Beyond Laplace: Closed-form wrapped Gaussian posterior approximations on statistical manifolds

2026-07-02

AI Summary

This work addresses the limitations of traditional Laplace approximations, which fail to capture posterior skewness, heavy tails, and narrow high-probability regions, as well as the computational expense of existing wrapped Gaussian methods that require evaluating geodesics, Christoffel symbols, or curvature tensors. By leveraging contrast function theory on a statistical manifold equipped with the Fisher–Rao metric and prior-induced geometry, the authors derive, for the first time, closed-form approximations of the exponential and logarithmic maps. This enables an efficient wrapped Gaussian approximation that avoids costly geometric computations. The proposed method substantially reduces computational complexity while accurately capturing complex posterior geometries across diverse models, achieving speedups of several orders of magnitude over current state-of-the-art approaches.

Abstract

In Bayesian statistics, the Laplace approximation provides a computationally efficient approximation to posterior distributions. However, its Gaussian form restricts it to elliptical shapes, limiting its ability to capture important posterior features such as skewness, heavy tails, and narrow high-probability regions. Recent work has addressed this limitation by exploiting Riemannian geometry to push forward Gaussian distributions from the tangent space to the manifold, referred to wrapped Gaussians. While offering greater flexibility, they introduce substantial computational challenges. Sampling requires solving geodesic equations through the exponential map and density evaluation additionally depends on the logarithmic map and Jacobi fields, involving costly differential equation solvers and geometric quantities such as inverse matrices, Christoffel symbols and curvature tensors. To overcome these limitations, we employ the theory of contrast functions to derive tractable approximations of the logarithmic and exponential maps on statistical manifolds endowed with the Fisher--Rao metric and the prior distribution geometry. The resulting methodology bypass the need to compute these geometric quantities and numerical solvers thereby removing the principal computational bottlenecks of existing wrapped Gaussian approaches. Empirical results across a range of models demonstrate that the proposed approximation captures complex posterior geometries while remaining orders of magnitude faster than current state-of-the-art approximation.

Problem

• Laplace approximation
• wrapped Gaussian
• posterior approximation
• statistical manifolds
• computational bottleneck

Innovation

• wrapped Gaussian
• contrast functions
• Fisher–Rao metric
• statistical manifold
• posterior approximation