Bayesian correction for covariate measurement error: a frequentist evaluation and comparison with regression calibration

Bayesian approaches for handling covariate measurement error are well established, and yet arguably are still relatively little used by researchers. For some this is likely due to unfamiliarity or disagreement with the Bayesian inferential paradigm. For others a contributory factor is the inability of standard statistical packages to perform such Bayesian analyses. In this paper we first give an overview of the Bayesian approach to handling covariate measurement error, and contrast it with regression calibration (RC), arguably the most commonly adopted approach. We then argue why the Bayesian approach has a number of statistical advantages compared to RC, and demonstrate that implementing the Bayesian approach is usually quite feasible for the analyst. Next we describe the closely related maximum likelihood and multiple imputation approaches, and explain why we believe the Bayesian approach to generally be preferable. We then empirically compare the frequentist properties of RC and the Bayesian approach through simulation studies. The flexibility of the Bayesian approach to handle both measurement error and missing data is then illustrated through an analysis of data from the Third National Health and Nutrition Examination Survey

Distance Estimation in Cosmology

In this paper we outline the framework of mathematical statistics with which one may study the properties of galaxy distance estimators. We describe, within this framework, how one may formulate the problem of distance estimation as a Bayesian inference problem, and highlight the crucial question of how one incorporates prior information in this approach. We contrast the Bayesian approach with the classical `frequentist' treatment of parameter estimation, and illustrate -- with the simple example of estimating the distance to a single galaxy in a redshift survey -- how one can obtain a significantly different result in the two cases. We also examine some examples of a Bayesian treatment of distance estimation -- involving the definition of Malmquist corrections -- which have been applied in recent literature, and discuss the validity of the assumptions on which such treatments have been based.Comment: Plain Latex version 3.1, 18 pages + 2 figures, `Vistas in Astronomy' in pres

Global consensus Monte Carlo

To conduct Bayesian inference with large data sets, it is often convenient or necessary to distribute the data across multiple machines. We consider a likelihood function expressed as a product of terms, each associated with a subset of the data. Inspired by global variable consensus optimisation, we introduce an instrumental hierarchical model associating auxiliary statistical parameters with each term, which are conditionally independent given the top-level parameters. One of these top-level parameters controls the unconditional strength of association between the auxiliary parameters. This model leads to a distributed MCMC algorithm on an extended state space yielding approximations of posterior expectations. A trade-off between computational tractability and fidelity to the original model can be controlled by changing the association strength in the instrumental model. We further propose the use of a SMC sampler with a sequence of association strengths, allowing both the automatic determination of appropriate strengths and for a bias correction technique to be applied. In contrast to similar distributed Monte Carlo algorithms, this approach requires few distributional assumptions. The performance of the algorithms is illustrated with a number of simulated examples

Bayesian Bootstrap Analysis of Systems of Equations

Research Methods/ Statistical Methods,

Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification

We consider the high energy physics unfolding problem where the goal is to estimate the spectrum of elementary particles given observations distorted by the limited resolution of a particle detector. This important statistical inverse problem arising in data analysis at the Large Hadron Collider at CERN consists in estimating the intensity function of an indirectly observed Poisson point process. Unfolding typically proceeds in two steps: one first produces a regularized point estimate of the unknown intensity and then uses the variability of this estimator to form frequentist confidence intervals that quantify the uncertainty of the solution. In this paper, we propose forming the point estimate using empirical Bayes estimation which enables a data-driven choice of the regularization strength through marginal maximum likelihood estimation. Observing that neither Bayesian credible intervals nor standard bootstrap confidence intervals succeed in achieving good frequentist coverage in this problem due to the inherent bias of the regularized point estimate, we introduce an iteratively bias-corrected bootstrap technique for constructing improved confidence intervals. We show using simulations that this enables us to achieve nearly nominal frequentist coverage with only a modest increase in interval length. The proposed methodology is applied to unfolding the $Z$ boson invariant mass spectrum as measured in the CMS experiment at the Large Hadron Collider.Comment: Published at http://dx.doi.org/10.1214/15-AOAS857 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1401.827