On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

By Eero Saksman, Matti Vihola. 2008

78 citations recorded by the source corpus.

This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen (9), for target distributions with a non-compact support. The conditions ensuring a strong law of large numbers and a central limit theorem require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, and independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines (1).

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/30b5b2355052a9b97daa22f01cabce7e84b15e5e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “thal (2005) and Atchade (2006) to verify its ergodicity under weaker assumptions and apply the mixingale approach to adaptive MALA. Another successful approach (Andrieu and Moulines (2006) rened in Saksman and Vihola (2010)) rests on martingale dierence approximations and martingale limit theorems t”
A tutorial on adaptive MCMC (2008, methodology): “In Saksman and Vihola (2008) a different approach to prove stability is used, and consists of proving that provided that {θi} does not drift too fast to bad values, then the algorithm preserves ergodicity.”
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (2012, unknown): “Now, if |x| > aM0 where a := exp(2π tan(arccos(βπ))), then [33], Lemma 22, implies { y ∈ R :π(y) = π(x)} ⊂ B(0, a|x|) \ B(0, a−1|x|).”
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.”
Adaptive Gibbs samplers and related MCMC methods (2011, background): “…and rapidly growing literature on both theory and practice of adaptive MCMC (see e.g. [18, 19, 5, 1, 20, 13, 44, 45, 25, 50, 51, 14, 8, 6, 7, 47, 49, 2, 3, 15]) which includes counterintuitive examples where X n fails to converge to the desired distribution π (c.f. [5, 44, 8, 25]), as well…”
Adaptive MCMC with Bayesian Optimization (2012, background): “That is, the Markov chain is made inhomogenous by the dependence of the parameter updates upon the history of the Markov chain, but its ergodicity can be ensured (Andrieu & Robert, 2001; Andrieu & Moulines, 2006; Saksman & Vihola, 2010).”