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Concept# Probabilistic numerics

Summary

Probabilistic numerics is an active field of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
A numerical method is an algorithm that approximates the solution to a mathematical problem (examples below include the solution to a linear system of equations, the value of an integral, the solution of a differential equation, the minimum of a multivariate function). In a probabilistic numerical algorithm, this process of approximation is thought of as a problem of estimation, inference or learning and realised in the framework of probabilistic inference (often, but not always, Bayesian inference).
Formally, this means casting the setup of the computational problem in terms of a prior distribution, formulating the relationship between numbers computed by the computer (e.g. matrix-vector multiplications in linear algebra, gradients in optimization, values of the integrand or the vector field defining a differential equation) and the quantity in question (the solution of the linear problem, the minimum, the integral, the solution curve) in a likelihood function, and returning a posterior distribution as the output. In most cases, numerical algorithms also take internal adaptive decisions about which numbers to compute, which form an active learning problem.
Many of the most popular classic numerical algorithms can be re-interpreted in the probabilistic framework. This includes the method of conjugate gradients, Nordsieck methods, Gaussian quadrature rules, and quasi-Newton methods. In all these cases, the classic method is based on a regularized least-squares estimate that can be associated with the posterior mean arising from a Gaussian prior and likelihood.

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