AI 中文总结
针对拉普拉斯近似无法捕捉后验偏斜、厚尾等复杂几何特征的问题，利用对比函数理论在Fisher-Rao度量和先验几何下推导对数映射和指数映射的易处理近似，实现统计流形上的闭式包裹高斯后验近似，显著降低计算成本。
AI 中文摘要
在贝叶斯统计学中，拉普拉斯近似提供了一种计算高效的后验分布近似方法。然而，其高斯形式限制了它只能处理椭圆形状，无法捕捉后验的重要特征，如偏斜、厚尾和狭窄的高概率区域。最近的研究通过利用黎曼几何将高斯分布从切空间推前到流形上（称为包裹高斯）来解决这一局限。虽然这种方法提供了更大的灵活性，但也引入了巨大的计算挑战。采样需要通过指数映射求解测地线方程，密度评估还依赖于对数映射和雅可比场，涉及昂贵的微分方程求解器以及逆矩阵、克里斯托费尔符号和曲率张量等几何量。为了克服这些局限，我们采用对比函数理论，在具有Fisher-Rao度量和先验分布几何的统计流形上推导对数映射和指数映射的易处理近似。所得到的方法避免了计算这些几何量和数值求解器的需要，从而消除了现有包裹高斯方法的主要计算瓶颈。跨多个模型的实证结果表明，所提出的近似能够捕捉复杂的后验几何，同时比当前最先进的近似方法快数个数量级。
英文摘要
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 a chosen 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 alongside full matrices inverse operations, 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 geometric state-of-the-art approximation.
