The Containment Condition and Adapfail Algorithms

Łatuszyński, Krzysztof; Rosenthal, Jeffrey S.

doi:10.1239/jap/1421763335

Cited by 5 publications

(4 citation statements)

References 20 publications

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“…Adaptive MCMC methods (Andrieu & Thoms 2008, Hoffman & Gelman 2014, Yang et al 2019, Pompe et al 2020) have been proven effective for sampling in high-dimensional spaces with unfriendly geometries. Injecting adaptive ideas into the world of sampling with intractable targets is hindered by stringent conditions that need to be satisfied by an adaptive transition kernel, e.g., the containment condition (Bai et al 2011, Łatuszy ński & Rosenthal 2014). Some inroads have been made into eliminating the latter by Craiu et al (2015) and Rosenthal & Yang (2018), so we expect to see more adaptive designs permeating in pseudodata-generation-type samplers.…”

Section: Discussionmentioning

confidence: 99%

Approximate Methods for Bayesian Computation

Craiu

Levi

2023

Annu. Rev. Stat. Appl.

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Rich data generating mechanisms are ubiquitous in this age of information and require complex statistical models to draw meaningful inference. While Bayesian analysis has seen enormous development in the last 30 years, benefitting from the impetus given by the successful application of Markov chain Monte Carlo (MCMC) sampling, the combination of big data and complex models conspire to produce significant challenges for the traditional MCMC algorithms. We review modern algorithmic developments addressing the latter and compare their performance using numerical experiments. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 10 is March 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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Section: Discussionmentioning

confidence: 99%

Approximate Methods for Bayesian Computation

Craiu

Levi

2023

Annu. Rev. Stat. Appl.

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“…Fort et al (2011)). However, it turns out ( Latuszyński and Rosenthal (2014)) that if Containment is not satisfied, then the algorithm may still converge, but with positive probability it will be asymptotically less efficient than any nonadaptive ergodic MCMC scheme. Hence algorithms that do not satisfy Containment are termed AdapFail and are best avoided.…”

Section: Optimal Scaling and Adaptive Mcmcmentioning

confidence: 99%

Bayesian computation: a summary of the current state, and samples backwards and forwards

et al. 2015

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Recent decades have seen enormous improvements in computational inference for statistical models; there have been competitive continual enhancements in a wide range of computational tools. In Bayesian inference, first and foremost, MCMC techniques have continued to evolve, moving from random walk proposals to Langevin drift, to Hamiltonian Monte Carlo, and so on, with both theoretical and algorithmic innovations opening new opportunities to practitioners. However, this impressive evolution in capacity is confronted by an even steeper increase in the complexity of the datasets to be addressed. The difficulties of modelling and then handling ever more complex datasets most likely call for a new type of tool for computational inference that dramatically reduces the dimension and size of the raw data while capturing its essential aspects. Approximate models and algorithms may thus be at the core of the next computational revolution.

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“…However, the second condition is notoriously difficult to verify (see, e.g., [5]) and thus a severe limitation (though an essential condition; cf. [16]). On the other hand, the containment condition (15) is reminiscent of the boundedness in probability property (4), which is implied by our various theorems above.…”

Section: Application To Adaptive Mcmc Algorithms Markov Chain Montementioning

confidence: 99%

“…Now let x ∈ C, and A ∈ F with A ∩ C = ∅. Using first (16) To continue, use (20), then (16) again and finally (19) to obtain P P * (x, A) ≥ βε z∈D(ε) y∈A∩C π(dy)P (y, dz) ≥ β 2 εν(D(ε))π(A ∩ C) ≥ β 2 ε(1 − ε)π(A ∩ C).…”

Section: Appendix: Replacing the Minorizing Measure By πmentioning

confidence: 99%

Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

Craiu¹,

Gray²,

Łatuszyński³

et al. 2015

Ann. Appl. Probab.

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We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.1. Introduction. This paper considers whether bounded modifications of stable Markov chains remain stable. Specifically, we let P be a fixed time-homogeneous ergodic Markov chain kernel with bounded step size, and let {X n } be a stochastic process which follows the transition probabilities P except on a bounded subset K where an "adversary" can make arbitrary bounded jumps. Under what conditions must such a process {X n } be bounded in probability?One might think that such boundedness would follow easily, at least under mild regularity and continuity assumptions, that is, that modifying a stable continuous Markov chain inside a bounded set K couldn't possibly lead to unstable behavior out in the tails. In fact the situation is rather more subtle, as we explore herein. We will provide counterexamples to show that boundedness may fail even for well-behaved continuous chains. We will then show that under various additional conditions, including bounds on transition probabilities and/or small set assumptions and/or geometric ergodicity, such boundedness does hold.

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The Containment Condition and Adapfail Algorithms

Cited by 5 publications

References 20 publications

Approximate Methods for Bayesian Computation

Approximate Methods for Bayesian Computation

Bayesian computation: a summary of the current state, and samples backwards and forwards

Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

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