Probabilistic Numerics and Uncertainty in Computations (1506.01326v1)

Abstract: 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.

• The paper's main contribution is to recast numerical tasks as inference problems using a probabilistic framework.
• It applies probabilistic models like Gaussian processes to classical problems such as integration, linear algebra, and differential equations.
• The work offers improved empirical accuracy and adaptive algorithms to address uncertainties inherent in traditional computational methods.

Probabilistic Numerics and Uncertainty in Computations

The paper "Probabilistic Numerics and Uncertainty in Computations" by Hennig, Osborne, and Girolami introduces an innovative approach to numerical analysis through the lens of probabilistic techniques. The primary focus is on quantifying and managing uncertainty that arises inherently in numerical computations, bridging a significant gap in traditional deterministic numerical methods.

Analytical Framework and Core Concepts

Traditional numerical methods have long been deterministic, providing point estimates or approximations for otherwise intractable mathematical problems. The authors propose a shift towards a probabilistic framework where numerical tasks are treated as inference problems. This paradigm treats the uncertainties in numerical computations explicitly, often incorporating them into a coherent probabilistic model with prior distributions, likelihoods, and posterior inference.

By framing numerical tasks such as linear algebra, integration, optimization, and differential equation solving as probabilistic inference problems, this approach allows for capturing uncertainties about calculated solutions more effectively. This is a crucial development in fields requiring high precision and certainty, such as climate science and astrometry, where computational outputs drive decision-making processes.

Novel Interpretations and Methodologies

The paper illustrates how classical numerical methods can be reinterpreted from a probabilistic standpoint. For example, classic quadrature methods are linked to Gaussian process inference, providing probability distributions over possible integrand outcomes rather than single point estimates. This probabilistic treatment allows the accommodation of uncertainties arising from approximations used in computation. Such interpretations extend to other numeric challenges, facilitating a unified language that enables numerical algorithms to communicate uncertainties, ultimately improving overall computational robustness.

The proposed methodologies offer improved empirical performance in certain cases and provide versatile adaptive algorithms tailored to the specifics of individual applications. This flexibility is particularly advantageous in addressing real-world scientific problems characterized by complex, multi-dimensional datasets.

Implications and Future Directions

The theoretical and practical implications of adopting probabilistic numerics are vast. Conceptually, it offers a refreshed viewpoint on computational algebra and statistical inference, equipping researchers with tools to better manage computational errors and improve model interpretations. Practically, implementing these frameworks could enhance the accuracy and reliability of scientific outputs, producing better-informed predictions and diagnostics.

Looking forward, further research and development could focus on the broader integration of probabilistic numerics into standard computational toolkits. This may involve refining computational techniques, exploring more comprehensive applications, and investigating the scalability of such methods across diverse computational environments.

Conclusion

Probabilistic numerics represents an advancement in how computations are approached, introducing a systematic probabilistic framework for dealing with uncertainties traditionally ignored or underestimated in numeric computation. By both extending and reinterpreting existing numerical methodologies within an inference paradigm, this paper lays a foundation for enhanced computational accuracy and reliability, encouraging broader adoption across scientific and engineering disciplines.