# Probabilistic numerics and uncertainty in computations

@article{Hennig2015ProbabilisticNA, title={Probabilistic numerics and uncertainty in computations}, author={Philipp Hennig and Michael A. Osborne and Mark A. Girolami}, journal={Proceedings. Mathematical, Physical, and Engineering Sciences / The Royal Society}, year={2015}, volume={471} }

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions…

## 248 Citations

Probabilistic Integration: A Role in Statistical Computation?

- Computer ScienceStatistical Science
- 2019

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.

Probabilistic Integration: A Role for Statisticians in Numerical Analysis?

- Computer Science
- 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

- Computer ScienceArXiv
- 2015

Probabilistic (Bayesian) versions of both Markov chain and Quasi Monte Carlo methods for integration are presented and rigorous theoretical guarantees for convergence rates are provided, in both posterior mean and posterior contraction.

Black Box Probabilistic Numerics

- Computer Science, MathematicsNeurIPS
- 2021

This paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method, which massively expands the range of tasks to which Probabilistic numerics can be applied, inherits the features and performance of state-of-the-art numerical methods, and enables provably higher orders of convergence to be achieved.

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models.

- Computer ScienceJournal of computational neuroscience
- 2022

It is shown that numerical uncertainty can affect the outcome of typical neuroscience simulations, e.g. jittering spikes by milliseconds or even adding or removing individual spikes from simulations altogether, and it is demonstrated that probabilistic solvers reveal these numerical uncertainties with only moderate computational overhead.

Statistical analysis of differential equations: introducing probability measures on numerical solutions

- MathematicsStat. Comput.
- 2017

It is shown that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of ordinary and partial differential equation models, and the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis are provided.

Bayesian Quadrature for Multiple Related Integrals

- Computer ScienceICML
- 2018

This paper proposes the first Bayesian probabilistic numerical method by extending the well-known Bayesian quadrature algorithm to the case where it is interested in computing the integral of several related functions, and demonstrates its efficiency in the context of multi-fidelity models for complex engineering systems, as well as a problem of global illumination in computer graphics.

Probability Measures for Numerical Solutions of Differential Equations

- Mathematics
- 2015

In this paper, we present a formal quantification of epistemic uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential…

A comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments

- Computer Science
- 2017

A critical comparison of polynomial chaos and Gaussian process emulation for a range of criteria and examples is provided, with particular focus on the approximation accuracy of the surrogates under changes in the size and type of the experimental design.

Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees

- Computer ScienceNIPS
- 2015

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.

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