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2020 On a Metropolis–Hastings importance sampling estimator
Daniel Rudolf, Björn Sprungk
Electron. J. Statist. 14(1): 857-889 (2020). DOI: 10.1214/20-EJS1680


A classical approach for approximating expectations of functions w.r.t. partially known distributions is to compute the average of function values along a trajectory of a Metropolis–Hastings (MH) Markov chain. A key part in the MH algorithm is a suitable acceptance/rejection of a proposed state, which ensures the correct stationary distribution of the resulting Markov chain. However, the rejection of proposals causes highly correlated samples. In particular, when a state is rejected it is not taken any further into account. In contrast to that we consider a MH importance sampling estimator which explicitly incorporates all proposed states generated by the MH algorithm. The estimator satisfies a strong law of large numbers as well as a central limit theorem, and, in addition to that, we provide an explicit mean squared error bound. Remarkably, the asymptotic variance of the MH importance sampling estimator does not involve any correlation term in contrast to its classical counterpart. Moreover, although the analyzed estimator uses the same amount of information as the classical MH estimator, it can outperform the latter in scenarios of moderate dimensions as indicated by numerical experiments.


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Daniel Rudolf. Björn Sprungk. "On a Metropolis–Hastings importance sampling estimator." Electron. J. Statist. 14 (1) 857 - 889, 2020.


Received: 1 October 2019; Published: 2020
First available in Project Euclid: 10 February 2020

zbMATH: 07200219
MathSciNet: MR4062161
Digital Object Identifier: 10.1214/20-EJS1680

Primary: 60J05 , 60J22 , 62-04
Secondary: 60F05 , 62F15

Keywords: central limit theorem , importance sampling , Markov chains , Metropolis–Hastings algorithm , variance reduction


Vol.14 • No. 1 • 2020
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