March 2023 Expansions for posterior distributions
Christopher Withers, Saralees Nadarajah
Author Affiliations +
Braz. J. Probab. Stat. 37(1): 73-100 (March 2023). DOI: 10.1214/22-BJPS561


Suppose that Xn is a sample of size n with log likelihood nl(θ), where θ is an unknown parameter in Rp having a prior distribution ξ(θ). We need not assume that the sample values are independent or even stationary. Let θˆ be the maximum likelihood estimate (MLE). We show that θ|Xn is asymptotically normal with mean θˆ and covariance n1l,(θˆ)1, where l,(θ)=2l(θ)/θθ. In contrast, θˆ|θ is asymptotically normal with mean θ and covariance n1[I(θ)]1, where I(θ)=E[l,(θˆ)|θ] 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,(θˆ)|θI(θ) in probability. So, Bayesian inference (that is, conditional on Xn) 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 Lt(θ)|Xn, as well as confidence regions for t(θ)|Xn of high accuracy. We also give expansions for the Bayes estimate (estimator) of t(θ) about t(θˆ), and for the maximum a posteriori estimate about θˆ, as well as their relative efficiencies with respect to squared error loss.


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Christopher Withers. Saralees Nadarajah. "Expansions for posterior distributions." Braz. J. Probab. Stat. 37 (1) 73 - 100, March 2023.


Received: 1 September 2022; Accepted: 1 December 2022; Published: March 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580885
zbMATH: 07692850
Digital Object Identifier: 10.1214/22-BJPS561

Keywords: Cumulant expansions , Edgeworth–Cornish–Fisher expansions , maximum likelihood estimates

Rights: Copyright © 2023 Brazilian Statistical Association


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Vol.37 • No. 1 • March 2023
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