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Probabilistic Numerics and Uncertainty in Computations
Philipp Hennig, Michael A Osborne, Mark Girolami
TL;DR
Probabilistic numerics addresses how to quantify uncertainty introduced by limited or incomplete numerical computation. The paper interprets classic numerical methods as probabilistic inference, develops and illustrates probabilistic algorithms, and proposes uncertainty propagation across computational pipelines. Its conclusion is that these methods can improve performance and provide new functionality, while calibration, computational overhead, formal analysis, and implementation remain open challenges.
Problem
Numerical calculations for tasks such as linear algebra, integration, optimization, and differential equations can lose precision under limited time or hardware, motivating explicit management of computational uncertainty.
Method
The paper interprets established numerical algorithms as probabilistic inference procedures and develops probabilistic formulations that encode problem structure and propagate information.
Results
The probabilistic view provides improved performance, new functionality, and conceptual clarity, with demonstrations in astrometry and computational photography.
Takeaways & Limitations
Probabilistic numerical methods offer a framework for representing and propagating uncertainty through computations that combine multiple numerical algorithms.
Takeaways & Limitations
Formal analysis is at an early stage, efficient and stable implementations remain in development, and convincing use cases across scientific disciplines are still emerging.
Abstract
from arXiv · showhide
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 on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.
1. Introduction
Probabilistic numerics applies probabilistic inference to well-posed deterministic computations, treating uncertainty as arising from incomplete or intractable numerical calculation. The framework reinterprets established methods, motivates new algorithms, and supports uncertainty propagation across computational pipelines.
- Motivation: The paper distinguishes probabilistic numerics from statistical uses of probability that combine data with model assumptions to infer physical quantities.It asks whether probability can meaningfully describe uncertainty in solutions to deterministic mathematical problems.
- Motivation: Probabilistic numerics assigns uncertainty to results of incomplete or intractable computations, such as integration from finitely many function evaluations.Iterative numerical algorithms accumulate information that improves running estimates, while probabilistic methods represent the remaining uncertainty.
- Contribution: Established numerical methods can be identified with families of probabilistic inference procedures, providing improved performance, new functionality, and conceptual clarity.The article connects this interpretation to recent results and applications in astrometry and computational photography.
- Scope: The probabilistic perspective differs from uncertainty quantification by focusing on well-posed deterministic problems and uncertainty arising from computation itself.Uncertainty quantification instead addresses ill-posed problems whose uncertainty arises from the computational setup.
- Motivation: Conventional optimization error estimates are informal local diagnostics that are typically not posterior variances, communicable to other algorithms, or integrated into broader error estimation.These estimates are generally added after the numerical method rather than informing its design.
2. Probabilistic numerical methods from classical ones
Probabilistic numerics recasts numerical algorithms as inference procedures, preserving classical estimates while adding uncertainty models. Examples from quadrature and linear algebra show that prior assumptions can improve performance and calibration, but stronger assumptions and model selection introduce trade-offs.
- Classical numerical algorithms can be understood as maximum-a-posteriori estimates under specific priors and likelihoods.
- Bayes-Hermite Quadrature: The exponentiated quadratic prior can yield faster convergence and better-calibrated error estimates than the linear spline model for suitable integrands.For the illustrated function, the trapezoid rule is optimal while a Gaussian-kernel GP is over-confident; the exponentiated quadratic error estimate tracks real error better than the linear spline estimate.
- Bayes-Hermite Quadrature: Bayesian quadrature with a linear spline prior produces a posterior mean identical to the trapezoid-rule estimate.The posterior mean is a linear spline interpolant constrained to pass through evaluated function values.
- Added value, and challenge, of probabilistic output: Probabilistic quadrature embeds rules with different assumptions in a hierarchical model space, enabling model selection without additional function evaluations.Selecting among models requires more computational work than fixing a regressor, but can improve uncertainty calibration.
- Bayes-Hermite Quadrature: For rougher integrands, the exponentiated quadratic prior can produce inaccurate, over-confident, and inefficient estimates, whereas a matched spline prior is calibrated.The spline prior is optimal by construction when the integrand is sampled from its corresponding Gaussian process.
- Empirical evaluation of Bayesian quadrature for astrometry: WSABI combines smoothness and positivity priors with active node selection and converges faster than simple Monte Carlo and annealed importance sampling.Bayesian Monte Carlo uses the same samples as simple Monte Carlo, while WSABI’s improvement over BMC highlights the role of active selection.
- Linear Algebra: In linear algebra, suitable Gaussian priors make the posterior mean reproduce conjugate gradients while providing a posterior covariance for uncertainty quantification.The construction uses observed matrix projections and yields a manageable Gaussian posterior over the unknown matrix.
(i) Further Areas
Probabilistic interpretations extend beyond quadrature and linear algebra to optimization and differential equations, while enabling information transfer across related computational problems.
- Broader implications: Probabilistic formulations connect numerical methods across problem boundaries and motivate propagating uncertainty or knowledge between computational tasks.These connections suggest applications beyond the examples discussed in detail.
- Optimization: Quasi-Newton BFGS methods admit a probabilistic interpretation related to conjugate gradients and can model noise in evaluated gradients.In linear problems, BFGS and conjugate gradients are the same algorithm.
- Differential equations: Runge–Kutta methods for initial value problems can be interpreted as linear extrapolation rules within a probabilistic framework.The method repeatedly constructs solution estimates and uses function evaluations as observations.
- Linear algebra: In astronomical blind deconvolution, communicating posterior means between sequential linear solves makes later problems progress increasingly faster.The initial residuals decrease by about one order of magnitude as information is reused.
- Linear algebra: The posterior-mean vectors for inverse blur operators converge toward a generic basis for point-spread functions.This supports reuse of information across the sequence of deconvolution problems.
3. Discussion
The discussion presents probabilistic numerics as a recipe built from a generative model and an action rule, then outlines its opportunities, practical challenges, and long-term vision for chained computations.
- (a) General Recipe for Probabilistic Numerical Algorithms: A probabilistic numerical method specifies a joint model for the target quantity and computed numerical observations, separating prior and likelihood.The prior encodes assumptions about solution structure, while the likelihood describes the numerical task.
- (a) General Recipe for Probabilistic Numerical Algorithms: The method also requires an action rule determining which computations to perform, ranging from fixed grids to adaptive decisions based on previous evaluations.For many classic methods, the decision rule is greedy or otherwise simple.
- (a) General Recipe for Probabilistic Numerical Algorithms: Classic algorithms for quadrature, linear algebra, optimization, and ordinary differential equations can often be cast as maximum a-posteriori inference under Gaussian priors.Their likelihoods are typically deterministic because observations are assumed noise-free.
- (c) New Paths for Research: The field remains limited by early-stage formal analysis, developing implementations, scarce convincing use cases, and difficult lightweight hyper-parameter adaptation.Fixed parameters may reproduce existing analytic bounds rather than provide actively fitted uncertainty estimates.
- (c) New Paths for Research: The probabilistic view supports extending numerical methods to noisy evaluations, including optimization of noisy functions encountered in large-scale machine learning.Noise can be represented by replacing point-mass likelihoods with Gaussian distributions.
- (d) A Vision: Chained numerical methods communicating uncertainty: Propagating probabilistic inputs and outputs through chained numerical methods could support runtime error diagnosis, computational-budget allocation, and early termination.The proposed representation turns chained computations into factor graphs connected by message passing.
4. Conclusion
The conclusion frames numerical tasks as inference problems whose probabilistic formulations can improve functionality and performance while supporting uncertainty propagation through computational pipelines.
- 4. Conclusion: Probabilistic numerical methods reinterpret numerical tasks as inference problems and establish connections among otherwise disparate algorithms.The paper presents this reinterpretation as a framework for linear algebra, integration, optimization, and differential equations.
- 4. Conclusion: These formulations can encode prior information about problem structure and propagate collected information between problem instances, improving performance on structured problems.The conclusion distinguishes these benefits from a not-yet-rigorous general uncertainty interpretation.
- 4. Conclusion: Well-calibrated priors and models are needed before posterior uncertainty can be interpreted as a statement about computational error.The paper identifies calibration as an important condition for uncertainty interpretation.
- 4. Conclusion: In the long run, uncertainty propagation through computational pipelines could enable active control of computational effort in hierarchical, modular computations.The conclusion presents this as a future possibility rather than an established capability.