Establishing some order amongst exact approximations of MCMCs
Andrieu, C., & Vihola, M. (2016). Establishing some order amongst exact approximations of MCMCs. Annals of Applied Probability, 26(5), 2661-2696. https://doi.org/10.1214/15-AAP1158
Julkaistu sarjassa
Annals of Applied ProbabilityPäivämäärä
2016Tekijänoikeudet
© Institute of Mathematical Statistics, 2016. Published in this repository with the kind permission of the publisher.
Exact approximations of Markov chain Monte Carlo (MCMC) algorithms
are a general emerging class of sampling algorithms. One of the main
ideas behind exact approximations consists of replacing intractable quantities
required to run standard MCMC algorithms, such as the target probability
density in a Metropolis–Hastings algorithm, with estimators. Perhaps
surprisingly, such approximations lead to powerful algorithms which are exact
in the sense that they are guaranteed to have correct limiting distributions.
In this paper, we discover a general framework which allows one to compare,
or order, performance measures of two implementations of such algorithms.
In particular, we establish an order with respect to the mean acceptance probability,
the first autocorrelation coefficient, the asymptotic variance and the
right spectral gap. The key notion to guarantee the ordering is that of the convex
order between estimators used to implement the algorithms. We believe
that our convex order condition is close to optimal, and this is supported by
a counterexample which shows that a weaker variance order is not sufficient.
The convex order plays a central role by allowing us to construct a martingale
coupling which enables the comparison of performance measures of Markov
chain with differing invariant distributions, contrary to existing results. We
detail applications of our result by identifying extremal distributions within
given classes of approximations, by showing that averaging replicas improves
performance in a monotonic fashion and that stratification is guaranteed to
improve performance for the standard implementation of the Approximate
Bayesian Computation (ABC) MCMC method.
...
Julkaisija
Institute of Mathematical StatisticsISSN Hae Julkaisufoorumista
1050-5164Asiasanat
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https://converis.jyu.fi/converis/portal/detail/Publication/26285312
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