Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It

Effectiveness of Bayesian filters: An information fusion perspective

“…An astonishing surge of various recursive filters/smoothers has been witnessed since [24,25]. 18 These recursive estimators, which have the Bayesian paradigm as the theoretically most elaborated base [26], perform well 19 as long as the models used are accurate, having few disturbances, and that the approximation (required in nonlinear systems) 20 is insignificant. Ideally, an optimality (e.g.…”

“…[19,34]), robust filtering (e.g. effective characteristics [7] Bayesian smoothers and predictors [1,6,18,48] as well as other recursive estimators e.g. optimization-based estimator [27,42,46].…”

Contextuality of Misspecification and Data-Dependent Losses

“…So, if the goal is to make good squared-error predictions, the Bayes posterior-if it concentrates at all-will concentrate on the squared-error risk-optimal θ in the model. In the terminology of Grünwald and Van Ommen (2014), the squared error is associated with the Gaussian regression model; in WH's terminology, the Gaussian regression model is suitable in the context of squared error prediction, even under misspecification. .…”

“…One then adds a second parameter η when determining the posterior. Grünwald and Van Ommen (2014) show that such an η is different from λ and σ −2 and cannot be absorbed, in general, into the likelihood itself; it needs to be added since setting η = 1 may cause the posterior never to converge at all under misspecification. While λ is then determined by standard Bayesian means (it is part of the prior and posterior), something different is needed for η: Grünwald and Van Ommen (2014) describe a data-dependent ("Safe Bayesian") method for finding it.…”

“…I was happy to see that it begins by pointing out that misspecification is contextual-a point that cannot be stressed enough, and that has also played a central part in my own work on Bayesian inconsistency under misspecification (Grünwald andLangford, 2007, Grünwald andVan Ommen, 2014). I will focus my comments on this aspect and on the developments in Section 4, which are the most closely related to my own work.…”