-- Compute and memory constraints have historically prevented traffic simulation software users from fully utilizing the predictive models underlying them. When calibrating car-following models, particularly, accommodations have included 1) using sensitivity analysis to limit the number of parameters to be calibrated, and 2) identifying only one set of parameter values using data collected from multiple car-following instances across multiple drivers. Shortcuts are further motivated by insufficient data set sizes, for which a driver may have too few instances to fully account for the variation in their driving behavior . In this paper, we demonstrate that recent technological advances can enable transportation researchers and engineers to overcome these constraints and produce calibration results that 1) outperform industry standard approaches, and 2) allow for a unique set of parameters to be estimated for each driver in a data set, even given a small amount of data. We propose a novel calibration procedure for car-following models based on Bayesian machine learning and probabilistic programming, and apply it to real-world data from a naturalistic driving study. We also discuss how this combination of mathematical and software tools can offer additional benefits such as more informative model validation and the incorporation of true-to-data uncertainty into simulation traces. Traffic simulation software packages are widely used in transportation engineering to estimate the impacts of potential changes to a roadway network and forecast system performance under future scenarios. These packages are underpinned by math-and physics-based models, which are designed to describe behavior at an aggregate (macroscopic) level or at the level of individual drivers (microscopic).
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.
The inference of parameters of expensive computer models from data is a classical problem in Statistics (Sacks et al., 1989). Such a problem is often referred to as calibration (Kennedy and O'Hagan, 2001), and the results of calibration are often of interest to draw conclusions on parameters that may have a direct interpretation of real physical quantities. There are many fundamental difficulties in calibrating expensive computer models, which we can distinguish between computational and statistical. Computational issues arise from the fact that traditional optimization and inference techniques require running the expensive computer model many times for different values of the parameters, which might be unfeasible within a given computational budget. Statistical limitations, instead, arise from the fact that computer models are abstractions of real processes, which might be inaccurate.
Bayesian hybrid models fuse physics-based insights with machine learning constructs to correct for systematic bias. In this paper, we compare Bayesian hybrid models against physics-based glass-box and Gaussian process black-box surrogate models. We consider ballistic firing as an illustrative case study for a Bayesian decision-making workflow. First, Bayesian calibration is performed to estimate model parameters. We then use the posterior distribution from Bayesian analysis to compute optimal firing conditions to hit a target via a single-stage stochastic program. The case study demonstrates the ability of Bayesian hybrid models to overcome systematic bias from missing physics with less data than the pure machine learning approach. Ultimately, we argue Bayesian hybrid models are an emerging paradigm for data-informed decision-making under parametric and epistemic uncertainty.
Deep Neural Networks (DNNs) have achieved state-of-the-art accuracy performance in many tasks. However, recent works have pointed out that the outputs provided by these models are not well-calibrated, seriously limiting their use in critical decision scenarios. In this work, we propose to use a decoupled Bayesian stage, implemented with a Bayesian Neural Network (BNN), to map the uncalibrated probabilities provided by a DNN to calibrated ones, consistently improving calibration. Our results evidence that incorporating uncertainty provides more reliable probabilistic models, a critical condition for achieving good calibration. We report a generous collection of experimental results using high-accuracy DNNs in standardized image classification benchmarks, showing the good performance, flexibility and robust behavior of our approach with respect to several state-of-the-art calibration methods. Code for reproducibility is provided.