Stability of Noisy Metropolis-Hastings

Medina-Aguayo, Felipe J.; Lee, Anthony; Roberts, Gareth O.

doi:10.48550/arxiv.1503.07066

Cited by 3 publications

(4 citation statements)

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“…We applied the iAPF to Bayesian parameter estimation in general state space HMMs by using it as an ingredient in a PMMH Markov chain. It could also conceivably be used in similar, but inexact, noisy Markov chains; Medina-Aguayo et al (2015) showed that control on the quality of the marginal likelihood estimates can provide theoretical guarantees on the behaviour of the noisy Markov chain. The performance of the iAPF marginal likelihood estimates also suggests they may be useful in simulated maximum likelihood procedures.…”

Section: Discussionmentioning

confidence: 99%

The Iterated Auxiliary Particle Filter

Guarniero

Johansen

Lee

2017

Journal of the American Statistical Association

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103

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We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions ψ and has an associated ψ-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ψ * that is optimal in the sense that the ψ * -auxiliary particle filter's estimate of L has zero variance. In practical applications, ψ * is unknown so the ψ * -auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate ψ * , and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm.

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Section: Discussionmentioning

confidence: 99%

The Iterated Auxiliary Particle Filter

Guarniero

Johansen

Lee

2017

Journal of the American Statistical Association

Self Cite

103

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“…Geometric ergodicity, whilst by no means guaranteeing sensible estimators in the non-asymptotic context, does give steps towards this in some generality, through (2). As mentioned earlier, it also appears to have other favourable consequences [16,21]. As such, we feel it is a property worth establishing.…”

Section: Discussionmentioning

confidence: 82%

“…[17]). Geometric ergodicity is also often a requirement in establishing the stability of noisy Markov chains in which P is approximated due to either intractability or computational convenience [20,21] (in other instances slightly weaker but related conditions are needed [22]).…”

Section: Markov Chains and Geometric Ergodicitymentioning

confidence: 99%

Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance

Livingstone¹

2015

Preprint

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We consider a Metropolis-Hastings method with proposal kernel N (x, hG −1 (x)), where x is the current state. After discussing specific cases from the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of G −1 (x) can change the ergodicity properties compared to the Random Walk Metropolis case N (x, hΣ), either for the better or worse. In higher dimensions we use a specific example to show that judicious choice of G −1 (x) can produce a chain which will converge at a geometric rate to its limiting distribution when probability concentrates on an ever narrower ridge as |x| grows, something which is not true for the Random Walk Metropolis.

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“…Such a procedure is termed Monte Carlo within Metropolis (Andrieu and Roberts 2009). Unfortunately this approach does not preserve the stationary distribution, and the resulting Markov chain may even not be ergodic (Medina-Aguayo et al 2015). If ergodic, the difference between stationary distribution, resulting from the noisy acceptance must be quantified, which is a highly nontrivial task and the bounds will rarely be tight (see also Alquier et al (2014); Pillai and Smith (2014); Rudolf and Schweizer (2015) for related methodology and theory).…”

Section: Estimated Likelihoods and Pseudo-marginalsmentioning

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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Stability of Noisy Metropolis-Hastings

Cited by 3 publications

References 0 publications

The Iterated Auxiliary Particle Filter

The Iterated Auxiliary Particle Filter

Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance

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

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