Bajgiran Amirsaman H, Mardikoraem Mahsa, Soofi Ehsan S
Department of Industrial and Manufacturing Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA.
Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA.
Eur J Oper Res. 2021 Apr 1;290(1):196-209. doi: 10.1016/j.ejor.2020.07.052. Epub 2020 Aug 8.
Quantiles are available in various problems for developing probability distributions. In some problems quantiles are elicited from experts and used for fitting parametric models, which induce non-elicited information. In some other problems comparisons are made with a quantile of an assumed model which is noncommittal to the quantile information. The maximum entropy (ME) principle provides models that avoid these issues. However, the information theory literature has been mainly concerned about models based on moment information. This paper explores the ME models that are the minimum elaborations of the uniform and moment-based ME models by quantiles. This property provides diagnostics for the utility of elaboration in terms of the information value of each type of information over the other. The ME model with quantiles and moments is represented as the mixture of truncated distributions on consecutive intervals whose shapes and existence are determined by the moments. Elaborations of several ME distributions by quantiles are presented. The ME model based only on quantiles elicited by the fixed interval method possesses a useful property for pooling information elicited from multiple experts. The elaboration of Laplace distribution is an extension of the information theory connection with minimum risk under symmetric loss functions to the asymmetric linear loss. This extension produces a new Asymmetric Laplace distribution. Application examples compare ME priors with a parametric model fitted to elicited quantiles, illustrate measuring uncertainty and disagreement of economic forecasters based on elicited probabilities, and adjust ME models for a fundamental quantile in an inventory management problem.
分位数在各种用于构建概率分布的问题中都有应用。在一些问题中，分位数是由专家得出并用于拟合参数模型的，这会引入未得出的信息。在其他一些问题中，则是与一个假设模型的分位数进行比较，该假设模型对分位数信息不做明确表态。最大熵（ME）原理提供了避免这些问题的模型。然而，信息论文献主要关注基于矩信息的模型。本文探讨了基于分位数的均匀和基于矩的ME模型的最小扩展的ME模型。这一特性根据每种信息相对于其他信息的信息价值，为扩展的效用提供了诊断方法。具有分位数和矩的ME模型表示为连续区间上截断分布的混合，其形状和存在性由矩决定。给出了几种通过分位数对ME分布进行扩展的情况。仅基于固定区间法得出的分位数的ME模型具有一个有用的特性，即可以汇总从多个专家那里得出的信息。拉普拉斯分布的扩展是将信息论与对称损失函数下的最小风险的联系扩展到非对称线性损失。这种扩展产生了一种新的非对称拉普拉斯分布。应用实例将ME先验与拟合得出的分位数的参数模型进行了比较，说明了基于得出的概率来衡量经济预测者的不确定性和分歧，并针对库存管理问题中的一个基本分位数对ME模型进行了调整。
