Bayesian influence analysis: a geometric approach

Abstract

Summary The aim of this paper is to develop a general framework of Bayesian influence analysis for assessing various perturbation schemes to the data, the prior and the sampling distribution for a class of statistical models. We introduce a perturbation model to characterize these various perturbation schemes. We develop a geometric framework, called the Bayesian perturbation manifold, and use its associated geometric quantities including the metric tensor and geodesic to characterize the intrinsic structure of the perturbation model. We develop intrinsic influence measures and local influence measures based on the Bayesian perturbation manifold to quantify the effect of various perturbations to statistical models. Theoretical and numerical examples are examined to highlight the broad spectrum of applications of this local influence method in a formal Bayesian analysis.

Local and Global Robustness With Conjugate and Sparsity Priors

“…They also give ways to calculate I for different exponential family distribution. In particular, for a normal distribution the approximation error In (Zhu et al, 2011, Definition 1) a perturbation manifold is defined to be a triplet consisting of the space of perturbations M, a proper metric closely related to Fisher Information, and the corresponding Levi-Civita connection. Examples of this perturbation space include an additive -contamination class and can include linear and non-linear perturbation schemes.…”

“…Specifically, in a Bayesian context, when priors have been chosen, the sensitivity of the posterior to prior choice is an important issue. A rich literature on sensitivity to perturbations of data, prior and sampling distribution exists, see for example: Cook (1986), McCulloch (1989), Lavine (1991), Ruggeri and Wasserman (1993), Blyth (1994), Gustafson (1996), Critchley and Marriott (2004), Linde (2007), Zhu et al (2007), Zhu, Ibrahim, and Tang (2011), Zhu, Ibrahim, and Tang (2014) and Kurtek and Bharath (2015).…”

“…We also can compare our method with the geometric approach of Zhu, Ibrahim, and Tang (2011) which uses a manifold based approach. Our, more linear approach, considStatistica Sinica: Newly accepted Paper (accepted author-version subject to English editing) erably improves interpretability and tractability while sharing a similar underlying geometric foundation.…”

Bayesian diagnostics of transformation structural equation models

“…Very few studies have been conducted in the context of the Bayesian approach. Recently, Zhu et al (2011) proposed a geometric approach to conduct Bayesian influence analysis. Given that outliers and/or influential points are usually not obvious for non-normal data, developing sound statistical diagnostic methods for such data is challenging and important.…”

“…Given that outliers and/or influential points are usually not obvious for non-normal data, developing sound statistical diagnostic methods for such data is challenging and important. Motivated by the work of Zhu et al (2011), we develop a Bayesian diagnostic procedure for the local influence analysis of transformation SEMs. Compared with conventional ML-based diagnostic procedures, the Bayesian approach could detect outliers and/or influential points in the observed data, as well as conduct model comparison and sensitivity analysis through various perturbations of the data, sampling distributions, or the prior distributions of model parameters.…”