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Control functionals for Monte Carlo integration
A non‐parametric extension of control variates is presented. These leverage gradient information on the sampling density to achieve substantial variance reduction. It is not required that the…
Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees
This paper presents the first probabilistic integrator that admits such theoretical treatment, called Frank-Wolfe Bayesian Quadrature (FWBQ), which is applied to successfully quantify numerical error in the solution to a challenging Bayesian model choice problem in cellular biology.
Probabilistic Integration: A Role for Statisticians in Numerical Analysis?
- François-Xavier Briol, C. Oates, M. Girolami, Michael A. Osborne, D. Sejdinovic
- Computer Science
- 3 December 2015
This paper examines thoroughly the case for probabilistic numerical methods in statistical computation and a specific case study is presented for Markov chain and Quasi Monte Carlo methods.
Probabilistic Integration: A Role in Statistical Computation?
- F. Briol, C. Oates, M. Girolami, Michael A. Osborne, D. Sejdinovic
- Computer ScienceStatistical Science
- 3 December 2015
These show that probabilistic integrators can in principle enjoy the "best of both worlds", leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assess the impact of numerical error on scientific conclusions.
Bayesian Probabilistic Numerical Methods
Average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods and found to be superior to worst- case error.
- W. Chen, Lester W. Mackey, Jackson Gorham, François-Xavier Briol, C. Oates
- Computer ScienceICML
- 27 March 2018
The empirical results demonstrate that Stein Points enable accurate approximation of the posterior at modest computational cost, and theoretical results are provided to establish convergence of the method.
The Controlled Thermodynamic Integral for Bayesian Model Evidence Evaluation
This article considers the reduction of variance that can be achieved by exploiting control variates in this setting and applies whenever the gradient of both the log-likelihood and thelog-prior with respect to the parameters can be efficiently evaluated.
Control Functionals for Quasi-Monte Carlo Integration
This strategy is explored herein and shown to be well-suited to modern statistical computation and to trade-off numerical integration with functional approximation.
Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems ∗
A probabilistic numerical method for solution of partial differential equations (PDEs) and application of that method to PDE-constrained inverse problems, which enables the solution of challenging inverse problems whilst accounting for the impact of discretisation error due to numerical solution of the PDE.
Stein Point Markov Chain Monte Carlo
This paper removes the need to solve this optimisation problem by selecting each new point based on a Markov chain sample path, which significantly reduces the computational cost of Stein Points and leads to a suite of algorithms that are straightforward to implement.