论文标题
贝叶斯逆问题中拉普拉斯近似的非反应误差估计值
Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems
论文作者
论文摘要
在本文中，我们研究了在非线性贝叶斯反问题中产生的后验分布的拉普拉斯近似的特性。我们的工作是由Schillings等人激励的。 （2020），在这种情况下，在这种设置中，hellinger距离中的拉普拉斯近似误差按噪声水平的顺序收敛至零。在这里，我们证明了给定噪声水平的新误差估计值，该误差估计还量化了由于正向映射的非线性和问题的尺寸而引起的效果。特别是，我们对线性向前映射受到小型非线性映射的扰动的设置感兴趣。我们的结果表明，在这种情况下，拉普拉斯近似误差是扰动的大小。本文提供了对非线性反问题中贝叶斯推论的见解，在非线性反问题中，正向映射的线性化具有合适的近似属性。
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. (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.
