Geometric ergodicity of Gibbs samplers for linear latent models with GIG variance mixtures

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.

💡 Research Summary

This paper investigates the geometric ergodicity of Gibbs samplers applied to linear latent non‑Gaussian models (LLnGMs), a hierarchical class where non‑Gaussianity is introduced through generalized inverse Gaussian (GIG) variance‑mixture augmentation. The authors develop two complementary routes to establish geometric ergodicity for the marginal Markov chain on the mixing variables V.

First, they prove that the associated Markov operator is trace‑class over a large interior region of the GIG parameter space (a>0, b>0, arbitrary p). By showing that the kernel of the operator is square‑integrable, they demonstrate that its singular values are summable, which implies compactness and the existence of a spectral gap. Consequently, the chain converges to its stationary distribution at an exponential rate on this region.

Second, for the remaining boundary regimes (inverse‑Gamma and Gamma limits) and heavy‑tail settings, the trace‑class argument fails. Here the authors employ a drift‑minorization framework. They construct a Lyapunov function V(x)=1+‖x‖² and verify a linear drift condition E