Coupling and Ergodicity of Adaptive MCMC

Convergence 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...

Recurrent and Ergodic Properties of Adaptive MCMC

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 erg...

On the Containment Condition for Adaptive Markov Chain Monte Carlo Algorithms

This paper considers ergodicity properties of certain adaptive Markov chain Monte Carlo (MCMC) algorithms for multidimensional target distributions, in particular Adaptive Metropolis and Adaptive Metropoliswithin-Gibbs. It was previously shown by Roberts and Rosenthal (2007) that Diminishing Adaptation and Containment imply ergodicity of adaptive MCMC. We derive various sufficient conditions to...

Simultaneous 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...

Some Results on the Ergodicity of Adaptive MCMC Algorithms