in Infinite Mixture Models
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中文总结 AI 辅助
该研究探讨无限混合模型的可识别性，证明核被特定算子零化时可存在无穷多混合测度产生相同密度，给出可验证条件并补充估计误差下界，同时描述三类保持可识别性的核。
中文摘要 AI 辅助
我们研究无限混合模型中混合测度的可识别性。我们证明，在许多常见情形下，可识别性缺失可通过核族关于其参数的特定微分结构来刻画。在主要结果中，我们证明当核被参数空间上的非平凡微分或差分-微分算子零化时，存在无穷多个不同的混合测度产生相同的混合密度。我们给出这类算子存在的可验证条件，涵盖位置-尺度高斯、位置-尺度学生t分布、Gamma、Beta、Dirichlet、负二项及非中心卡方等常见核族。此外，我们的条件适用于参数维数超过充分统计量维数的任意指数族，更一般地适用于得分函数多项式增长的核。我们补充了非可识别情形下混合测度在Wasserstein距离下估计误差的极小极大下界，同时描述了三类可保持可识别性且仍能进行非参数统计推断的核。
英文摘要
We study identifiability of mixing measures in infinite mixture models. We show that, in many common cases, lack of identifiability can be characterized in terms of certain differential structures of the kernel family with respect to its parameters. In our main results, we prove that when the kernel is annihilated by a non-trivial differential or difference-differential operator over the parameter space, there exist infinitely many distinct mixing measures yielding the same mixture density. We give verifiable conditions for such operators to exist, covering many common cases, including the location-scale Gaussian, location-scale Student-t, Gamma, Beta, Dirichlet, negative binomial and non-central Chi-squared families. Furthermore, our conditions apply to any exponential family whose parameter dimension exceeds the dimension of its sufficient statistic and, more generally, to kernels with polynomially-growing score functions. We complement our results with a minimax lower bound on the estimation error for the mixing measure in the Wasserstein distance under non-identifiability. On the flips side, we describe three classes of kernels for which identifiability is preserved and nonparametric statistical inference remains possible.
