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，该方法计算高效且在贝叶斯推断中渐近精确，但对于复杂目标及有限数据后验分布而言，该近似往往过于粗糙。近期对拉普拉斯近似的一类泛化方法，根据所选黎曼几何对高斯近似进行变换，在保持计算效率的同时提供了更丰富的近似族。然而，本文研究显示，该方法的性质高度依赖于所选度量——实际上，先前工作中采用的度量会导致近似分布过度狭窄，甚至在无限数据极限下存在偏差。我们通过进一步扩展该近似家族来修正这一缺陷，推导出两种能在无限数据极限下保持精确的替代变体，深化了对该方法的理论分析，并通过一系列实验证明了其在实际应用中的改进效果。