A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the true kernel is slow to evaluate. A relatively limited set of quantitative tools exist to determine whether the performance of such approximations will be well behaved and to assess the quality of approximation. We derive a set a tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, @math . The focus of our work is on determining whether the approximating kernel (i.e. perturbation) will preserve the geometric ergodicity of the chain, and whether the approximating stationary distribution will be close to the original stationary distribution. Our results directly generalise the results of johndrow2015approximations from the uniformly ergodic case to the geometrically ergodic case. We then apply our results to the class of Noisy MCMC' algorithms.
Close to the present paper, @cite_2 derive perturbation bounds to assess the robustness of approximate MCMC algorithms. The assumptions upon which their results rely are: the original chain is uniformly contractive in the total variation norm (this implies uniform ergodicity); and the perturbation is sufficiently small (in the operator norm induced by the total variation norm). The main results of their paper are: the perturbed kernel is uniformly contractive in the total variation norm; the perturbed stationary distribution is close to the original stationary distribution in total variation; explicit bounds on the total variation distance between finite time approximate sampling distributions and the original stationary distribution; explicit bounds on total variation difference between the original stationary distribution and the mixture of finite time approximate sampling distributions; and explicit bounds on the MSE for integral approximation using approximate kernel and the true kernel. The results derived by @cite_2 are applied within the same paper to a wide variety of approximate MCMC problems including low rank approximation to Gaussian processes and subsampling approximations.
@cite_2
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In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the L2-risk of the nonparametric spectral estimators. When the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.