This work deals with parameter identification problems in which uncertainties are modelled using random sets (RS), i.e. set-valued random variables. Dempster’s rule of combination is applied for replacing the role of Bayes’ rule to infer the posterior, which is also a RS. The considered framework allows accounting for mixed epistemic-aleatory uncertainty descriptions such as probability boxes and intervals. In this paper, we aim at an efficient computational method to sample the posterior RS using stochastic methods developed for Bayesian inverse problems. To this end, by applying the capacity transformation method, the considered problem is translated into a Bayesian inverse problem, and the region at which the posterior RS concentrates is exploited using a Markov Chain Monte Carlo (MCMC) algorithm. To sample the posterior RS, we approximate it as a random finite set whose domain consists of points obtained from the MCMC algorithm. Because the forward model could be computationally expensive and is required to be evaluated at many points, we construct a polynomial chaos expansion-based surrogate model for it. The developed approach is demonstrated with a numerical example in which measurement errors are noisy and also contain unknown but bounded biases.

中文翻译：

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贝叶斯反演在随机集理论背景下参数识别的有效计算方法

这项工作涉及参数识别问题，其中使用随机集（RS）（即集值随机变量）对不确定性进行建模。Dempster的组合法则用于替换贝叶斯法则以推断后验的作用，后者也是RS。考虑的框架允许考虑混合的认知-不确定不确定性描述，例如概率框和间隔。在本文中，我们针对一种有效的计算方法，采用为贝叶斯逆问题开发的随机方法对后RS进行采样。为此，通过应用容量变换方法，将考虑的问题转换为贝叶斯逆问题，并使用马尔可夫链蒙特卡洛（MCMC）算法来利用后方RS集中的区域。要采样后路RS，我们将其近似为一个随机有限集，其域包含从MCMC算法获得的点。由于前向模型的计算量可能很大，并且需要在许多点进行评估，因此我们为它构建了一个基于多项式混沌扩展的替代模型。数值示例演示了开发的方法，其中测量误差很大，并且还包含未知但有界的偏差。