A note on formal constructions of sequential conditional couplings

Roberts, Gareth O.; Rosenthal, Jeffrey S.

doi:10.1016/j.spl.2013.05.012

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…To prove our results presented in Section 3 we will use the coupling construction analogous to [42] (see also [45] for a more rigorous presentation).…”

Section: Proofs For Sectionmentioning

confidence: 99%

A framework for adaptive MCMC targeting multimodal distributions

Pompe¹,

Holmes²,

Łatuszyński³

2020

Ann. Statist.

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“…To prove our results presented in Section 3 we will use the coupling construction analogous to [42] (see also [45] for a more rigorous presentation).…”

Section: Proofs For Sectionmentioning

confidence: 99%

A framework for adaptive MCMC targeting multimodal distributions

Pompe¹,

Holmes²,

Łatuszyński³

2020

Ann. Statist.

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show abstract

“…Proofs for Section 3. To prove our results presented in Section 3 we will use the coupling construction analogous to [42] (see also [45] for a more rigorous presentation).…”

Section: 3mentioning

confidence: 99%

A Framework for Adaptive MCMC Targeting Multimodal Distributions

Pompe¹,

Holmes²,

Łatuszyński³

2018

Preprint

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We propose a new Monte Carlo method for sampling from multimodal distributions. The idea of this technique is based on splitting the task into two: finding the modes of a target distribution π and sampling, given the knowledge of the locations of the modes. The sampling algorithm relies on steps of two types: local ones, preserving the mode; and jumps to regions associated with different modes. Besides, the method learns the optimal parameters of the algorithm while it runs, without requiring user intervention. Our technique should be considered as a flexible framework, in which the design of moves can follow various strategies known from the broad MCMC literature.In order to design an adaptive scheme that facilitates both local and jump moves, we introduce an auxiliary variable representing each mode and we define a new target distribution π on an augmented state space X × I, where X is the original state space of π and I is the set of the modes. As the algorithm runs and updates its parameters, the target distribution π also keeps being modified. This motivates a new class of algorithms, Auxiliary Variable Adaptive MCMC. We prove general ergodic results for the whole class before specialising to the case of our algorithm.

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“…Proof. The proof utilises a construction similar to the coupling proof of [18,Theorem 1] (see also [20]). First, by the diminishing adaptation property, for any fixed δ c > 0, ε c > 0, and integer M ≥ 1, we can choose n large enough such that…”

Section: Definition 32 (Diminishing Adaptation)mentioning

confidence: 99%

The Containment Condition and Adapfail Algorithms

Łatuszyński

Rosenthal

2014

Journal of Applied Probability

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This short note investigates convergence of adaptive MCMC algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the Containment condition introduced in [RR07]. We show that if the Containment condition is not satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used.

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A note on formal constructions of sequential conditional couplings

Abstract: Abstract. We discuss a formal mathematical framework for certain coupling constructions via minorisation conditions, which are often used to prove bounds on convergence to stationarity of stochastic processes and MCMC algorithms.

Cited by 5 publications

References 23 publications

A framework for adaptive MCMC targeting multimodal distributions

A framework for adaptive MCMC targeting multimodal distributions

A Framework for Adaptive MCMC Targeting Multimodal Distributions

The Containment Condition and Adapfail Algorithms

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