Probabilistic Numerics

Numerical methods for tasks like quadrature, optimization, linear algebra and the solution of differential equations estimate latent quantities from the observed result of tractable computations. In this sense, they are learning machines, and accessible to the framework of probabilistic inference.

What started as an entertaining observation has, over the past few years, turned up increasingly precise links between computation and inference, and begun to produce practically useful functionality. I will give a brief overview of recent developments, emphasizing a string of results identifying classic numerical methods -- Gaussian quadrature, conjugate gradients, BFGS , Runge-Kutta -- with maximum a posteriori estimators. I will also highlight several stirring potential applications that, coupled with a stack of fundamental questions, make probabilistic numerics an exciting area at the boundary between mathematics and computer science.