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 depend heavily 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.

翻译：拉普拉斯方法通过在目标密度的众数处使用高斯分布进行近似。由于伯恩斯坦-冯·米塞斯定理，该方法计算效率高且在贝叶斯推断中具有渐近精确性，但对于复杂目标分布和有限数据后验分布，其近似往往过于粗糙。近期提出的拉普拉斯近似推广方法根据选定的黎曼几何变换高斯近似，提供了更丰富的近似族，同时仍保持计算效率。然而，如本文所示，其性质高度依赖于所选度量——事实上，先前研究采用的度量会导致近似结果过于狭窄，即使在无限数据极限下也存在偏差。我们通过进一步发展该近似族来修正这一缺陷，推导出两种在无限数据极限下精确的替代变体，扩展了该方法的理论分析，并通过一系列实验证明了实际改进效果。