We consider the Bayesian inference of nonlinear, large-scale regression problems in which the parameters model the spatial distribution of some property. A homoscedastic Gaussian sampling distribution is supposed as well as certain assumptions about the regression function. Propriety of the posterior and the existence of its moments are explored when using improper prior distributions expressing different levels of prior knowledge, ranging from a purely noninformative prior over intrinsic Gaussian Markov random field priors to a partition prior. The considered class of problems includes magnetic resonance fingerprinting (MRF). We apply an approximate Bayesian inference to this particular application and demonstrate its practicability in dimensions up to \(10^5\) or larger. The benefit of incorporating substantial prior knowledge is illustrated. By analyzing simulated realistic MRF data, it is shown that MAP estimates can significantly improve the results achieved with maximum likelihood estimation.

中文翻译：

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近似大规模贝叶斯空间建模及其在定量磁共振成像中的应用

我们考虑非线性，大规模回归问题的贝叶斯推断，其中参数对某些属性的空间分布进行建模。假定同高斯抽样分布以及关于回归函数的某些假设。当使用表示不同水平的先验知识的不正确先验分布时，探讨后验的适当性及其存在的时刻，从纯粹的非信息先验而不是固有的高斯马尔可夫随机场先验再到分区先验。考虑的问题类别包括磁共振指纹图谱（MRF）。我们对此应用程序应用了近似贝叶斯推断，并证明了其在高达\（10 ^ 5 \）的维度中的实用性或更大。说明了合并大量先验知识的好处。通过分析模拟的现实MRF数据，可以看出MAP估计可以极大地改善使用最大似然估计获得的结果。