Numerical algorithms, such as methods for the numerical solution of integrals and ordinary differential equations, as well as optimization algorithms can be interpreted as estimation rules. They estimate the value of a latent, intractable quantity – the value of an integral, the solution of a differential equation, the location of an extremum – given the result of tractable computations (“observations”, such as function values of the integrand, evaluations of the differential equation, function values of the gradient of an objective). So these methods perform inference, and are accessible to the formal frameworks of probability theory. They are learning machines.
Taking this observation seriously, a probabilistic numerical method is a numerical algorithm that takes in a probability distribution over its inputs, and returns a probability distribution over its output. Recent research shows that it is in fact possible to directly identify existing numerical methods, including some real classics, with specific probabilistic models.
Interpreting numerical methods as learning algorithms offers various benefits. It can offer insight into the algebraic assumptions inherent in existing methods. As a joint framework for methods developed in separate communities, it allows transfer of knowledge among these areas. But the probabilistic formulation also explicitly provides a richer output than simple convergence bounds. If the probability measure returned by a probabilistic method is well-calibrated, it can be used to monitor, propagate and control the quality of computations.
This site collects information pertaining to the development, analysis and use of numerical algorithms with probabilistic interpretations. We retain a growing list of academic publications on the subject, collect open and central research questions, and publish a blog discussing recent developments.
Although the mathematical idea of uncertainty about a computation was discussed by several authors in the past, the first formal meeting of our community was a workshop at Neural Information Processing Systems 2012. (the original website of this workshop can be found here).
Would you like to get involved with PN research community? If you feel that your own published work fits the above description, and should be mentioned on the literature site, then please don’t hesitate to contact us. The fastest way to get your documents onto the site is to clone our github repository, add your documents to the relevant BibTeX-file in /_bibliography, then either send us a pull-request, or an email with the updated file (see box on top right for our contacts).
Most importantly, if you would like to contribute and meet other people interested in related questions, join us for one of our meetings and workshops.
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