Abstract
A delayed-acceptance version of a Metropolis–Hastings algorithm can be useful for Bayesian inference when it is computationally expensive to calculate the true posterior, but a computationally cheap approximation is available; the delayed-acceptance kernel targets the same posterior as its associated “parent” Metropolis-Hastings kernel. Although the asymptotic variance of the ergodic average of any functional of the delayed-acceptance chain cannot be less than that obtained using its parent, the average computational time per iteration can be much smaller and so for a given computational budget the delayed-acceptance kernel can be more efficient. When the asymptotic variance of the ergodic averages of all L^2 functionals of the chain are finite, the kernel is said to be variance bounding. It has recently been noted that a delayed-acceptance kernel need not be variance bounding even when its parent is. We provide sufficient conditions for inheritance: for non-local algorithms, such as the independence sampler, the discrepancy between the log density of the approximation and that of the truth should be bounded; for local algorithms, two alternative sets of conditions are provided. As a by-product of our initial, general result we also supply sufficient conditions on any pair of proposals such that, for any shared target distribution, if a Metropolis-Hastings kernel using one of the proposals is variance bounding then so is the Metropolis-Hastings kernel using the other proposal.
Highlights
The Metropolis-Hastings (MH) algorithm is widely used to approximately compute expectations with respect to complicated high-dimensional posterior distributions (e.g. Gilks et al (1996); Geyer (2011))
When a computationally-cheap approximation, or surrogate, is available, the delayed-acceptance Metropolis-Hastings (DAMH) algorithm known as the twostage algorithm, and a special case of the surrogate-transition method Liu (2001); Christen and Fox (2005); Higdon et al (2011) leverages the surrogate to produce a new Markov chain that still targets the original distribution of interest
We investigate the conditions under which a DAMH kernel inherits variance bounding from its MH parent and, as a by product, discover conditions under which two different proposals produce MH kernels that are equivalent in terms of whether or not they are variance bounding
Summary
The Metropolis-Hastings (MH) algorithm is widely used to approximately compute expectations with respect to complicated high-dimensional posterior distributions (e.g. Gilks et al (1996); Geyer (2011)). Using our Proposition 1, the lack of inheritance of geometric ergodicity in the example in Banterle et al (2019) is equivalent to a lack of inheritance of the variance bounding property: even though the asymptotic variance using the parent MH kernel is finite for all h ∈ L2( ) , there exist h ∈ L2( ) for which the asymptotic variance using the DA kernel is infinite For such h, estimated quantities such as effective sample size (e.g. Hoff (2009) are invalid, and consequent, standard CLT-based intuitions about the sizes of typical errors in estimates of [h] from the chain do not hold. The target and surrogate distributions are denoted by and π , respectively, and they are assumed to have densities of (x) and π(x) with respect to Lebesgue measure
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
