Bayesian conjugate analysis for federated statistical inference

Abstract

In many research settings, sufficiently large sample sizes can only be achieved by combining data from multiple collection sites. However, pooling individual participant data in a central server is often restricted due to regulatory constraints. Federated inference addresses this challenge by distributing the statistical analysis across local sites, allowing pooled inference in a central server using privacy-preserving summary statistics. Although federated inference methods exist in a frequentist framework, the full potential of Bayesian approaches has not yet been explored. Bayesian methods offer distinct advantages, including the ability to incorporate prior knowledge and perform predictive checks for model criticism. A recently published Bayesian method for federated inference relies on approximate solutions even in linear regression scenarios where exact solutions are available. We therefore propose a different approach to federated inference using Bayesian conjugate analysis (BCA), which is communication-efficient and mathematically convenient. For linear regression problems, BCA yields lossless parameter inference, that is, producing the same posterior distribution as if the pooled data had been analyzed. For problems where the parameters estimates are asymptotically normal (such as generalized linear models), BCA is equivalent to a multivariate fixed-effects meta-analysis, up to prior specification. We further show that BCA naturally lends itself to Reverse-Bayes analysis, which allows for computationally efficient predictive checks and identification of outlier sites. An implementation of BCA is available through the open-source confeR package (conjugate federated analysis in R). Our proposed framework thus facilitates privacy-preserving multi-center trials in a Bayesian setting.