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Bayesian modelling and computational inference by Markov chain Monte Carlo (MCMC) is a principled framework for large-scale uncertainty quantification, though is limited in practice by computational cost when implemented in the simplest form that requires simulating an accurate computer model at each iteration of the MCMC. The delayed acceptance Metropolis--Hastings MCMC leverages a reduced model for the forward map to lower the compute cost per iteration, though necessarily reduces statistical efficiency that can, without care, lead to no reduction in the computational cost of computing estimates to a desired accuracy. Randomizing the reduced model for the forward map can dramatically improve computational efficiency, by maintaining the low cost per iteration but also avoiding appreciable loss of statistical efficiency. Randomized maps are constructed by a posteriori adaptive tuning of a randomized and locally-corrected deterministic reduced model. Equivalently, the approximated posterior distribution may be viewed as induced by a modified likelihood function for use with the reduced map, with parameters tuned to optimize the quality of the approximation to the correct posterior distribution. Conditions for adaptive MCMC algorithms allow practical approximations and algorithms that have guaranteed ergodicity for the target distribution. Good statistical and computational efficiencies are demonstrated in examples of calibration of large-scale numerical models of geothermal reservoirs and electrical capacitance tomography.
We propose a new kernel for Metropolis Hastings called Directional Metropolis Hastings (DMH) with multivariate update where the proposal kernel has state dependent covariance matrix. We use the derivative of the target distribution at the current sta
MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper an approach to reduce the computational costs of such algorithms by a si
The ability to generate samples of the random effects from their conditional distributions is fundamental for inference in mixed effects models. Random walk Metropolis is widely used to conduct such sampling, but such a method can converge slowly for
Can we make Bayesian posterior MCMC sampling more efficient when faced with very large datasets? We argue that computing the likelihood for N datapoints in the Metropolis-Hastings (MH) test to reach a single binary decision is computationally ineffic
In this article we propose multiplication based random walk Metropolis Hastings (MH) algorithm on the real line. We call it the random dive MH (RDMH) algorithm. This algorithm, even if simple to apply, was not studied earlier in Markov chain Monte Ca