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Paper No. 12-18

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A Lee and K Latuszynski

Variance bounding and geomtric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

Abstract: Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable like-lihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the re-liability of estimates in practice. We then prove that a recently introduced Markov kernel in this setting can be variance bounding and geometrically ergodic whenever its intractable Metropolis-Hastings counterpart is, under reasonably weak and man-ageable conditions. We indicate that the computational cost of the latter kernel is bounded whenever the prior is proper, and present indicative results on an example where spectral gaps and asymptotic variances can be computed.

Keywords: Approximate Bayesian computation; Markov chain Monte Carlo; Variance bounding; Geometric ergodicity; Local adaptation.