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Today, we’re gonna learn about probabilistic numerics — what they are, what they are good for, and how they relate computation and inference in artificial intelligent systems.

To do this, I have the honor of hosting Philipp Hennig, a distinguished expert in this field, and the Chair for the Methods of Machine Learning at the University of Tübingen, Germany. Philipp studied in Heidelberg, also in Germany, and at Imperial College, London. Philipp received his PhD from the University of Cambridge, UK, under the supervision of David MacKay, before moving to Tübingen in 2011. 

Since his PhD, he has been interested in the connection between computation and inference. With international colleagues, he helped establish the idea of probabilistic numerics, which describes computation as Bayesian inference. His book, Probabilistic Numerics — Computation as Machine Learning, co-authored with Mike Osborne and Hans Kersting, was published by Cambridge University Press in 2022 and is also openly available online. 

So get comfy to explore the principles that underpin these algorithms, how they differ from traditional numerical methods, and how to incorporate uncertainty into the decision-making process of these algorithms.

Our theme music is « Good Bayesian », by Baba Brinkman (feat MC Lars and Mega Ran). Check out his awesome work at https://bababrinkman.com/ !

Thank you to my Patrons for making this episode possible!

Yusuke Saito, Avi Bryant, Ero Carrera, Giuliano Cruz, Tim Gasser, James Wade, Tradd Salvo, William Benton, James Ahloy, Robin Taylor, Chad Scherrer, Zwelithini Tunyiswa, Bertrand Wilden, James Thompson, Stephen Oates, Gian Luca Di Tanna, Jack Wells, Matthew Maldonado, Ian Costley, Ally Salim, Larry Gill, Ian Moran, Paul Oreto, Colin Caprani, Colin Carroll, Nathaniel Burbank, Michael Osthege, Rémi Louf, Clive Edelsten, Henri Wallen, Hugo Botha, Vinh Nguyen, Raul Maldonado, Marcin Elantkowski, Adam C. Smith, Will Kurt, Andrew Moskowitz, Hector Munoz, Marco Gorelli, Simon Kessell, Bradley Rode, Patrick Kelley, Rick Anderson, Casper de Bruin, Philippe Labonde, Michael Hankin, Cameron Smith, Tomáš Frýda, Ryan Wesslen, Andreas Netti, Riley King, Yoshiyuki Hamajima, Sven De Maeyer, Michael DeCrescenzo, Fergal M, Mason Yahr, Naoya Kanai, Steven Rowland, Aubrey Clayton, Jeannine Sue, Omri Har Shemesh, Scott Anthony Robson, Robert Yolken, Or Duek, Pavel Dusek, Paul Cox, Trey Causey, Andreas Kröpelin, Raphaël R, Nicolas Rode, Gabriel Stechschulte, Arkady, Kurt TeKolste, Gergely Juhasz, Marcus Nölke, Maggi Mackintosh, Grant Pezzolesi, Avram Aelony, Joshua Meehl, Javier Sabio, Kristian Higgins, Alex Jones, Gregorio Aguilar and Matt Rosinski.

Visit https://www.patreon.com/learnbayesstats to unlock exclusive Bayesian swag ;)

Links from the show:

• Philipp on Twitter: https://twitter.com/PhilippHennig5
• Philipp on Github: https://github.com/philipphennig
• Philipp on LinkedIn: https://www.linkedin.com/in/philipp-hennig-635832278/
• An introductory course on Probabilistic Numerics, taught collaboratively by Philipp’s Group: https://youtube.com/playlist?list=PL05umP7R6ij2lwDdj7IkuHoP9vHlEcH0s 
• An introductory tutorial on Probabilistic Numerics: https://youtu.be/0Q1ZTLHULcw 
• Philipp’s book: https://www.probabilistic-numerics.org/textbooks/
• ProbNum python package: https://probnum.readthedocs.io/en/latest/
• Probabilistic solvers for differential equations in JAX: https://pnkraemer.github.io/probdiffeq/
• Probabilistic Numerical Differential Equation Solvers in Julia: https://nathanaelbosch.github.io/ProbNumDiffEq.jl/stable/#Probabilistic-Numerical-Differential-Equation-Solvers
• Philipp’s research: https://www.probabilistic-numerics.org/
• Philipp’s academic page: https://uni-tuebingen.de/en/fakultaeten/mathematisch-naturwissenschaftliche-fakultaet/fachbereiche/informatik/lehrstuehle/methods-of-machine-learning/start/ 
• Tübingen Machine Learning on YouTube: https://www.youtube.com/c/TübingenML 
• LBS #74 Optimizing NUTS and Developing the ZeroSumNormal Distribution, with Adrian Seyboldt: https://learnbayesstats.com/episode/74-optimizing-nuts-developing-zerosumnormal-distribution-adrian-seyboldt/
• LBS #12 Biostatistics and Differential Equations, with Demetri Pananos: https://learnbayesstats.com/episode/12-biostatistics-and-differential-equations-with-demetri-pananos/

Abstract

by Christoph Bamberg

In episode 88 with Philipp Henning, chair of Methods in Machine Learning at the Eberhard Karls University Tübingen, we learn about new, technical areas for the Bayesian way of thinking: Probabilistic numerics.

Philipp gives us a conceptual introduction to Machine Learning as “refining a model through data” and explains what challenges Machine Learning phases due to the intractable nature of data and the used computations. 

The Bayesian approach, emphasising uncertainty over estimates and parameters, naturally lends itself for handling these issues. 

In his research group, Philipp tries to find more general implementations of classically used algorithms, while maintaining computational efficiency. They successfully achieve this goal by bringing in the Bayesian approach to inferences. 

Philipp explains probabilistic numerics as “redescrbiing everything a computer does as Bayesian inference” and how this approach is suitable for advancing Machine Learning.

We expand on how to handle uncertainty in machine learning and Philipp details his teams approach for handling this issue.

We also collect many resources for those interested in probabilistic numerics and finally talk about the future of this field.

Transcript

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