The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows the formulation of expert knowledge or physical constraints in a probabilistic fashion and plays an important role for the success of the inference. Recently, Bayesian inverse problems were solved using generative models as highly informative priors. Generative models are a popular tool in machine learning to generate data whose properties closely resemble those of a given database. Typically, the generated distribution of data is embedded in a low-dimensional manifold. For the inverse problem, a generative model is trained on a database that reflects the properties of the sought solution, such as typical structures of the tissue in the human brain in magnetic resonance (MR) imaging. The inference is carried out in the low-dimensional manifold determined by the generative model which strongly reduces the dimensionality of the inverse problem. However, this proceeding produces a posterior that admits no Lebesgue density in the actual variables and the accuracy reached can strongly depend on the quality of the generative model. For linear Gaussian models we explore an alternative Bayesian inference based on probabilistic generative models which is carried out in the original high-dimensional space. A Laplace approximation is employed to analytically derive the required prior probability density function induced by the generative model. Properties of the resulting inference are investigated. Specifically, we show that derived Bayes estimates are consistent, in contrast to the approach employing the low-dimensional manifold of the generative model. The MNIST data set is used to construct numerical experiments which confirm our theoretical findings.
翻译:Bayesian 解决反问题的方法取决于对先前问题的选择。 这种关键成分使得能够以概率化的方式形成专家知识或物理限制,并且对推论的成功起到重要作用。 最近, Bayesian 反问题用基因化模型作为高度信息化的前科解决了。 创用模型是机器学习生成数据的一种流行工具,其性质与给定数据库的特性非常相似。 通常, 生成的数据分布是嵌入一个低维的多元体。 对于反向问题, 一种基因化模型在反映所寻求的解决办法的特性的数据库上进行了基因化模型的培训, 例如, 人类大脑的磁共振(MR)成像典型的组织结构结构, 并且对于成功发挥作用。 这种推导出模型的基因化模型的典型性能结构化结构化, 也就是我们所利用的原生的原生的原基因化模型, 也就是我们所利用的原生机变精度, 其原生的精确性能, 是我们所利用的原生机变精度, 其原生的精确性分析结果, 由我们所利用的原生的原变精度分析结果性模型 显示的精确性 。