Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data

Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. We further develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models, including preferential sampling models with an intractable spatial normalizing integral. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application further demonstrates the utility of the mixed-domain construction for spatial preferential modeling.

Simulation-based parameter estimation via a combination of embedded normalizing flows and implied empirical probabilities under moment restrictions

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

FedSPM: Routing-Enabled Federated Learning under Dual Heterogeneity via Semiparametric Mixture

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 $\mathcal{O}(1/\sqrt{T})$ 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.