Better together? Statistical learning in models made of modules

Abstract

In modern applications, statisticians are faced with integrating heterogeneous data modalities relevant for an inference, prediction, or decision problem. In such circumstances, it is convenient to use a graphical model to represent the statistical dependencies, via a set of connected 'modules', each relating to a specific data modality, and drawing on specific domain expertise in their development.In principle, given data, the conventional statistical update then allows for coherent uncertainty quantification and information propagation through and across the modules. However, misspecification of any module can contaminate the estimate and update of others, often in unpredictable ways. In various settings, particularly when certain modules are trusted more than others, practitioners have preferred to avoid learning with the full model in favor of approaches that restrict the information propagation between modules, for example by restricting propagation to only particular directions along the edges of the graph. In this article, we investigate why these modular approaches might be preferable to the full model in misspecified settings. We propose principled criteria to choose between modular and full-model approaches. The question arises in many applied settings, including large stochastic dynamical systems, meta-analysis, epidemiological models, air pollution models, pharmacokinetics-pharmacodynamics, and causal inference with propensity scores.

Asymptotics of cut distributions and robust modular inference using Posterior Bootstrap

“…As shown by Zigler et al [2013], performing the full Bayesian inference is possible, but leads to bias in the estimation of causal effects; see also Zigler [2016]. A review of various applications of modular inference is presented in Jacob et al [2017].…”

“…As shown in Lemma 1 of [Yu et al, 2021], the cut posterior π cut (θ 1 , θ 2 ) minimizes the Kullback-Leibler divergence between the families of densities q(θ 1 , θ 2 ) having π 1,cut as their marginal distribution with respect to θ 1 , and the full posterior π(θ 1 , θ 2 ). Both Jacob et al [2017] and Yu et al [2021] propose data-driven methods to assess whether to cut feedback or not. Carmona and Nicholls [2020] extend the modular methodology to "semimodular" inference (SMI), where the user can regulate the amount of feedback passed on from the second module to the first one.…”

Causal inference under mis-specification: adjustment based on the propensity score

“…Methods that posit the capability of recovering the correct components of the outcome model using flexible modelling without reference to the propensity score also provide valid routes to inference about this target, but these methods often carry a heavier computational burden. There are also links to modularized Bayesian inference (Bayarri et al, 2009;Jacob et al, 2017) which also depend on a 'conscious misspecification' formulation, and in the causal setting (the main examples and the examples in section 8) existing frequentist semiparametric theory can give insight into the operating characteristics of such Bayesian analyses; see Pompe and Jacob (2021) for initial explorations in this direction.…”

“…Bayarri et al (2009) argue for a form of Bayesian inference based on 'modularization' of the model, where a form of stagewise analysis in complex models is used. Motivated by formulations based on Bayesian mis-specification, Jacob et al (2017) provide extensive evidence that such modularized inference can be advantageous in Bayesian settings, including a study of the empirical properties of propensity score regression estimators using the methods from section 3.1.…”

“…where γ OPT is itself a loss-minimizing quantity, say the posterior mode or mean, derived using the variational solution from ( 22), which under an independent prior specification is the conventional posterior for γ. The formulation in (39) evidently leads to a form of 'modularized' inference as advocated by Bayarri et al (2009); Zigler (2016) and Jacob et al (2017). In this setting, however, due to the requirement to use a 'best estimate' of γ in order to produce consistent estimation of τ , the modularization is a necessary step rather than a choice the Bayesian analyst may opt to make.…”