A General Framework for Cutting Feedback within Modularised Bayesian Inference

TL;DR

This work provides a general, graph-based framework for cutting feedback in modularised Bayesian inference by defining self-contained Bayesian modules within arbitrary DAGs. It formalizes module construction, ordering, and cut distributions, and proves that cut distributions minimize a KL divergence from the joint distribution, ensuring principled decoupling of misspecified components. The authors extend the two-module cut paradigm to three and multiple modules via sequential splitting, with theoretical guarantees and illustrative examples in epidemiology and longitudinal modeling demonstrating bias reduction under misspecification. The framework enables robust, modular inference in complex models where partial misspecification is unavoidable, with practical guidance on module grouping, ordering, and cut-subgraph construction. Overall, the paper contributes a rigorous, scalable approach to modular Bayesian inference with concrete procedures and proofs for ensuring well-defined, interpretable inferences under misspecification.

Abstract

Standard Bayesian inference can build models that combine information from various sources, but this inference may not be reliable if components of a model are misspecified. Cut inference, as a particular type of modularized Bayesian inference, is an alternative which splits a model into modules and cuts the feedback from the suspect module. Previous studies have focused on a two-module case, but a more general definition of a "module" remains unclear. We present a formal definition of a "module" and discuss its properties. We formulate methods for identifying modules; determining the order of modules; and building the cut distribution that should be used for cut inference within an arbitrary directed acyclic graph structure. We justify the cut distribution by showing that it not only cuts the feedback but also is the best approximation satisfying this condition to the joint distribution in the Kullback-Leibler divergence. We also extend cut inference for the two-module case to a general multiple-module case via a sequential splitting technique and demonstrate this via illustrative applications.

Figures (9)

• Figure 1: Self-contained Bayesian module. Squares denote observable random variables and circles denote parameters. The dashed part is a minimally self-contained Bayesian module for $Y_{1:m}$ (see Definition \ref{['DE1']}).
• Figure 2: Venn diagram illustrating the partition of $\Psi$ and the links between these sets. Arrows (in green) indicate links that may exist, whereas crossed arrows (in red) indicate links that cannot exist. The crossed links (in blue) indicate a v-structure that cannot exist. The numbers on links indicate the part of Lemma \ref{['l1']} that proves the corresponding property.
• Figure 3: Venn diagram illustrating the splitting of module $T$ into modules $B$ and $C$. The arrows represent all directed edges from modules to $S=(A\cup T)^\mathsf{c}$.
• Figure 4: Module order graphs when splitting two modules into three modules. Circles represent modules and arrows reflect the direction of ordering between two modules. Each panel depicts the possible module orderings of three modules when splitting module $T$. Panel (a) shows the orderings when the original order is $A\rightharpoonup T$: from top to bottom $(A, B)\rightharpoonup C$; $A\rightharpoonup B\rightharpoonup C$; $A\rightharpoonup (B, C)$; and $(C, (A\rightharpoonup B))$. Panel (b) shows the orderings when the original two modules $A$ and $T$ are unordered: from top to bottom $(A, (B\rightharpoonup C))$; and $(A, B, C)$.
• Figure 5: Grouped module order graphs when splitting module $T$ with both ancestor modules $A$ and descendant modules $D$. Circles represent modules or grouped modules. Solid arrows indicate the direction of the ordering between two modules or module groups. Dashed arrows represent a possible direction of ordering between two modules or module groups. (a) the original grouped module order graph before splitting module $T$. Panels (b)-(i) depict all possible grouped module order graphs after splitting module $T$ in modules $B$ and $C$.

Theorems & Definitions (23)

• Definition 1: $X^\ast$-associated parameters and observables
• Example 1
• Definition 2: Self-contained Bayesian module for $X^\ast$
• Example 1: continued
• Lemma 1
• Lemma 2
• Example 1: continued
• Theorem 1
• Example 1: continued
• Lemma 3