Cutting feedback and modularized analyses in generalized Bayesian inference

Abstract

Even in relatively simple settings, model misspecification can make the application and interpretation of Bayesian inference difficult. One approach to make Bayesian analyses fit-for-purpose in the presence of model misspecification is the use of cutting feedback methods. These methods modify conventional Bayesian inference by limiting the influence of one part of the model by "cutting" the link between certain components. We examine cutting feedback methods in the context of generalized posterior distributions, i.e., posteriors built from arbitrary loss functions, and provide novel results on their behaviour. A direct product of our results are diagnostic tools that allow for the quick, and easy, analysis of two key features of cut posterior distributions: one, how uncertainty about the model unknowns in one component impacts inferences about unknowns in other components; two, how the incorporation of additional information impacts the cut posterior distribution.

Guaranteed Accuracy of Semi-Modular Posteriors

“…In the case of Bayesian inference, this philosophy has led researchers to conduct inference using the so-called cut posterior distribution (see, Plummer, 2015, Jacob et al, 2017. As shown in Carmona and Nicholls (2020) and Nicholls et al (2022a), the cut posterior is a "generalized" posterior distribution (see, e.g., Bissiri et al, 2016) that restricts the information flow to guard against model misspecification (Frazier and Nott, 2022). In the canonical two module system, the cut posterior takes the form…”

“…Similar to their analysis, see, e.g., Section 2.1 in Hjort and Claeskens (2003), Assumption 1 creates a level of ambiguity as to whether or not the model is misspecified. This misspecification framework differs from the designs in Pompe and Jacob (2021), and Frazier and Nott (2022), where it is known with probability converging to one if the model is misspecified. The key feature of Assumption 1 is that the rate at which we learn about misspecification, √ n, is the same rate with which sample information accumulates, thus ensuring that there is always a level of ambiguity regarding whether the model is correctly specified.…”

“…Semi-modular inference interpolates between cut and full posterior distributions continuously using a tuning parameter ω ∈ [0, 1], where ω = 0 corresponds to the cut posterior and ω = 1 to the full posterior. Further developments and applications are discussed in Liu and Goudie (2021), Carmona and Nicholls (2022), Nicholls et al (2022a) and Frazier and Nott (2022). The motivation for semi-modular inference is the existence of a bias-variance trade-off: as Carmona and Nicholls (2020) explain, "In Cut-model inference, feedback from the suspect module is completely cut.…”

Modularized Bayesian analyses and cutting feedback in likelihood-free inference

“…There are many other applications of cutting feedback methods -see Bayarri et al (2009) and Jacob et al (2017) for further discussion of these and modularized Bayesian analyses more generally. Pompe and Jacob (2021) and Frazier and Nott (2022) have recently studied the theoretical behaviour of the cut posterior distribution.…”