高斯混合的随机重加权NPMLE的多对数稀疏性

中文摘要 AI 辅助

非参数最大似然估计（NPMLE）用于高斯位置混合，在混合分布的无限维空间上最大化似然。最大化混合分布可能不唯一，其原子数量的经典界随样本量n线性增长。我们证明，对似然进行一个趋于零的随机扰动可产生精确的多对数稀疏性。由此得到的随机重加权NPMLE最大化一个加权似然，其独立权重（在我们的分析中取为Gamma分布）随n增大而集中于1附近。以高概率，该估计量是唯一的，在维度d下具有O{(log n/log log n)^d+log n}个原子，几乎最大化普通似然，并以参数速率（直至对数因子）估计混合密度，该速率在Hellinger距离下达到参数阶。这种稀疏性适用于估计量本身，而非其近似，且无需支持惩罚。证明依赖于正核混合的有效维数原理：拟合值的低维变化控制最大化者集合中每个极值点的支持。数值示例验证了重加权NPMLE的Hellinger风险和支持大小与普通NPMLE相当。

英文摘要

The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian location mixture maximizes the likelihood over the infinite-dimensional space of mixing distributions. The maximizing mixing distribution can be nonunique, and the classical bound on its number of atoms grows linearly with the sample size $n$. We show that a vanishingly small random perturbation of the likelihood yields exact polylogarithmic sparsity. The resulting randomly reweighted NPMLE maximizes a weighted likelihood whose independent weights, taken to be Gamma in our analysis, concentrate around one as $n$ grows. With high probability, it is unique, has $O\{(\log n/\log\log n)^d+\log n\}$ atoms in dimension $d$, nearly maximizes the ordinary likelihood, and estimates the mixture density at a Hellinger rate that is parametric up to logarithmic factors. This sparsity holds for the estimator itself, not for an approximation of it, and requires no support penalty. The proof rests on an effective-dimension principle for positive kernel mixtures: low-dimensional variation of the fitted values controls the support of every extreme point of the set of maximizers. Numerical illustrations verify that the reweighted NPMLE has Hellinger risk and support size comparable to those of the ordinary NPMLE.

发表机构

• University of Toronto（多伦多大学）

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