反问题的贝叶斯推断
Bayesian inference for inverse problems
摘要：
传统上，MaxEnt研讨会从辅导日开始。本文总结了我在约翰霍普金斯大学2001年研讨会上的发言。本次演讲的主要思想是展示贝叶斯推断如何自然地为我们提供解决实际逆问题所需的所有必要工具：从简单的反演开始，我们假设确切地知道正演模型和所有输入模型参数，直到更现实的近视或盲反演高级问题，我们可能对正演模型不确定，我们可能有噪声数据。首先通过几个例子介绍反问题，并解释其不适定性的性质，我简要介绍了主要的经典确定性方法，如数据匹配和经典正则化方法，以显示其局限性。然后介绍了基于似然、信息论和最大熵的主要经典概率方法以及解决此类问题的贝叶斯推断框架。我表明，贝叶斯框架，不仅概括了所有这些方法，但也给了我们自然的工具，例如，推断计算的解决方案的不确定性，估计的超参数或处理近视或盲反演问题。最后，通过一个反卷积问题的例子，介绍了几种基于贝叶斯推断的方法，特别是针对一些质谱数据处理问题而设计的。
英文摘要：
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.
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