The Laplace's method approximates a target density with a Gaussian
distribution at its mode. It is computationally efficient and asymptotically
exact for Bayesian inference due to the Bernstein-von Mises theorem, but for
complex targets and finite-data posteriors it is often too crude an
approximation. A recent generalization of the Laplace Approximation transforms
the Gaussian approximation according to a chosen Riemannian geometry providing
a richer approximation family, while still retaining computational efficiency.
However, as shown here, its properties heavily depend on the chosen metric,
indeed the metric adopted in previous work results in approximations that are
overly narrow as well as being biased even at the limit of infinite data. We
correct this shortcoming by developing the approximation family further,
deriving two alternative variants that are exact at the limit of infinite data,
extending the theoretical analysis of the method, and demonstrating practical
improvements in a range of experiments.

拉普拉斯方法是一种近似目标密度函数的高斯分布方法，通过选择黎曼几何进行变换，可以提供更丰富的近似函数族，具有计算效率高的优点，并且通过引入两个可选的变种提高了无限数据极限下的准确性。

利用 Fisher 度量的黎曼拉普拉斯近似

Riemannian Laplace Approximation with the Fisher Metric

Laplace's method, a family of asymptotic methods used to approximate
integrals, is presented as a potential candidate for the tool box of techniques
used for knowledge acquisition and probabilistic inference in belief networks
with continuous variables. This technique approximates posterior moments and
marginal posterior distributions with reasonable accuracy [errors are O(n^-2)
for posterior means] in many interesting cases. The method also seems promising
for computing approximations for Bayes factors for use in the context of model
selection, model uncertainty and mixtures of pdfs. The limitations, regularity
conditions and computational difficulties for the implementation of Laplace's
method are comparable to those associated with the methods of maximum
likelihood and posterior mode analysis.

本文介绍了拉普拉斯方法，一种用于逼近积分的渐近方法，作为连续变量信念网络中知识获取和概率推断的技术工具箱的潜在候选项。该方法在许多有趣的情况下以合理的精度逼近后验矩和边际后验分布，此方法还似乎有助于计算用于模型选择、模型不确定性和 pdf 混合的贝叶斯因子的近似值。拉普拉斯方法的局限性、正则性条件和计算难点与最大似然和后验模式分析方法相当。