Motivated by the privacy, sensitivity and sharing limitations of health data, we present a comprehensive pipeline for inference of Bayesian mixture models within a federated learning setting, i.e. when data cannot be fully shared or pooled across compute nodes. We adopt a Consensus Monte Carlo (CMC) approach, in which an MCMC algorithm is run independently within each data silo to estimate local posterior distributions, which are then aggregated to approximate the posterior over the full data. The variational CMC approach of Rabinovich, Angelino and Jordan (2015) [1] frames the aggregation step as a variational inference problem, but their application to mixtures assumes the number of clusters and key mixture parameters to be known. Our main methodological contributions are: (i) an extension of variational CMC to over-fitted Bayesian mixture models that infer the number of clusters and all model parameters, without requiring conjugacy; (ii) novel cluster-matching algorithms suitable for cross-silo settings in which not every cluster appears in each local dataset; (iii) a number of inference strategies for the aggregation step, matched to different federated learning constraints; and (iv) guidelines for choosing among these in practice. A comprehensive simulation study validates the framework and allows us to compare to state-of-the-art federated learning alternatives. Notably, we show that when the composition of local datasets reflects the underlying clustering structure in the data, our approach can recover small clusters with greater accuracy than standard MCMC applied to the pooled data. We illustrate the framework on large-scale electronic health record data, identifying multi-morbidity patterns in a British geriatric population.
Federated Learning (FL) facilitates collaborative model training across decentralized clients while preserving data privacy by avoiding raw data exchange. Despite its potential, FL performance is often compromised by data heterogeneity across clients. To address this, Clustered Federated Learning (CFL) groups clients with similar data distributions to improve model performance, but constrained by intra-cluster heterogeneity. Conversely, Personalized Federated Learning (PFL) tailors models to individual clients, but usually neglects the underlying structural similarities among clients. In this work, we investigate a probabilistic mixture (PM) scenario, where each client's local data distribution is modeled as a convex combination of several shared inherent distributions. To effectively model this structure, we propose FedGMI, a framework that utilizes Variational Autoencoders (VAEs) as generative density estimators to represent these inherent distributions and infer the mixture components of clients' local data distributions. This approach enables structured personalization without sacrificing the benefits of collaborative learning. Extensive experiments demonstrate that FedGMI effectively characterizes and discriminate the inherent distributions, as well as accurately estimates mixture proportions. Furthermore, FedGMI maintains robust performance even under communication cost constraints.
Federated learning trains a shared model from private client data. In practice, data-generating distributions may differ, and the true mixture across clients is often unknown, making the underlying group distribution difficult to specify. Existing approaches address cross-client mixture uncertainty by optimizing against the worst-case mixture, yet assume accurate client-wise distribution estimates. However, these estimates can be unreliable when based on finite samples. To handle both cross-client mixture uncertainty and within-client distributional ambiguity, we construct a global ambiguity set as the union of admissible mixtures of local ambiguity sets. The construction allows client-specific ambiguity radii and admits a client-wise separable reformulation. Leveraging this structure, we establish a high-probability out-of-sample performance guarantee. We further develop a federated algorithm for a penalty-based reformulation and prove its convergence under milder regularity conditions. Simulations validate the algorithm's effectiveness.
Routing-prediction federated learning has emerged as a new paradigm that reframes inter-client heterogeneity as a resource for system-level intelligence: at inference time, the server routes each external query to the best-matched client for prediction. Existing approaches, however, typically treat each client as internally homogeneous, overlooking latent subpopulations within local data. For example, patients with the same diagnosis at one hospital may exhibit morphologically distinct disease subtypes. The coexistence of inter-client and intra-client heterogeneity, which we call dual heterogeneity, can impair both routing and prediction. To address this challenge, we propose FedSPM, a routing-enabled semiparametric mixture framework that represents each client using client-specific latent components. Each component combines a predictive distribution for classification with a feature distribution for routing. To flexibly model feature distributions while effectively sharing information across clients, FedSPM models their density ratios relative to a common nonparametric measure estimated via empirical likelihood. We develop a federated expectation-maximization algorithm that optimizes a tractable surrogate and prove convergence of the exact profiled objective at the standard O(1/T) rate when the surrogate errors are properly controlled. Experiments on controlled benchmarks and real-world medical data demonstrate consistent improvements in routing and prediction under dual heterogeneity. Code is available at https://github.com/zijianwang0510/FedSPM.