On the ergodicity properties of some adaptive MCMC algorithms

By Christophe Andrieu, '. Moulines. 2006

323 citations recorded by the source corpus.

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 theorem. We prove that the conditions required are satisfied for the independent Metropolis–Hastings algorithm and the random walk Metropolis algorithm with symmetric increments. Finally, we propose an application of these results to the case where the proposal distribution of the Metropolis–Hastings update is a mixture of distributions from a curved exponential family.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c7a41ebb9a055e164926fb00efcbcafb40f0a16f

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Citations in the corpus

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains (2008, methodology): “Andrieu and Moulines [1] made important progress by generalizing the Poisson equation and martingale approximation techniques to the adaptive setting.”
On Perturbed Proximal Gradient Algorithms (2014, methodology): “Such conditions have been worked out for general families of MCMC kernels including HastingsMetropolis dynamics, Gibbs samplers, and hybrid MCMC algorithm; see for example Proposition 12 in Andrieu and Moulines (2006), the proof of Theorem 3.”
Analysis of nonsmooth stochastic approximation: the differential inclusion approach (2018, result): “commonly used in nonsmooth analysis. This is not a serious limitation for the minimizationproblems we have primarily in mind. The assumptions (A2, A3) are usual in stochastic approximation literature [2, 22]. It is worth noting, that condition (A3) allows perturbations sequences which have random an”
An introduction to sampling via measure transport (2016, background): “While these eﬀorts are too diverse to summarize easily, they often rest on the design of improved (and structure-exploiting) proposal mechanisms within the algorithms [37, 3, 26, 22, 62, 53, 34, 24].”
Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms (2007, methodology): “(Andrieu and Moulines (2006) also proved a central limit theorem result.)”
Towards Automatic Model Comparison: An Adaptive Sequential Monte Carlo Approach (2013, methodology): “They applied an idea used within adaptive MCMC methods (Andrieu and Moulines, 2006) to SMC samplers by using variance of parameters estimated from its particle system approximation as the proposal scale for the next iteration, suitably scaled with reference to the dimension of the parameters to be…”
On the stability of sequential Monte Carlo methods in high dimensions (2011, methodology): “Now, using the Marcinkiewicz–Zygmund inequality (there is a version with ∈ [1,2); see, for example, [21], Chapter 7), the L -norm of the first summand in (3.”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “Another successful approach (Andrieu and Moulines (2006) reﬁned in Saks-man and Vihola (2010)) rests on martingale diﬀerence approximations and martingale limit theorems to obtain, under suitable technical assumptions, versions of LLN and CLTs.”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, methodology): “Another successful approach (Andrieu and Moulines 2006 reﬁned in Saksman and Vihola 2010) rests on martingale difference approximations and martingale limit theorems to obtain, under suitable technical assumptions, versions of LLN and CLTs.”
Skew mixture models for loss distributions: a Bayesian approach (2012, background): “The presence of the fixed component guarantees the convergence of the algorithm even in cases where the parameter space is unbounded, see Andrieu and Moulines (2006) and Haario et al. (2001), and is very useful in exploring multimodal posterior distributions when the number of modes of the proposal…”
Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC) (2010, background): “Several recent papers proposing theoretical conditions that must be satisfied to ensure ergodicity of adaptive algorithms include, [Andrieu and Moulines, 2006] and [Haario et al., 2005].”
DRAM: Efficient adaptive MCMC (2006, background): “We expect the result to hold under quite minimal assumptions, especially without the extra smoothness and decay of π assumed in [2]).”
Bayesian inference of chemical kinetic models from proposed reactions (2015, methodology): “The method we use in this paper for posterior exploration is an adaptive across-model sampling scheme, wherein the proposal distribution in the MCMC algorithm continuously adapts based on previous posterior samples to avoid manual tuning (Ji and Schmidler, 2013).”
Adaptive Gibbs samplers and related MCMC methods (2011, methodology): “eP iis denoted as Pi, hence, Pα = d i=1 αiPi. We consider a class of adaptive random scan Gibbs samplers where selection probabilities α= (α1,...,αd) are subject to optimization within some subset Y ⊆[0,1]d of possible choices. Therefore a single step of our generic adaptive algorithm for drawing Xn”
Bayesian calibration of a large‐scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm (2011, methodology): “Adaptive MCMC sampling techniques [Haario et al., 2001; Atchade and Rosenthal, 2005; Andrieu and Moulines, 2006; Haario et al., 2006; Roberts and Rosenthal, 2007, 2009] offer great flexibility in the automatic tuning of these variables.”
Bayesian nonparametric vector autoregressive models (2018, unknown): “The second algorithm is Adaptive scaling within the Adaptive Metropolis–Hastings (ASWAM) algorithm (Andrieu and Moulines, 2006; Atchadé and Fort, 2010).”
A tutorial on adaptive MCMC (2008, methodology): “Consider the following toy example, suggested in Andrieu and Moulines (2006), where X = {1,2} and π = (1/2,1/2) (it is understood here that for such a case we will abuse notation and use π for the vector of values of π and Pθ for the transition matrix) and where the family of transition…”