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Sample-based Bayesian inference provides a route to uncertaintyquantification in the geosciences, and inverse problems in general, though isvery computationally demanding in the naive form that requires simulating anaccurate computer model at each iteration. We present a new approach thatconstructs a stochastic correction to the error induced by a reduced model,with the correction improving as the algorithm proceeds. This enables samplingfrom the correct target distribution at reduced computational cost periteration, as in existing delayed-acceptance schemes, while avoidingappreciable loss of statistical efficiency that necessarily occurs when using areduced model. Use of the stochastic correction significantly reduces thecomputational cost of estimating quantities of interest within desireduncertainty bounds. In contrast, existing schemes that use a reduced modeldirectly as a surrogate do not actually improve computational efficiency in ourtarget applications. We build on recent simplified conditions for adaptiveMarkov chain Monte Carlo algorithms to give practical approximation schemes andalgorithms with guaranteed convergence. The efficacy of this new approach isdemonstrated in two computational examples, including calibration of alarge-scale numerical model of a real geothermal reservoir, that show goodcomputational and statistical efficiencies on both synthetic and measured datasets.

A posteriori stochastic correction of reduced models in delayed acceptance MCMC, with application to multiphase subsurface inverse problems