关于反问题贝叶斯统计方法的教程

Faaiq G. Waqar, Swati Patel, Cory M. Simon
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引用次数: 0

摘要

逆问题在科学和工程中普遍存在。关于物理系统的两类逆问题是:(1)根据观察到的输入输出对估计系统模型中的参数;(2)给定一个系统模型,重构引起某些观察到的输出的输入。应用逆问题具有挑战性,因为解决方案可能(i)不存在,(ii)不是唯一的,或(iii)对污染数据的测量噪声敏感。贝叶斯统计反演(BSI)是一种解决不适定和/或病态逆问题的方法。有利的是,BSI提供了一种“解决方案”,它(i)通过为未知参数/输入的每个可能值分配概率来量化不确定性,(ii)结合了关于参数/输入的先验信息和信念。在这里,我们通过举例说明从环境空气到冷酸橙水果的热量传递,为逆问题提供了一个BSI的教程。首先,我们使用BSI从石灰温度随时间的测量中推断石灰温度动态模型中的参数。其次,我们使用BSI从稍后的温度测量中重建石灰的初始条件。我们展示了先验信息的结合,可视化了参数/初始条件的后验分布,并展示了模型中石灰温度轨迹的后验样本。我们的教程旨在达到广泛的科学家和工程师。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A tutorial on the Bayesian statistical approach to inverse problems
Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers.
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