SUMMARY

We develop a theoretical framework for framing and solving probabilistic linear(ized) inverse problems in function spaces. This is built on the statistical theory of Gaussian Processes, and allows results to be obtained independent of any basis, avoiding any difficulties associated with the fidelity of representation that can be achieved. We show that the results of Backus–Gilbert theory can be fully understood within our framework, although there is not an exact equivalence due to fundamental differences of philosophy between the two approaches. Nevertheless, our work can be seen to unify several strands of linear inverse theory, and connects it to a large body of work in machine learning. We illustrate the application of our theory using a simple example, involving determination of Earth’s radial density structure.

中文翻译：

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高斯过程模型-I。概率连续逆理论的框架

概要

我们建立了一个理论框架，用于框架和解决函数空间中的概率线性（化）逆问题。这是建立在高斯过程的统计理论基础上的，它允许独立于任何基础而获得结果，从而避免了与可以实现的表示的逼真度相关的任何困难。我们展示了Backus–Gilbert理论的结果可以在我们的框架内得到完全理解，尽管由于这两种方法在哲学上存在根本差异，因此没有确切的对等关系。然而，可以看出我们的工作将线性逆理论的几条线统一起来，并将其与机器学习中的大量工作联系起来。我们用一个简单的例子说明了我们理论的应用，其中涉及确定地球的径向密度结构。