Variational Inference with Vine Copulas: An efficient Approach for Bayesian Computer Model Calibration

The ever-growing access to high performance computing in scientific communities has enabled development of complex computer models in fields such as nuclear physics, climatology, and engineering that produce massive amounts of data. These models need real-time calibration with quantified uncertainties. Bayesian methodology combined with Gaussian process modeling has been heavily utilized for calibration of computer models due to its natural way to account for various sources of uncertainty; see Higdon et al. (2015), and King et al. (2019) for examples in nuclear physics, Sexton et al. (2012) and Pollard et al. (2016) for examples in climatology, and Lawrence et al. (2010), Plumlee et al. (2016) and Zhang et al. (2019) for applications in engineering, astrophysics, and medicine. The original framework for Bayesian calibration of computer models was developed by Kennedy and O'Hagan (2001) with extensions provided by Higdon et al. (2005, 2008); Bayarri et al. (2007); Plumlee (2017, 2019), and Gu and Wang (2018), to name a few. Despite its popularity, however, Bayesian calibration becomes infeasible in big-data scenarios with complex and many-parameter models because it relies on Markov chain Monte Carlo (MCMC) algorithms to approximate posterior densities. This text presents a scalable and statistically principled approach to Bayesian calibration of computer models. We offer an alternative approximation to posterior densities using variational Bayesian inference (VBI), which originated as a machine learning algorithm that approximates a target density through optimization. Statisticians and computer scientists (starting with Peterson and Anderson (1987); Jordan et al. (1999)) have been widely using variational techniques because they tend to be faster and easier to scale to massive datasets. Moreover, the recently published frequentist consistency of variational Bayes by Wang and Blei (2018) established VBI as a theoretically valid procedure.