Archive for Bradley–Terry–Luce model

SEINE AI

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , on March 23, 2026 by xi'an

Ten days ago I took part in the SEINE AI 2026 workshop in Jouy-en-Josas, near Paris (homestead of HEC), organised by the Huawei Paris Research Center.. In which I was invited to speak, even though I felt sort of an outlier given the deeply machine-learning, entreprenarial orientation of the meeting, with its theme being Building the Agentic Future of ICT, given that I chose to present our most recent Bayesian adversarial privacy paper. Hence, I stood within a game-theoretic, Bayesian, formal landscape, presumably loosing most of the audience and keeping them away from their lunch!

Other speakers included Simon Lucas from Queen Mary London on Simulation-based AI, which I had trouble distinguishing from building a statistical model by goodness of fit (and using bandits used for update), while focussing on competing on some computer game challenges. And Volker Tresp from LMU München on a tensor brain model that he opposes to a Bayesian brain (with a related paper entitled Bayes or Heisenberg: Who(se) rules? which we discussed in general terms over lunch, namely Bayesian learning vs. quantum updating. And Michal Valko from INRIA Paris (and other companies), who went full blast against the Bradley-Terry model!, with a title of Nash and Nemirovski walk into a bar! With a half-time technique approximating Nash equilibria that reminded me of leapfrog. Much entertaining talk that further provided a game-theoretic transition to mine’s.

As an aside, I played yesterday with ChatGPT composing my talk slides out of our arXiv document and it proved a disaster, with hallucinations of results and concepts not in the paper and a complete mess of handling graphs, first creating generic, fake, unrelated pictures, then inserting actual graphs haphazardly throughout the slides. The sorry result I obviously did not use as the workshop did not seem the ideal place for this sort of prank! The actual version only recycles a few of its summarising slides. (With ye Norse farce proper colour choice!)

 

Elo rating systems via Markov Chains

Posted in pictures, Running, Statistics, University life with tags , , , , , , , , , , , on December 18, 2025 by xi'an

In preparation for meeting with a national sport association towards a Bayesian approach to ranking (I can only confirm this not for volleyball!), I was searching for advanced studies of the Elo (not ELO!) rating system and came across this 2024 arXival by Sam Olesker-Taylor (U Warwick) and Luca Zanetti. Where they analyse the online behaviour of the player ratings and their convergence (in Wasserstein distance), with fairly intricate proofs.

“Elo is not a reversible Markov chain and, while it has a unique stationary distribution, assuming a minor and natural condition, it does not converge to it in total variation.”

The analysis is based on the Bradley–Terry–Luce model of the probability of player i winning a game against player j. Which amounts to a logit or sigmoïd transform of the difference between the players’ true ratings. The practical Elo updates amounts to one stochastic gradient step for the associated likelihood. The authors also propose a random allocation of players into pairs that achieves optimal convergence to the true rating, by maximising the spectral gap of the allocation matrix. Although I did not spot how to derive it in practice.