Recurrent and Ergodic Properties of Adaptive MCMC
نویسنده
چکیده
We will discuss the recurrence on the state space of the adaptive MCMC algorithm using some examples. We present the ergodicity properties of adaptive MCMC algorithms under the minimal recurrent assumptions, and show the Weak Law of Large Numbers under the same conditions. We will analyze the relationship between the recurrence on the product space of state space and parameter space and the ergodicity, give a counter-example to open problem 21 in Roberts and Rosenthal’s paper, and try to give the positive results under some stronger conditions.
منابع مشابه
On the ergodicity properties of someadaptive MCMC algorithms
In this paper we study the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms (MCMC) that have been recently proposed in the literature. We prove that under a set of verifiable conditions, ergodic averages calculated from the output of a so-called adaptive MCMC sampler converge to the required value and can even, under more stringent assumptions, satisfy a central limit ...
متن کاملSimultaneous drift conditions for Adaptive Markov Chain Monte Carlo algorithms
In the paper, we mainly study ergodicity of adaptive MCMC algorithms. Assume that under some regular conditions about target distributions, all the MCMC samplers in {Pγ : γ ∈ Y} simultaneously satisfy a group of drift conditions, and have the uniform small set C in the sense of the m-step transition such that each MCMC sampler converges to target at a polynomial rate. We say that the family {Pγ...
متن کاملAn Adaptive Metropolis algorithm
A proper choice of a proposal distribution for MCMC methods, e.g. for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis Algorithm (AM), where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the pr...
متن کاملConvergence of Adaptive Markov Chain Monte Carlo Algorithms
In the thesis, we study ergodicity of adaptive Markov Chain Monte Carlo methods (MCMC) based on two conditions (Diminishing Adaptation and Containment which together imply ergodicity), explain the advantages of adaptive MCMC, and apply the theoretical result for some applications. First we show several facts: 1. Diminishing Adaptation alone may not guarantee ergodicity; 2. Containment is not ne...
متن کاملLimit theorems for some adaptive MCMC algorithms with subgeometric kernels
Abstract. This paper deals with the ergodicity (convergence of the marginals) and the law of large numbers for adaptive MCMC algorithms built from transition kernels that are not necessarily geometrically ergodic. We develop a number of results that broaden significantly the class of adaptive MCMC algorithms for which rigorous analysis is now possible. As an example, we give a detailed analysis...
متن کامل