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.

翻译：拉普拉斯方法利用高斯分布在其众数处近似目标密度。由于伯恩斯坦-冯·米塞斯定理，该方法计算高效且贝叶斯推断渐近精确，但对于复杂目标及有限数据后验而言，该近似通常过于粗糙。近期一种拉普拉斯近似推广方法根据特定黎曼几何变换高斯近似，提供了更丰富的近似族，同时仍保持计算高效性。然而如本文所示，其性质高度依赖于所选度量——先前工作中采用的度量在无限数据极限下仍会导致近似过于狭窄且存在偏差。我们通过进一步扩展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，拓展该方法的理论分析，并通过系列实验展示实际改进效果。