8109 2020 eng article 0 -- -- -- Markov chain (MC) algorithms are ubiquitous in machine learning and statistics and many other disciplines. Typically, these algorithms can be formulated as acceptance rejection methods. In this work we present a novel estimator applicable to these methods, dubbed Markov chain importance sampling (MCIS), which efficiently makes use of rejected proposals. For the unadjusted Langevin algorithm, it provides a novel way of correcting the discretization error. Our estimator satisfies a central limit theorem and improves on error per CPU cycle, often to a large extent. As a by-product it enables estimating the normalizing constant, an important quantity in Bayesian machine learning and statistics. Journal of Computational and Graphical Statistics 10.1080/10618600.2020.1826953 yes epub ahead of print 2020-09-04 false true Ilja Klebanov Ilja Klebanov Ingmar Schuster Numerical Mathematics Schuster, Ingmar MathPlus - TrU-2 Modeling and Simulation of Complex Processes