Geometric Ergodicity Research Articles

Randomized Hamiltonian Monte Carlo

Tuning the durations of the Hamiltonian flow in Hamiltonian Monte Carlo (also called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational cost and sampling quality, which is typically challenging to resolve in a satisfactory way. In this article we present and analyze a randomized HMC method (RHMC), in which these durations are i.i.d. exponential random variables whose mean is a free parameter. We focus on the small time step size limit, where the algorithm is rejection-free and the computational cost is proportional to the mean duration. In this limit, we prove that RHMC is geometrically ergodic under the same conditions that imply geometric ergodicity of the solution to underdamped Langevin equations. Moreover, in the context of a multi-dimensional Gaussian distribution, we prove that the sampling efficiency of RHMC, unlike that of constant duration HMC, behaves in a regular way. This regularity is also verified numerically in non-Gaussian target distributions. Finally we suggest variants of RHMC for which the time step size is not required to be small.

Ergodicity of Markov chain Monte Carlo with reversible proposal

Abstract We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.

Smooth Transition Spatial Autoregressive Models

This paper introduces a new model for spatial time series in which cross-sectional dependence varies nonlinearly over space by means of smooth transitions. We refer to our model as the Smooth Transition Spatial Autoregressive (ST-SAR). We establish consistency and asymptotic Gaussianity for the MLE under misspecification and provide additional conditions for geometric ergodicity of the model. Simulation results justify the use of limit theory in empirically relevant settings. The model is applied to study spatio-temporal dynamics in two cases that differ in spatial and temporal extent. We study clustering in urban densities in a large number of neighborhoods in the Netherlands over a 10-year period. We pay particular focus to the advantages of the ST-SAR as an alternative to linear spatial models. In our second study, we apply the ST-SAR to monthly long term interest rates of 15 European sovereigns over 25-year period. We develop a strategy to assess financial stability across the Eurozone based on attraction of individual sovereigns toward the common stochastic trend. Our estimates reveal that stability attained a low during the Greek sovereign debt crisis, and that the Eurozone has remained to struggle in attaining stability since the onset of the financial crisis. The results suggest that the European Monetary System has not fully succeeded in aligning the economies of Ireland, Portugal, Italy, Spain, and Greece with the rest of the Eurozone, while attraction between other sovereigns has continued to increase. In our applications linearity of spatial dependence is overwhelmingly rejected in terms of model fit and forecast accuracy, estimates of control variables improve, and residual correlation is better neutralized.

On the Tail Behavior of a Class of Multivariate Conditionally Heteroskedastic Processes

Conditions for geometric ergodicity of multivariate ARCH processes, with the so-called BEKK parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation, in the spirit of Perfekt (1997). The characterization of the tail behavior of the processes is used for deriving the asymptotic distribution of the sample covariance matrices.

Geometric ergodicity of Gibbs samplers in Bayesian penalized regression models

We consider three Bayesian penalized regression models and show that the respective deterministic scan Gibbs samplers are geometrically ergodic regardless of the dimension of the regression problem. We prove geometric ergodicity of the Gibbs samplers for the Bayesian fused lasso, the Bayesian group lasso, and the Bayesian sparse group lasso. Geometric ergodicity along with a moment condition results in the existence of a Markov chain central limit theorem for Monte Carlo averages and ensures reliable output analysis. Our results of geometric ergodicity allow us to also provide default starting values for the Gibbs samplers.

Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors

The Bayesian probit regression model (Albert and Chib (1993)) is popular and widely used for binary regression. While the improper flat prior for the regression coefficients is an appropriate choice in the absence of any prior information, a proper normal prior is desirable when prior information is available or in modern high dimensional settings where the number of coefficients ($p$) is greater than the sample size ($n$). For both choices of priors, the resulting posterior density is intractable and a Data Dugmentation (DA) Markov chain is used to generate approximate samples from the posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for constructing standard errors for Markov chain based estimates of posterior quantities. In this paper, we first show that in case of proper normal priors, the DA Markov chain is geometrically ergodic *for all* choices of the design matrix $X$, $n$ and $p$ (unlike the improper prior case, where $n \geq p$ and another condition on $X$ are required for posterior propriety itself). We also derive sufficient conditions under which the DA Markov chain is trace-class, i.e., the eigenvalues of the corresponding operator are summable. In particular, this allows us to conclude that the Haar PX-DA sandwich algorithm (obtained by inserting an inexpensive extra step in between the two steps of the DA algorithm) is strictly better than the DA algorithm in an appropriate sense.

Geometric ergodicity of Rao and Teh’s algorithm for Markov jump processes and CTBNs

Rao and Teh (2012, 2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Teh’s algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set. A similar result is also proved for a special version of the algorithm, used for probabilistic inference in Continuous Time Bayesian Networks.

A Correction to “A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions''

In A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions, [SIAM J. Control Optim., 50 (2012), pp. 1886--1902], convergence of the relative value iteration for the ergodic control problem for a nondegenerate diffusion controlled through its drift was established, under the assumption of geometric ergodicity, using two methods: (a) the theory of monotone dynamical systems and (b) the theory of reverse martingales. However, in the proof using (a) it is wrongly claimed that the semiflow is strong order preserving. In this note, we provide a simple generic proof and also comment on how to relax the uniform geometric ergodicity hypothesis.

High Order Conformal Symplectic and Ergodic Schemes for the Stochastic Langevin Equation via Generating Functions

In this paper, we consider the stochastic Langevin equation with additive noises, which possesses both conformal symplectic geometric structure and ergodicity. We propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic Hamiltonian system and modifying the associated generating function. To illustrate this approach, we construct a specific second order numerical scheme, and prove that its symplectic form dissipates exponentially. Moreover, for the linear case, the proposed scheme is also shown to inherit the ergodicity of the original system, and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time. Numerical experiments are given to verify these theoretical results.

Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard–Jones-like repulsive potential

Molecular dynamics simulation is among the most important and widely used tools in the study of molecular systems, providing fundamental insights into molecular mechanisms at a level of detail unattainable by experimental methods [AT87, Lea96, FS96, Sch02]. Usage of molecular dynamics spans a diverse array of fields, from physics and chemistry, to molecular and cellular biology, to engineering and materials science. Due to their size and complexity, simulations of large systems such as biological macromolecules (DNA, RNA, proteins, carbohydrates, and lipids) are typically performed under a classical mechanics representation. A critical requirement of such simulations is ergodicity, or convergence in the limit to the equilibrium (typically canonical) Boltzmann measure μ(dq, dp) = Z(β)−1e−βH(q,p)dqdp. Although ergodicity is commonly assumed, recently [CS08] showed that all commonly used deterministic dynamics methods for simulating the canonical (constanttemperature) ensemble fail to be ergodic. They also showed that introduction of a stochastic hybrid Monte Carlo (HMC) correction guarantees ergodicity; however, HMC scales poorly with system dimension and is rarely used for macromolecules. [CS08] also show empirically that more commonly used stochastic Langevin dynamics [Pas94] appear to exhibit ergodic behavior, but were unable to provide rigorous proof. The key difficulty in applying existing arguments [MSH02] is the appearance of singularities in the potential U(q). Most modern molecular mechanics forcefields [PCC95, BBO83, JTR88] take the form

On a Generalization of the Preconditioned Crank–Nicolson Metropolis Algorithm

Metropolis algorithms for approximate sampling of probability measures on infinite dimensional Hilbert spaces are considered and a generalization of the preconditioned Crank-Nicolson (pCN) proposal is introduced. The new proposal is able to incorporate information of the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independent of the state space dimension and the variance of the observational noise. Moreover, a qualitative convergence result is provided by a comparison argument for spectral gaps. In particular, it is shown that the generalization inherits geometric ergodicity from the Metropolis algorithm with pCN proposal.

Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates

A diffusion with random switching is a Markov process that consists of a stochastic differential equation part $$X_t$$ and a continuous Markov jump process part $$Y_t$$ . Such systems have a wide range of applications, where the transition rates of $$Y_t$$ may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework.

On geometric ergodicity of additive and multiplicative transformation-based Markov Chain Monte Carlo in high dimensions

Recently Dutta and Bhattacharya (Statistical Methodology 16 (2014) 100–116) introduced a novel Markov Chain Monte Carlo methodology that can simultaneously update all the components of high-dimensional parameters using simple deterministic transformations of a one-dimensional random variable drawn from any arbitrary distribution defined on a relevant support. The methodology, which the authors refer to as transformation-based Markov Chain Monte Carlo (TMCMC), greatly enhances computational speed and acceptance rate in high-dimensional problems. Two significant transformations associated with TMCMC are additive and multiplicative transformations. Combinations of additive and multiplicative transformations are also of much interest. In this work, we investigate geometric ergodicity associated with additive and multiplicative TMCMC, along with their combinations, assuming that the target distribution is multi-dimensional and belongs to the super-exponential family; we also illustrate their efficiency in practice with simulation studies.

On Buffered Double Autoregressive Time Series Models

A buffered double autoregressive (BDAR) time series model is proposed in this paper to depict the buffering phenomenon of conditional mean and conditional variance in time series. To build this model, a novel flexible regime switch mechanism is introduced to modify the classical threshold AR model by capturing the stickiness of signal. Besides, considering the inadequacy of traditional models under the lack of information, a signal retrospection is run in this model to provide a more accurate judgement. Moreover, formal proofs suggest strict stationarity and geometric ergodicity of BDAR model under several sufficient conditions. A Gaussian quasi-maximum likelihood estimation (QMLE) is employed to compute the estimators in this model. The strict proof reveals the asymptotic normality of the estimated coefficients of double AR models. More importantly, it has been demonstrated that the estimated threshold of the BDAR model weakly converges to a functional of a two-sided compound Poisson process. Furthermore, a model selection criteria and asymptotic property have been established. Simulation studies are constructed to evaluate the finite sample performance of QMLE and model selection criteria. Finally, an empirical analysis of Hang Seng Index (HSI) using BDAR model reveals the asymmetry of investors’ preference over losses and gains as well as the asymmetry of volatility structure.

On solutions of mean field games with ergodic cost

A general class of mean field games are considered where the governing dynamics are controlled diffusions in Rd. The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of the mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated N-player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as N→∞.

On the Ergodicity of General Mixture of Linear Autoregressive Time Series

In this paper, a general framework of mixture time series is introduced as General Mixture of Linear Autoregressive. This framework contains well known mixture time series such as Mixture Autoregressive, Local Linear Global Neural Network and Logistic Mixture Autoregressive models. The sufficient conditions for geometric ergodicity of such model are derived.

Ergodicity of Truncated Stochastic Navier Stokes with Deterministic Forcing and Dispersion

Turbulence in idealized geophysical flows is a very rich and important topic. The anisotropic effects of explicit deterministic forcing, dispersive effects from rotation due to the \(\beta \)-plane and F-plane, and topography together with random forcing all combine to produce a remarkable number of realistic phenomena. These effects have been studied through careful numerical experiments in the truncated geophysical models. These important results include transitions between coherent jets and vortices, and direct and inverse turbulence cascades as parameters are varied, and it is a contemporary challenge to explain these diverse statistical predictions. Here we contribute to these issues by proving with full mathematical rigor that for any values of the deterministic forcing, the \(\beta \)- and F-plane effects and topography, with minimal stochastic forcing, there is geometric ergodicity for any finite Galerkin truncation. This means that there is a unique smooth invariant measure which attracts all statistical initial data at an exponential rate. In particular, this rigorous statistical theory guarantees that there are no bifurcations to multiple stable and unstable statistical steady states as geophysical parameters are varied in contrast to claims in the applied literature. The proof utilizes a new statistical Lyapunov function to account for enstrophy exchanges between the statistical mean and the variance fluctuations due to the deterministic forcing. It also requires careful proofs of hypoellipticity with geophysical effects and uses geometric control theory to establish reachability. To illustrate the necessity of these conditions, a two-dimensional example is developed which has the square of the Euclidean norm as the Lyapunov function and is hypoelliptic with nonzero noise forcing, yet fails to be reachable or ergodic.

Geometric ergodicity of Rao and Teh’s algorithm for homogeneous Markov jump processes

Rao and Teh (2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Teh’s algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set. We work under the assumption that the parameters of the hidden process are known and the goal is to restore its trajectory.

Nonlinear stability and ergodicity of ensemble based Kalman filters

The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are data assimilation methods used to combine high dimensional, nonlinear dynamical models with observed data. Despite their widespread usage in climate science and oil reservoir simulation, very little is known about the long-time behavior of these methods and why they are effective when applied with modest ensemble sizes in large dimensional turbulent dynamical systems. By following the basic principles of energy dissipation and controllability of filters, this paper establishes a simple, systematic and rigorous framework for the nonlinear analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the dynamical properties of boundedness and geometric ergodicity. The time uniform boundedness guarantees that the filter estimate will not diverge to machine infinity in finite time, which is a potential threat for EnKF and ESQF known as the catastrophic filter divergence. Geometric ergodicity ensures in addition that the filter has a unique invariant measure and that initialization errors will dissipate exponentially in time. We establish these results by introducing a natural notion of observable energy dissipation. The time uniform bound is achieved through a simple Lyapunov function argument, this result applies to systems with complete observations and strong kinetic energy dissipation, but also to concrete examples with incomplete observations. With the Lyapunov function argument established, the geometric ergodicity is obtained by verifying the controllability of the filter processes; in particular, such analysis for ESQF relies on a careful multivariate perturbation analysis of the covariance eigen-structure.

Adaptive estimation for bifurcating Markov chains

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^p$ -loss error, for $1\leq p<\infty$. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.