{"title":"Information About the Moments or the Likelihood Model Parameters? A Chicken and Egg Problem","authors":"Omid M. Ardakani, Majid Asadi, Ehsan S. Soofi","doi":"10.1002/asmb.2931","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>This article compares the information content of a sample for two competing Bayesian approaches. One approach follows Dennis Lindley's Bayesian standpoint, where one begins by formulating a prior for a parameter related to the problem in question and incorporates a likelihood to transition to a posterior. This contrasts with the usual Bayesian approach, where one starts with a likelihood model, formulates a prior distribution for its parameters, and derives the corresponding posterior. In both cases, the sample information content is measured using the difference between the prior and posterior entropies. We investigate this contrast in the context of learning about the moments of a variable. The maximum entropy principle is used to construct the likelihood model consistent with the given moment parameters. This likelihood model is then combined with the prior information on the parameters to derive the posterior. The model parameters are the Lagrange multipliers for the moment constraints. A prior for the moments induces a prior for the model parameters; however, the data provides differing amounts of information about them. The results obtained for several problems show that the information content using the two formulations can differ significantly. Additional information measures are derived to assess the effects of operating environments on the lifetimes of system components.</p>\n </div>","PeriodicalId":55495,"journal":{"name":"Applied Stochastic Models in Business and Industry","volume":"41 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Stochastic Models in Business and Industry","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/asmb.2931","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
This article compares the information content of a sample for two competing Bayesian approaches. One approach follows Dennis Lindley's Bayesian standpoint, where one begins by formulating a prior for a parameter related to the problem in question and incorporates a likelihood to transition to a posterior. This contrasts with the usual Bayesian approach, where one starts with a likelihood model, formulates a prior distribution for its parameters, and derives the corresponding posterior. In both cases, the sample information content is measured using the difference between the prior and posterior entropies. We investigate this contrast in the context of learning about the moments of a variable. The maximum entropy principle is used to construct the likelihood model consistent with the given moment parameters. This likelihood model is then combined with the prior information on the parameters to derive the posterior. The model parameters are the Lagrange multipliers for the moment constraints. A prior for the moments induces a prior for the model parameters; however, the data provides differing amounts of information about them. The results obtained for several problems show that the information content using the two formulations can differ significantly. Additional information measures are derived to assess the effects of operating environments on the lifetimes of system components.
期刊介绍:
ASMBI - Applied Stochastic Models in Business and Industry (formerly Applied Stochastic Models and Data Analysis) was first published in 1985, publishing contributions in the interface between stochastic modelling, data analysis and their applications in business, finance, insurance, management and production. In 2007 ASMBI became the official journal of the International Society for Business and Industrial Statistics (www.isbis.org). The main objective is to publish papers, both technical and practical, presenting new results which solve real-life problems or have great potential in doing so. Mathematical rigour, innovative stochastic modelling and sound applications are the key ingredients of papers to be published, after a very selective review process.
The journal is very open to new ideas, like Data Science and Big Data stemming from problems in business and industry or uncertainty quantification in engineering, as well as more traditional ones, like reliability, quality control, design of experiments, managerial processes, supply chains and inventories, insurance, econometrics, financial modelling (provided the papers are related to real problems). The journal is interested also in papers addressing the effects of business and industrial decisions on the environment, healthcare, social life. State-of-the art computational methods are very welcome as well, when combined with sound applications and innovative models.