An official website of the United States government
Abstract When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the$$\chi^2$$metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.
more »
« less
Two-time-scale regime-switching stochastic Kolmogorov systems with wideband noises
In our recent work, in lieu of using white noise, we examined Kolmogorov systems driven by wideband noise. Such systems naturally arise in statistical physics, biological and ecological systems, and many related fields. One of the motivations of our study is to treat more realistic models than the usually assumed stochastic differential equation models. The rationale is that a Brownian motion is an idealization used in a wide range of models, whereas wideband noise processes are much easier to be realized in the actual applications. This paper further investigates the case that in addition to the wideband noise process, there is a singularly perturbed Markov chain. The added Markov chain is used to model discrete events. Although it is a more realistic formulation, because of the non-Markovian formulation due to the wideband noise and the singularly perturbed Markov chain, the analysis is more difficult. Using weak convergence methods, we obtain a limit result. Then we provide several examples for the utility of our findings.
more »
« less
- Award ID(s):
- 1710827
- PAR ID:
- 10250201
- Date Published:
- Journal Name:
- Annals
- Volume:
- 12
- Issue:
- 1-2/2020
- ISSN:
- 2066-5997
- Page Range / eLocation ID:
- 62-89
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.more » « less
-
This paper is devoted to the detection of contingencies in modern power systems. Because the systems we consider are under the framework of cyber-physical systems, it is necessary to take into consideration of the information processing aspect and communication networks. A consequence is that noise and random disturbances are unavoidable. The detection problem then becomes one known as quickest detection. In contrast to running the detection problem in a discretetime setting leading to a sequence of detection problems, this work focuses on the problem in a continuous-time setup. We treat stochastic differential equation models. One of the distinct features is that the systems are hybrid involving both continuous states and discrete events that coexist and interact. The discrete event process is modeled by a continuous-time Markov chain representing random environments that are not resented by a continuous sample path. The quickest detection then can be written as an optimal stopping problem. This paper is devoted to finding numerical solutions to the underlying problem. We use a Markov chain approximation method to construct the numerical algorithms. Numerical examples are used to demonstrate the performance.more » « less
-
Abstract We introduce a new Markov Chain Monte Carlo (MCMC) algorithm with parallel tempering for fitting theoretical models of horizon-scale images of black holes to the interferometric data from the Event Horizon Telescope (EHT). The algorithm implements forms of the noise distribution in the data that are accurate for all signal-to-noise ratios. In addition to being trivially parallelizable, the algorithm is optimized for high performance, achieving 1 million MCMC chain steps in under 20 s on a single processor. We use synthetic data for the 2017 EHT coverage of M87 that are generated based on analytic as well as General Relativistic Magnetohydrodynamic (GRMHD) model images to explore several potential sources of biases in fitting models to sparse interferometric data. We demonstrate that a very small number of data points that lie near salient features of the interferometric data exert disproportionate influence on the inferred model parameters. We also show that the preferred orientations of the EHT baselines introduce significant biases in the inference of the orientation of the model images. Finally, we discuss strategies that help identify the presence and severity of such biases in realistic applications.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
