stability of noisy Metropolis-Hastings
Felipe Medina-Aguayo, Anthony Lee and Gareths Roberts, all from Warwick, arXived last Thursday a paper on the stability properties of noisy Metropolis-Hastings algorithms. The validation of unbiased estimators of the target à la Andrieu and Roberts (2009, AoS)—often discussed here—is in fact obvious when following the auxiliary variable representation of Andrieu and Vihola (2015, AoAP). Assuming the unbiased estimator of the target is generated conditional on the proposed value in the original Markov chain. The noisy version of the above means refreshing the unbiased estimator at each iteration. It also goes under the name of Monte Carlo within Metropolis. The difficulty with this noisy version is that it is not exact, i.e., does not enjoy the true target as its marginal stationary distribution. The paper by Medina-Aguayo, Lee and Roberts focusses on its validation or invalidation (with examples of transient noisy versions). Under geometric ergodicity of the marginal chain, plus some stability in the weights, the noisy version is also geometrically ergodic. A drift condition on the proposal kernel is also sufficient. Under (much?) harder conditions, the limiting distribution of the noisy chain is asymptotically in the number of unbiased estimators the true target. The result is thus quite interesting in that it provides sufficient convergence conditions, albeit not always easy to check in realistic settings.