  Uncertainty Quantification through Markov Chain Monte Carlo (MCMC) can be
prohibitively expensive for target probability densities with expensive
likelihood functions, for instance when the evaluation it involves solving a
Partial Differential Equation (PDE), as is the case in a wide range of
engineering applications. Multilevel Delayed Acceptance (MLDA) with an Adaptive
Error Model (AEM) is a novel approach, which alleviates this problem by
exploiting a hierarchy of models, with increasing complexity and cost, and
correcting the inexpensive models on-the-fly. The method has been integrated
within the open-source probabilistic programming package PyMC3 and is available
in the latest development version. In this paper, the algorithm is presented
along with an illustrative example.
