Valid belief updates for prequentially additive loss functions arising in Semi-Modular Inference

Abstract

Model-based Bayesian evidence combination leads to models with multiple parameteric modules. In this setting the effects of model misspecification in one of the modules may in some cases be ameliorated by cutting the flow of information from the misspecified module. Semi-Modular Inference (SMI) is a framework allowing partial cuts which modulate but do not completely cut the flow of information between modules. We show that SMI is part of a family of inference procedures which implement partial cuts. It has been shown that additive losses determine an optimal, valid and order-coherent belief update. The losses which arise in Cut models and SMI are not additive. However, like the prequential score function, they have a kind of prequential additivity which we define. We show that prequential additivity is sufficient to determine the optimal valid and order-coherent belief update and that this belief update coincides with the belief update in each of our SMI schemes.

Cutting feedback and modularized analyses in generalized Bayesian inference

“…The (marginal) semi-modular posterior π γ (ϕ|z, w) attenuates the impact of the feedback term p(w|ϕ) by tempering its contribution in the (marginal) posterior. Moreover, similar to the (joint) semi-modular posteriors presented in Carmona and Nicholls (2020) and Nicholls et al (2022), the (marginal) semi-modular posterior π γ (ϕ|z, w) can be seen as interpolating between the cut posterior π cut (ϕ|z), at γ = 0, and the full (marginal) posterior π(ϕ|z, w), at γ = 1.…”

“…Recently, Carmona and Nicholls (2020) have proposed the use of semi-modular posterior distributions for cut-model inference; Nicholls et al (2022) extend this construction to prequentially additive loss functions, and Carmona and Nicholls (2022) investigate the use of normalizing flows in the construction of semi-modular posteriors. Semi-modular posterior inference allow us to robustly incorporate information from potentially misspecified modules within our posterior beliefs.…”

“…Such generalized Bayesian inference approaches, see, e.g., Bissiri et al (2016), are now common in statistical inference, and allow the researcher to hedge against misspecification of a full statistical model. There is also recent work (Nicholls et al, 2022) on generalized Bayes justifications of parametric modular methods, which we discuss later.…”

Modularized Bayesian analyses and cutting feedback in likelihood-free inference

“…Carmona and Nicholls (2020) suggest choosing γ using predictive methods. More recently, Nicholls et al (2022) consider validity of semi-modular inference in a generalized Bayesian inference framework, and consider alternative forms of semi-modular inference. We consider one more alternative below for the likelihood-free setting.…”

“…To the best of our knowledge, our work is the first time that cutting feedback methods have been considered for likelihood-free inference. A semi-modular inference approach where feedback is partially cut is also developed, extending work by Carmona and Nicholls (2020) and Nicholls et al (2022) to the case of likelihood-free inference.…”