Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

Helin Tapio; Kretschmann Remo

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Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

机译：贝叶斯逆问题中拉普拉斯近似的非渐近误差估计

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In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915-971, 2020. ), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

机译：在本文中，我们研究了在非线性贝叶斯逆问题中产生的后验分布的拉普拉斯近似的性质。我们的工作受到 Schillings 等人（Numer Math 145：915-971， 2020.）的启发，结果表明，在这样的设置中，Hellinger 距离中的拉普拉斯近似误差按噪声水平的顺序收敛到零。在这里，我们证明了给定噪声水平的新误差估计，该估计也量化了由于前向映射的非线性和问题的维度而产生的影响。特别是，我们对线性前向映射受到小型非线性映射干扰的设置感兴趣。我们的结果表明，在这种情况下，拉普拉斯近似误差的大小与扰动的大小相同。本文深入探讨了非线性逆问题中的贝叶斯推理，其中前向映射的线性化具有合适的逼近特性。

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《Numerische Mathematik》 |2022年第2期|521-549|共29页
作者
Helin Tapio; Kretschmann Remo;
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作者单位

LUT Univ;

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原文格式 PDF
正文语种英语
中图分类数值分析;
关键词
INFERENCE; DESIGN;

机译：推理;设计;

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Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

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