### Abstract

We consider a pseudo-marginal Metropolis–Hastings kernel
Pm
that is constructed using an average of
m
exchangeable random variables, and an analogous kernel
Ps
that averages
s<m
of these same random variables. Using an embedding technique to facilitate comparisons, we provide a lower bound for the asymptotic variance of any ergodic average associated with
Pm
in terms of the asymptotic variance of the corresponding ergodic average associated with
Ps
. We show that the bound is tight and disprove a conjecture that when the random variables to be averaged are independent, the asymptotic variance under
Pm
is never less than
s/m
times the variance under
Ps
. The conjecture does, however, hold for continuous-time Markov chains. These results imply that if the computational cost of the algorithm is proportional to
m
, it is often better to set
m=1
. We provide intuition as to why these findings differ so markedly from recent results for pseudo-marginal kernels employing particle filter approximations. Our results are exemplified through two simulation studies; in the first the computational cost is effectively proportional to
m
and in the second there is a considerable start-up cost at each iteration.

Original language | English |
---|---|

Pages (from-to) | 727-734 |

Number of pages | 8 |

Journal | Biometrika |

Volume | 104 |

Issue number | 3 |

Early online date | 21 Jun 2017 |

DOIs | |

Publication status | Published - 1 Sep 2017 |

### Keywords

- Importance sampling
- Pseudo-marginal
- Markov chain
- Monte Carlo

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## Cite this

Sherlock, C., Thiery, A., & Lee, A. (2017). Pseudo-marginal Metropolis–Hastings sampling using averages of unbiased estimators.

*Biometrika*,*104*(3), 727-734. https://doi.org/10.1093/biomet/asx031