Bayesian Inference on Multivariate Medians and Quantiles

Bhattacharya, Indrabati; Ghosal, Subhashis

doi:10.5705/ss.202020.0108

Cited by 5 publications

(2 citation statements)

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“…BvM for median and quantiles under classical and Gibbs posterior [4], [5]. Interestingly, [46] show that Bayesian neural networks show inconsistency similar to that discussed in [16], applying variational Bayes leads to BNN becoming consistent; it would be interesting to study whether it is possible to achieve asymptotic efficiency.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

Bernstein - von Mises theorem and misspecified models: a review

Bochkina¹

2022

Preprint

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IntroductionConsider a family of probability models P (Y | θ) indexed by parameter θ ∈ Θ for observations y, and a prior distribution π on the parameter space Θ. In a classical Bayesian approach, the posterior distributionDenote the true distribution of observations P 0 , and we consider the casethe model is misspecified. Such case arises in many applications, particularly in complex models where the numerical evaluation of posterior distribution under the ideal probability model takes a long time to run, leading to increased use of approximate models with faster computing time. A typical example is approximating complex dependence structure by pairwise dependence only [2], [40].For well-specified regular models, the classical Bernstein -von Mises theorem states that for n independent identically distributed (iid) observations, for large n, the posterior distribution behaves approximately as a normal distribution centered on the true value of the parameter with a random shift, with both the posterior variance and the variance of the random shift being asymptotically equal to the inverse Fisher information, thus making the Bayesian inference asymptotically consistent and efficient.This was extended to locally asymptotically normal (LAN) models for Θ ∈ R p with fixed p [39], for LAN models with growing p, etc.A version of Bernstein -von Mises theorem is also available for nonregular models such as locally asymptotically exponential models ([20], [9]), models with parameter on the boundary of the parameter space [7] which hold under misspecified models. Here, however, we focus on "regular" models where estimators are asymptotically Gaussian.However, under model misspecification, Bayesian model is no longer optimal, as the posterior variance does not match the minimal lower bound on the variance of unbiased estimators [22] and [29]).Therefore, the standard way of constructing a posterior distribution following the Bayes theorem may not be appropriate for particular purposes, e.g. inference or prediction [27]. Different ways to construct a distribution of θ given Y that produces inference appropriate for the purpose of the analysis have been proposed. A natural aim for such a method is to behave like a standard Bayesian method when the

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Section: Discussion and Open Questionsmentioning

confidence: 99%

Bernstein - von Mises theorem and misspecified models: a review

Bochkina¹

2022

Preprint

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“…Spatial variations of the PP parameters are modelled using GAM; GAM is like linear regression, but replaces linear predictors with smoothing functions. We also use an experimental method from Ribatet et al (2012) to partially account for spatiotemporal correlation. The method is implemented through the R (R Core Team, 2016) package "evgam" (Youngman, 2022), which is available for public download via the R CRAN repository.…”

Section: Updated Approach To Estimate Return Levelmentioning

confidence: 99%