Geometric ergodicity of Gibbs samplers for linear latent models with GIG variance mixtures
Elsiddig Awadelkarim, David Bolin, Xiaotian Jin, Alexandre B. Simas, Jonas Wallin
TL;DR
This work establishes robust geometric ergodicity for Gibbs samplers in LLnGMs with GIG variance mixtures by proving trace-class properties in interior regimes and deploying drift–minorization in boundary regimes, including explicit null-smallness conditions. The results cover the full GIG parameter space and special GH cases such as NIG and GAL, ensuring the validity of Gibbs-based stochastic-gradient estimators for maximum likelihood. A non-centered parameterization is shown to improve integrability of score functions, enabling principled Rao–Blackwellized SGD with ergodic guarantees. Numerical experiments corroborate the theory, illustrating how mixing varies across parameter regimes and how null-smallness affects convergence along null directions. This framework provides a scalable, theoretically grounded path for inference in a broad class of latent non-Gaussian models while offering practical guidance for algorithm design.
Abstract
We study geometric ergodicity of the Gibbs sampler for linear latent non-Gaussian models (LLnGMs), a class of hierarchical models in which conditional Gaussian structure is preserved through generalized inverse Gaussian (GIG) variance-mixture augmentation. Two complementary routes to geometric ergodicity are developed for the marginal chain on the mixing variables. First, we show that the associated Markov operator is trace-class, and hence admits a spectral gap, over a large portion of the GIG parameter space. Second, for the remaining boundary and heavy-tail regimes, we establish geometric ergodicity via drift and minorization, subject to an explicit null-smallness condition that quantifies how the drift interacts with the null space of the observation operator. Together, these results cover the full GIG parameter space, including the normal-inverse Gaussian, generalized asymmetric Laplace, and Student-$t$ special cases. The geometric ergodicity of this chain underpins the consistency of Gibbs-based stochastic-gradient estimators for maximum likelihood estimation, and we provide conditions that make the required integrability checks transparent. Numerical experiments illustrate the theoretical findings, contrasting mixing efficiency across parameter regimes and probing the role of the null-smallness constant.