Multilevel Delayed Acceptance MCMC with an Adaptive Error Model in PyMC3

Uncertainty Quantification using Markov Chain Monte Carlo (MCMC) can be prohibitively expensive for target probability densities with expensive likelihood functions, for instance when 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 with the open-source probabilistic programming package PyMC3 and is available in the latest development version. In this paper, we present the algorithm along with an illustrative example.