Geometric Ergodicity Research Articles

Geometric Ergodicity of a Stochastic Reaction–Diffusion Tuberculosis Model with Varying Immunity Period

Geometric Ergodicity of a Stochastic Reaction–Diffusion Tuberculosis Model with Varying Immunity Period

Geometric ergodicity of SGLD via reflection coupling

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact sets. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of [Formula: see text] distance. The generalization to non-gradient drifts is also included.

Split-and-merge model selection of mixtures of Gaussian processes with RJMCMC

The mixture of Gaussian processes is a powerful statistical learning model that can be effectively applied to curve clustering and prediction. However, the corresponding model selection problem, that is, selecting an appropriate number of components in the mixture, is rather difficult to solve. In our previous work, we established the split-and-merge automatic model selection algorithm for mixtures of Gaussian processes along the output space under the framework of Reversible Jump Markov Chain Monte Carlo (RJMCMC), which can not only determine the number of actual Gaussian processes but also dynamically adjust the Gaussian process components to avoid dependence on parameter initialization and initial partitioning of the dataset during the parameter learning on a given dataset. In this study, we propose two algorithms: Penalized Likelihood RJMCMC and Penalized Prior RJMCMC. The former integrates a penalized term into the likelihood, while the latter incorporates a penalized term into the prior and operates within the full Bayesian inference framework, both aiming to focus more sharply on determining the number of components in the convergence process. Furthermore, we prove the geometric ergodicity of the RJMCMC algorithm for the mixture of Gaussian processes model, ensuring convergence of the posterior distribution with sufficient iterations. The experimental results further demonstrate the robustness of our PP-RJMCMC algorithm in model selection, showing superior performance compared to traditional approaches in curve classification and clustering. Additionally, the prediction performance is comparable to the EM algorithm. Although not directly explored in this study, the RJMCMC results can be used to initialize the EM algorithm, which could potentially improve prediction accuracy and accelerate computation.

Convergence of hybrid slice sampling via spectral gap

Abstract It is known that the simple slice sampler has robust convergence properties; however, the class of problems where it can be implemented is limited. In contrast, we consider hybrid slice samplers which are easily implementable and where another Markov chain approximately samples the uniform distribution on each slice. Under appropriate assumptions on the Markov chain on the slice, we give a lower bound and an upper bound of the spectral gap of the hybrid slice sampler in terms of the spectral gap of the simple slice sampler. An immediate consequence of this is that the spectral gap and geometric ergodicity of the hybrid slice sampler can be concluded from the spectral gap and geometric ergodicity of the simple version, which is very well understood. These results indicate that robustness properties of the simple slice sampler are inherited by (appropriately designed) easily implementable hybrid versions. We apply the developed theory and analyze a number of specific algorithms, such as the stepping-out shrinkage slice sampling, hit-and-run slice sampling on a class of multivariate targets, and an easily implementable combination of both procedures on multidimensional bimodal densities.

A dynamic count process

The current paper aims to complement the recent development of the observation-driven models of dynamic counts with a parametric-driven one for a general case, particularly discrete two parameters exponential family distributions. The current paper proposes a finite semiparametric exponential mixture of SETAR processes of the conditional mean of counts to capture the nonlinearity and complexity. Because of the intrinsic latency of the conditional mean, the general additive state-space representation of dynamic counts is firstly proposed then stationarity and geometric ergodicity are established under a mild set of conditions. We also propose to estimate the unknown parameters by using quasi maximum likelihood estimation and establishes the asymptotic properties of the quasi maximum likelihood estimators (QMLEs), particularly T-consistency and normality under the relatively mild set of conditions. Furthermore, the finite sample properties of the QMLEs are investigated via simulation exercises and an illustration of the proposed process is presented by applying the proposed method to the intraday transaction counts per minute of AstraZeneca stock.

On the existence of stationary threshold bilinear processes

This article investigates some statistical and probabilistic properties of general threshold bilinear processes. Sufficient conditions for the existence of a causal, strictly and weak stationary solution for the equation defining a self-exciting threshold superdiagonal bilinear SETBL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( SETBL\\right) $$\\end{document} process are derived. Then it is shown that under well-specified hypotheses the higher-order moments of the SETBL process are finite. As a result, the skewness and kurtosis indexes are explicitly computed. The exact autocorrelation function is derived with an arbitrarily fixed number of regimes. Also, the covariance functions of the process and its powers are evaluated and the second (respectively, higher)-order structure is shown to be similar to that of a linear process. This implies that the considered process admits an ARMA representation. Finally, necessary and sufficient conditions for the invertibility and geometric ergodicity of a SETBL model are established. Some examples illustrate the obtained theoretical results.

Geometric ergodicity of Gibbs samplers for Bayesian error-in-variable regression

Geometric ergodicity of Gibbs samplers for Bayesian error-in-variable regression

Geometric Ergodicity for Hamiltonian Monte Carlo on Compact Manifolds

Geometric Ergodicity for Hamiltonian Monte Carlo on Compact Manifolds

Geometric ergodicity and ultimate boundedness of a stochastic chemostat model with general nutrient uptake function

The ergodicity of solutions for stochastic systems has been widely concerned by many scholars in recent years. Thus, in this paper, we further analyze the geometric ergodicity of a stochastic chemostat model with general nutrient uptake function. By verifying minorization condition and Lyapunov condition, geometric ergodicity of stochastic chemostat system is presented, which means the transfer probability of Markov process will converge to its limiting distribution at an exponentially rate. Meanwhile, we prove that the solution of stochastic system is stochastically ultimately bounded.

Posterior Computation with the Gibbs Zig-Zag Sampler

An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). We propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be updated in blocks with a zig-zag sampler applied to certain parameters and traditional MCMC-style updates to others. We demonstrate the flexibility of this framework on posterior sampling for logistic models with shrinkage priors for high-dimensional regression and random effects, and provide conditions for geometric ergodicity and the validity of a central limit theorem.

Explicit Convergence Rates for the M/G/1 Queue under Perturbation

This paper establishes convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are caculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi–linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction–diffusion equation. Together with the uniform geometric ergodicity, we obtain the validity of the small mass limit for the solutions on the infinite time horizon [0,∞). This extends previously known results established for the damped wave equations under Lipschitz nonlinearities.

Shrinkage with shrunken shoulders: Gibbs sampling shrinkage model posteriors with guaranteed convergence rates.

Use of continuous shrinkage priors - with a "spike" near zero and heavy-tails towards infinity - is an increasingly popular approach to induce sparsity in parameter estimates. When the parameters are only weakly identified by the likelihood, however, the posterior may end up with tails as heavy as the prior, jeopardizing robustness of inference. A natural solution is to "shrink the shoulders" of a shrinkage prior by lightening up its tails beyond a reasonable parameter range, yielding a regularized version of the prior. We develop a regularization approach which, unlike previous proposals, preserves computationally attractive structures of original shrinkage priors. We study theoretical properties of the Gibbs sampler on resulting posterior distributions, with emphasis on convergence rates of the Pólya-Gamma Gibbs sampler for sparse logistic regression. Our analysis shows that the proposed regularization leads to geometric ergodicity under a broad range of global-local shrinkage priors. Essentially, the only requirement is for the prior on the local scale to satisfy . If further satisfies for , as in the case of Bayesian bridge priors, we show the sampler to be uniformly ergodic.

Equivalences of Geometric Ergodicity of Markov Chains

This paper gathers together different conditions which are all equivalent to geometric ergodicity of time-homogeneous Markov chains on general state spaces. A total of 34 different conditions are presented (27 for general chains plus 7 for reversible chains), some old and some new, in terms of such notions as convergence bounds, drift conditions, spectral properties, etc., with different assumptions about the distance metric used, finiteness of function moments, initial distribution, uniformity of bounds, and more. Proofs of the connections between different conditions are provided, somewhat self-contained but using some results from the literature where appropriate.

Geometric ergodicity and [formula omitted]-limit set of an SIRm epidemic model with regime switching

In this paper, a piecewise deterministic SIRm epidemic model with human mobility and regime switching is studied. First, we show that the epidemic threshold of the model can be represented by the basic reproduction number R0, that is, if R0<1, the epidemic will be eradicated almost surely, while if R0>1, the epidemic will continue to spread. Further, we study the ergodicity and the Ω-limit set of the model and discuss the relationship between the ergodicity and the Ω-limit set.

Geometric ergodicity and conditional self‐weighted M‐estimator of a GRCAR(p) model with heavy‐tailed errors

We establish the geometric ergodicity for general stochastic functional autoregressive (linear and nonlinear) models with heavy‐tailed errors. The stationarity conditions for a generalized random coefficient autoregressive model (GRCAR()) are presented as a corollary. And then, a conditional self‐weighted M‐estimator for parameters in the GRCAR() is proposed. The asymptotic normality of this estimator is discussed by allowing infinite variance innovations. Simulation experiments are carried out to assess the finite‐sample performance of the proposed methodology and theory, and a real heavy‐tailed data example is given as illustration.

Convergence properties of data augmentation algorithms for high-dimensional robit regression

The logistic and probit link functions are the most common choices for regression models with a binary response. However, these choices are not robust to the presence of outliers/unexpected observations. The robit link function, which is equal to the inverse CDF of the Student’s t-distribution, provides a robust alternative to the probit and logistic link functions. A multivariate normal prior for the regression coefficients is the standard choice for Bayesian inference in robit regression models. The resulting posterior density is intractable and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the desired posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for asymptotic validity of MCMC standard errors for desired posterior expectations/quantiles. Previous work [16] established geometric ergodicity of this robit DA Markov chain assuming (i) the sample size n dominates the number of predictors p, and (ii) an additional constraint which requires the sample size to be bounded above by a fixed constant which depends on the design matrix X. In particular, modern high-dimensional settings where n<p are not considered. In this work, we show that the robit DA Markov chain is trace-class (i.e., the eigenvalues of the corresponding Markov operator are summable) for arbitrary choices of the sample size n, the number of predictors p, the design matrix X, and the prior mean and variance parameters. The trace-class property implies geometric ergodicity. Moreover, this property allows us to conclude that the sandwich robit chain (obtained by inserting an inexpensive extra step in between the two steps of the DA chain) is strictly better than the robit DA chain in an appropriate sense, and enables the use of recent methods to estimate the spectral gap of trace class DA Markov chains.

Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

We study the geometric ergodicity and the long-time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB–IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles [Formula: see text], the time step [Formula: see text] for batch divisions or the batch size [Formula: see text]. Moreover, the Wasserstein-1 distance between the invariant distributions of the IPS and the RB–IPS is bounded by [Formula: see text], showing that the RB–IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.

Asynchronous and Distributed Data Augmentation for Massive Data Settings

Data augmentation (DA) algorithms are slow in massive data settings due to multiple passes through the entire data. We address this problem by developing a DA extension that exploits asynchronous and distributed computing. The extended DA algorithm is called Asynchronous and Distributed (AD) DA with the original DA as its parent. Any ADDA is indexed by a parameter r ∈ ( 0 , 1 ) and starts by dividing the entire data into k disjoint subsets and storing them on k processes. Every iteration of ADDA augments only an r-fraction of the k data subsets with some positive probability and leaves the remaining ( 1 − r ) -fraction of the augmented data unchanged. The parameter draws are obtained using the r-fraction of new and ( 1 − r ) -fraction of old augmented data. We show that the ADDA Markov chain is Harris ergodic with the desired stationary distribution under mild conditions on the parent DA algorithm. We demonstrate that ADDA is significantly faster than its parent for many (k, r) choices in three representative models. We also establish the geometric ergodicity of the ADDA Markov chain for all the three models, which yields asymptotically valid standard errors for estimates of desired posterior quantities. Supplementary materials for this article are available online.

Convergence of Position-Dependent MALA with Application to Conditional Simulation in GLMMs

We establish conditions under which Metropolis-Hastings (MH) algorithms with a position-dependent proposal covariance matrix will or will not have the geometric rate of convergence. Some of the diffusions based MH algorithms like the Metropolis adjusted Langevin algorithm (MALA) and the pre-conditioned MALA (PCMALA) have a position-independent proposal variance. Whereas, for other modern variants of MALA like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for geometric ergodicity of different variations of the Langevin algorithms. These results have important practical implications as these provide crucial justification for the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates. The general conditions are verified in the context of conditional simulation from the two most popular generalized linear mixed models (GLMMs), namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. Empirical comparison in the framework of some spatial GLMMs shows that the computationally less expensive PCMALA with an appropriately chosen pre-conditioning matrix may outperform the MMALA.