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Bayesian inference for inverse problems

机译：反问题的贝叶斯推断

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Traditionally, the MaxEnt workshops start by a tutorial day. This paper summarizes my talk during 2001 'th workshop at John Hopkins University. The main idea in this talk is to show how the Bayesian inference can naturally give us all the necessary tools we need to solve real inverse problems: starting by simple inversion where we assume to know exactly the forward model and all the input model parameters up to more realistic advanced problems of myopic or blind inversion where we may be uncertain about the forward model and we may have noisy data. Starting by an introduction to inverse problems through a few examples and explaining their ill posedness nature, I briefly presented the main classical deterministic methods such as data matching and classical regularization methods to show their limitations. I then presented the main classical probabilistic methods based on likelihood, information theory and maximum entropy and the Bayesian inference framework for such problems. I show that the Bayesian framework, not only generalizes all these methods, but also gives us natural tools, for example, for inferring the uncertainty of the computed solutions, for the estimation of the hyperparameters or for handling myopic or blind inversion problems. Finally, through a deconvolution problem example, I presented a few state of the art methods based on Bayesian inference particularly designed for some of the mass spectrometry data processing problems.
机译：传统上，MaxEnt研讨会是在教学日开始的。本文总结了我在约翰霍普金斯大学2001年研讨会期间的演讲。本演讲的主要思想是展示贝叶斯推理如何自然地为我们提供解决实际反问题所需的所有必要工具：从简单反演开始，在此我们假定确切地知道正向模型和所有输入模型参数，直至近视或盲目倒置的更现实的高级问题，在这些问题中我们可能不确定正向模型，并且可能会有嘈杂的数据。首先通过一些示例介绍反问题，并解释其不适性，然后简要介绍了主要的经典确定性方法，例如数据匹配和经典正则化方法，以显示其局限性。然后，我介绍了基于似然，信息论和最大熵以及针对此类问题的贝叶斯推理框架的主要经典概率方法。我展示了贝叶斯框架，不仅概括了所有这些方法，而且还为我们提供了自然的工具，例如用于推断计算出的解决方案的不确定性，用于估计超参数或用于处理近视或盲目反演问题。最后，通过反卷积问题示例，我提出了一些基于贝叶斯推断的最新技术方法，这些方法是专门为某些质谱数据处理问题而设计的。

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