How to extend Elo: a Bayesian perspective

The Elo rating system, originally designed for rating chess players, has since become a popular way to estimate competitors’ time-varying skills in many sports. Though the self-correcting Elo algorithm is simple and intuitive, it lacks a probabilistic justification which can make it hard to extend. In this paper, we present a simple connection between approximate Bayesian posterior mode estimation and Elo. We provide a novel justification of the approximations made by linking Elo to steady-state Kalman filtering. Our second key contribution is to observe that the derivation suggests a straightforward procedure for extending Elo. We use the procedure to derive versions of Elo incorporating margins of victory, correlated skills across different playing surfaces, and differing skills by tournament level in tennis. Combining all these extensions results in the most complete version of Elo presented for the sport yet. We evaluate the derived models on two seasons of men’s professional tennis matches (2018 and 2019). The best-performing model was able to predict matches with higher accuracy than both Elo and Glicko (65.8% compared to 63.7 and 63.5%, respectively) and a higher mean log-likelihood (−0.615 compared to −0.632 and −0.633, respectively), demonstrating the proposed model’s ability to improve predictions.

Inclusion of Aircraft Crash into NPP Design Bases and Probabilistic Justification of Loads on Civil Structures and Equipment

The article is about nuclear power plant (NPP) safety analysis in case of aircraft crash. Specifically, the article considers the following problems: inclusion of aircraft crash into NPP design bases regarding calculation of frequency of an aircraft crash into NPP; aspects of justification of loads on NPP structures, systems and components (SSCs) caused by mechanical action of a primary missile – aircraft fuselage impact. Probabilistic characteristics of such random parameters as frequency of aircraft crash and direction of aircraft trajectory are determined by the results of analysis of world statistics of aviation accidents. Method of calculation of aircraft crash frequency on structures, buildings and NPP as a whole is presented. It takes into account options of accidental and intentional aircraft crashes and various aircraft approach scenarios. Procedure of probabilistic justification of loads on civil structures under aircraft impact is described. The loads are specified so as not to exceed allowable value of failure probability of NPP as a whole. Calculation of failure frequency of civil structures of existing NPP is given as an example to show analysis in case of a crash of an aircraft heavier than considered in NPP design. Procedure of probabilistic justification of dynamic loads on NPP equipment in case of aircraft impact is described. Method of floor response spectra (FRS) calculation with the required non-exceedance probability is given. Probabilistically justified loads in case of intentional aircraft impact (act of terrorism) are also considered. Additionally it is presented how internal forces calculated with the use of FRS with the required non-exceedance probability can be summed to provide analysis of subsystems.

Probabilistic justification logic

We present a probabilistic justification logic, $\mathsf{PPJ}$, as a framework for uncertain reasoning about rational belief, degrees of belief and justifications. We establish soundness and strong completeness for $\mathsf{PPJ}$ with respect to the class of so-called measurable Kripke-like models and show that the satisfiability problem is decidable. We discuss how $\mathsf{PPJ}$ provides insight into the well-known lottery paradox.