Expansions for posterior distributions

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
C. Withers, S. Nadarajah
{"title":"Expansions for posterior distributions","authors":"C. Withers, S. Nadarajah","doi":"10.1214/22-bjps561","DOIUrl":null,"url":null,"abstract":": Suppose that X n is a sample of size n with log likelihood nl ( θ ), where θ is an unknown parameter in R p having a prior distribution ξ ( θ ). We need not assume that the sample values are independent or even stationary. Let (cid:98) θ be the maximum likelihood estimate (MLE). We show that θ | X n is asymptotically normal with mean (cid:98) θ and covariance − n − 1 l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) − 1 , where l (cid:5) , (cid:5) ( θ ) = ∂ 2 l ( θ ) /∂θ∂θ ′ . In contrast (cid:98) θ | θ is asymptotically normal with mean θ and covariance n − 1 [ I ( θ )] − 1 , where I ( θ ) = − E (cid:104) l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105) is Fisher’s information. So, frequentist inference conditional on θ cannot be used to approximate Bayesian inference, except for exponential families. However, under mild conditions − l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ → I ( θ ) in probability. So, Bayesian inference (that is, conditional on X n ) can be used to approximate frequentist inference. For t ( θ ) any smooth function, we obtain posterior cumulant expansions, posterior Edgeworth-Cornish-Fisher (ECF) expansions and posterior tilted Edgeworth expansions for L t ( θ ) | X n , as well as confidence regions for t ( θ ) | X n of high accuracy. We also give expansions for the Bayes estimate (estimator) of t ( θ ) about t (cid:16)(cid:98) θ (cid:17) , and for the maximum a posteriori estimate about (cid:98) θ , as well as their relative efficiencies with respect to squared error loss.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/22-bjps561","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

: Suppose that X n is a sample of size n with log likelihood nl ( θ ), where θ is an unknown parameter in R p having a prior distribution ξ ( θ ). We need not assume that the sample values are independent or even stationary. Let (cid:98) θ be the maximum likelihood estimate (MLE). We show that θ | X n is asymptotically normal with mean (cid:98) θ and covariance − n − 1 l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) − 1 , where l (cid:5) , (cid:5) ( θ ) = ∂ 2 l ( θ ) /∂θ∂θ ′ . In contrast (cid:98) θ | θ is asymptotically normal with mean θ and covariance n − 1 [ I ( θ )] − 1 , where I ( θ ) = − E (cid:104) l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105) is Fisher’s information. So, frequentist inference conditional on θ cannot be used to approximate Bayesian inference, except for exponential families. However, under mild conditions − l (cid:5) , (cid:5) (cid:16)(cid:98) θ (cid:17) | θ → I ( θ ) in probability. So, Bayesian inference (that is, conditional on X n ) can be used to approximate frequentist inference. For t ( θ ) any smooth function, we obtain posterior cumulant expansions, posterior Edgeworth-Cornish-Fisher (ECF) expansions and posterior tilted Edgeworth expansions for L t ( θ ) | X n , as well as confidence regions for t ( θ ) | X n of high accuracy. We also give expansions for the Bayes estimate (estimator) of t ( θ ) about t (cid:16)(cid:98) θ (cid:17) , and for the maximum a posteriori estimate about (cid:98) θ , as well as their relative efficiencies with respect to squared error loss.
后验分布的展开式
:设X n是一个大小为n的具有对数似然nl (θ)的样本,其中θ是R p中具有先验分布ξ (θ)的未知参数。我们不需要假设样本值是独立的,甚至是平稳的。设(cid:98) θ为最大似然估计(MLE)。我们证明了θ | X n是渐近正态的,具有均值(cid:98) θ和协方差- n−1 l (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17)−1,其中l (cid:5), (cid:5) (θ) =∂2 l (θ) /∂θ∂θ '。相反,(cid:98) θ | θ渐近正态,均值θ和协方差n−1 [I (θ)]−1,其中I (θ) =−E (cid:104) l (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17) | θ (cid:105)为Fisher信息。所以,以θ为条件的频率推理不能用来近似贝叶斯推理,除了指数族。然而,在温和条件下,−1 (cid:5), (cid:5) (cid:16)(cid:98) θ (cid:17) | θ→I (θ)的概率。因此,贝叶斯推理(即以X n为条件)可以用来近似频率推理。对于t (θ)任意光滑函数,我们得到了L t (θ) |xn的后向累积展开式、后向Edgeworth- cornish - fisher (ECF)展开式和后向倾斜Edgeworth展开式,以及t (θ) |xn高精度的置信区域。我们还给出了t (θ)关于t (cid:16)(cid:98) θ (cid:17)的贝叶斯估计(估计量)的展开式,以及关于(cid:98) θ的最大后验估计,以及它们相对于平方误差损失的相对效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
1.60
自引率
10.00%
发文量
30
审稿时长
>12 weeks
期刊介绍: The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes. More specifically, the following types of contributions will be considered: (i) Original articles dealing with methodological developments, comparison of competing techniques or their computational aspects. (ii) Original articles developing theoretical results. (iii) Articles that contain novel applications of existing methodologies to practical problems. For these papers the focus is in the importance and originality of the applied problem, as well as, applications of the best available methodologies to solve it. (iv) Survey articles containing a thorough coverage of topics of broad interest to probability and statistics. The journal will occasionally publish book reviews, invited papers and essays on the teaching of statistics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信