半参数 Bernstein-von Mises 定理：基于 Stein 方法
Semiparametric Bernstein-von Mises theorems from Stein's method
- Statistical Laboratory, University of Cambridge（剑桥大学统计实验室）
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AI总结：
本文提出基于 Stein 方法与信息几何框架的半参数 Bernstein-von Mises 定理证明新策略，给出非渐近上界，并应用于高斯白噪声与直方图模型中的各类泛函，通过泛函修正消除后验偏差。
AI中文摘要：
我们提出了一种基于 Stein 方法与信息几何框架相结合的新型证明策略，用于证明半参数 Bernstein-von Mises 定理。我们不是通过控制先验在合适的扰动下对积分似然稳定性的影响来限制先验对感兴趣泛函的渐近边际后验分布的作用，而是通过统计模型上合适向量场的先验加权散度来刻画其影响。在此设置中应用 Stein 方法而非通常的拉普拉斯变换方法，可以得到边际后验分布与半参数效率理论预测的相应高斯极限之间的有界 Lipschitz 距离的显式非渐近上界。这些上界完全由与泛函、先验和所选向量场（通常与有效影响函数相关）相关的局部标量微分量组成，并在后验收缩集上进行评估。我们将该理论应用于高斯白噪声模型中的二次泛函，以及直方图密度模型中的线性、二次和一般积分泛函，同时考虑共轭和非共轭先验。对于线性和二次泛函，同一理论识别出导致后验偏差的微分项，并允许我们通过自然的泛函修正将其移除，从而在原始未修正泛函可能失效的机制下获得 Bernstein-von Mises 型定理。
英文摘要：
We introduce a novel proof strategy for semiparametric Bernstein-von Mises theorems based on Stein's method combined with an information-geometric framework. Rather than controlling the effect of the prior on the asymptotic marginal posterior distribution of the functional of interest through the stability of an integrated likelihood under a suitable perturbation, we characterise its influence through the prior-weighted divergence of suitable vector fields over the statistical model. Applying Stein's method in this setting instead of the usual Laplace-transform approach yields explicit non-asymptotic upper bounds on the bounded-Lipschitz distance between marginal posterior distributions and the corresponding Gaussian limits predicted by semiparametric efficiency theory. These bounds consist entirely of local scalar differential quantities associated with the functional, the prior and a chosen vector field-typically related to the efficient influence function-evaluated over posterior contraction sets. We apply the theory to quadratic functionals in Gaussian white-noise models and to linear, quadratic and general integral functionals in histogram density models, with both conjugate and non-conjugate priors. For linear and quadratic functionals, the same theory identifies the differential term responsible for posterior bias and allows us to remove it through a natural functional correction, yielding Bernstein-von Mises-type theorems in regimes where they may fail for the original uncorrected functional.
