论文标题
高尺寸拉普拉斯近似准确性上的无维度非轴突界限
Dimension free non-asymptotic bounds on the accuracy of high dimensional Laplace approximation
论文作者
论文摘要
本说明试图以现代的非反应和无维度形式以拉普拉斯近似的经典结果重新审视经典结果。这样的扩展是由应用于高维统计和优化问题的应用。已建立的结果在总变化距离上以所谓的\ emph {有效尺寸} \（p_g \）为单位的高斯分布的高斯近似质量提供了明确的非反应界限。该值定义为数据中包含的信息与先前分布中的信息之间的相互作用。与著名的伯恩斯坦 - 冯·米塞斯的结果相反，即使真实的参数维度很大或无限，先前的影响也不可以忽略不计，并且即使真实的参数维度很大或无限。我们还解决了使用不精确参数使用高斯近似值的问题，重点是将最大后验（MAP）值替换为后平均值并设计基于拉普拉斯迭代的贝叶斯优化算法。结果指定为非线性逆问题。
This note attempts to revisit the classical results on Laplace approximation in a modern non-asymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The established results provide explicit non-asymptotic bounds on the quality of a Gaussian approximation of the posterior distribution in total variation distance in terms of the so called \emph{effective dimension} \( p_G \). This value is defined as interplay between information contained in the data and in the prior distribution. In the contrary to prominent Bernstein - von Mises results, the impact of the prior is not negligible and it allows to keep the effective dimension small or moderate even if the true parameter dimension is huge or infinite. We also address the issue of using a Gaussian approximation with inexact parameters with the focus on replacing the Maximum a Posteriori (MAP) value by the posterior mean and design the algorithm of Bayesian optimization based on Laplace iterations. The results are specified to the case of nonlinear inverse problem.
