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

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 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.

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BibTeX RIS

BibTeX

@misc{hartmann2026beyond,
  title = {{Beyond Laplace: Closed-form wrapped Gaussian posterior approximations on statistical manifolds}},
  author = {Marcelo Hartmann and Luu Hoang Phuc Hau and Anton Mallasto and Albert Kjøller Jacobsen and Georgios Arvanitidis and Søren Hauberg and Håvard Rue and Mark Girolami},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.01909},
  eprint = {2607.01909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.01909},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Beyond Laplace: Closed-form wrapped Gaussian posterior approximations on statistical manifolds
AU  - Marcelo Hartmann
AU  - Luu Hoang Phuc Hau
AU  - Anton Mallasto
AU  - Albert Kjøller Jacobsen
AU  - Georgios Arvanitidis
AU  - Søren Hauberg
AU  - Håvard Rue
AU  - Mark Girolami
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2607.01909
UR  - https://arxiv.org/abs/2607.01909
AB  - 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.
ER  -

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