Multilevel Delayed Acceptance MCMC with an Adaptive Error Model in PyMC3
Mikkel B. Lykkegaard, Grigorios Mingas, Robert Scheichl, Colin Fox, Tim J. Dodwell
备注：
8 pages, 4 figures, accepted for Machine Learning for Engineering Modeling, Simulation, and Design Workshop at Neural Information Processing Systems 2020
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.
In the quest for scalable Bayesian computational algorithms we need to exploit the full potential of existing methodologies. In this note we point out that message passing algorithms, which are very well developed for inference in graphical models, appear to be largely unexplored for scalable inference in Bayesian multilevel regression models. We show that nested multilevel regression models with Gaussian errors lend themselves very naturally to the combined use of belief propagation and MCMC. Specifically, the posterior distribution of the regression parameters conditionally on covariance hyp…
We develop a novel Markov chain Monte Carlo (MCMC) method that exploits a hierarchy of models of increasing complexity to efficiently generate samples from an unnormalized target distribution. Broadly, the method rewrites the Multilevel MCMC approach of Dodwell et al. (2015) in terms of the Delayed Acceptance (DA) MCMC of Christen & Fox (2005). In particular, DA is extended to use a hierarchy of models of arbitrary depth, and allow subchains of arbitrary length. We show that the algorithm satisfies detailed balance, hence is ergodic for the target distribution. Furthermore, multilevel variance…
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University of Amsterdam（阿姆斯特丹大学）；Higher School of Economics（高等经济学院）；CIFAR（加拿大高等研究院）
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Zalando Research, Zalando SE（Zalando研究部，Zalando公司）；Zuse Institute Berlin（柏林Zuse研究所）
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 esti…
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