Multilevel Delayed Acceptance MCMC with an Adaptive Error Model in PyMC3

TL;DR

This paper introduces a novel multilevel delayed acceptance MCMC method with an adaptive error model integrated into PyMC3, significantly reducing computational costs for expensive likelihood evaluations like PDEs.

Contribution

The paper presents a new multilevel delayed acceptance MCMC algorithm with an adaptive error model, implemented in PyMC3, to improve efficiency in uncertainty quantification tasks.

Findings

Reduces computational cost for PDE-based likelihood evaluations

Successfully integrated into PyMC3 for broader accessibility

Demonstrated effectiveness through an illustrative example

Abstract

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.