On the Robustness of Bayesian Modelling of Location and Scale Structures Using Heavy-Tailed Distributions

Robustness to outliers in location–scale parameter model using log-regularly varying distributions

“…We first consider µ ∈ R and δ/M ≤ σ < ∞, where M is the constant of monotonicity given in equation (1).…”

“…In step b, we use 1/f (x i ) ≤ |b i |D(|a i /b i |, |b i |)/f (ω). In step c, we set b i = 0 if k i = 1 and we use f (−µ/σ) = f (µ/σ) by symmetry of f .It suffices to show thatω/σ ωf (ω) l+r [f (µ/σ)] k−(b i ω − µ)/σ)] l i +r i < ∞, since (1/σ) k−1/2 (1/σ)f (µ/σ) is an integrable function on Quadrant 1, ∞ /σ)f (µ/σ) dµ dσ ≤ ∞To achieve this, we split the region of σ into three parts between 1 < ω 1/2 < ω/(2M ) < ∞, where M is defined in equation(1). Note that since ω ≥ x 0 , this is well defined if x 0 > max(1, (2M ) 2 ).Consider 0 ≤ µ < ∞ and ω/(2M ) ≤ σ < ∞.…”

“…To achieve this, we split the region of σ into three parts between 1 < ω 1/2 < ω/(2M ) < ∞, where M is defined in equation (1). Note that since ω ≥ x 0 , this is well defined if…”

A General Framework for Cutting Feedback within Modularized Bayesian Inference

“…One class of approaches adopts a tempered likelihood where the likelihood is raised to a power between 0 and 1, leading to a power posterior or fractional posterior, and a Bayesian update is conducted thereafter (e.g., Friel and Pettitt, 2008;Bissiri et al, 2016;Holmes and Walker, 2017;Bhattacharya et al, 2019;Miller and Dunson, 2019). Another class of approaches replace the distribution of the likelihood with heavy-tailed distributions, for example via individual-specific variance parameters, to account for conflicting information sources (e.g., O'Hagan and Pericchi, 2012;Andrade et al, 2013;Wang and Blei, 2018). However, especially when dealing with a complex true generating process, it is impossible to allocate equal confidence in all aspects of the model.…”