Active Learning of Model Evidence Using Bayesian Quadrature

By Michael A. Osborne, David Kristjanson Duvenaud, Roman Garnett, Carl E. Rasmussen, Stephen J. Roberts, Zoubin Ghahramani. 2012

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Control functionals for Monte Carlo integration (2014, background): “n and uses this information to model the estimation error. Early work in this direction, albeit from a slightly dierent angle, includes O’Hagan (1991) followed by Rasmussen and Ghahramani (2003) and Osborne et al. (2012). In each case the authors proceeded by imposing structure on the state space Xv”
Probabilistic numerics and uncertainty in computations (2015, unknown): “A practical example of the benefits of approximate Bayesian inference for the (hyper-)parameters of a Bayesian quadrature procedure is given by [28].”
Scalable Joint Models for Reliable Uncertainty-Aware Event Prediction (2017, methodology): “Similar ideas have been used in Bayesian quadrature [45], inter-domain sparseGPs [46], and Fourier features for sparseGPs [47].”
Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature (2014, methodology): “In [12], where the log-likelihood is modelled by a GP, a dependency is introduced between the uncertainty over the function at any point and the function value at that point.”
Adaptive Gaussian Process Approximation for Bayesian Inference with Expensive Likelihood Functions (2017, methodology): “ture (or Bayesian Monte Carlo) approaches [16, 19, 8], which were designed to perform numerical integrations in a Bayesian fashion (for example, to compute the evidence in Bayesian inference problems [18]). Unlike the Bayesian quadrature methods, the goal of this work is to construct an approximatio”
Active Learning of Linear Embeddings for Gaussian Processes (2013, result): “The MGP and BBQ were used to approximately marginalize over all GP hyperparameters (the output scale and D input scales), computing posteriors for the test points.”
The Thermodynamic Variational Objective (2019, methodology): “This does, however, point out opportunities for future work wherein we adaptively select optimal positions of the β 1: K points, perhaps borrowing techniques from the Bayesian numerical quadrature literature (O’Hagan, 1991; Rasmussen and Ghahramani, 2003; Osborne et al., 2012).”
Bayesian Solution Uncertainty Quantification for Differential Equations (2013, methodology): “sics (Conti and O’Hagan, 2010; Kennedy and O’Hagan, 2001; Tokmakian et al., 2012). Other recent examples of applying Bayesian approaches to numerical integration include Kennedy (1998); Minka (2000); Osborne et al. (2012), whose methods allow for a more informed probabilistic approach to the choice ”