One measurement, many names
Inferring ability from who wins, rediscovered field by field for a century.
One question keeps getting asked: how good is each competitor, given only the record of who beat whom? A race records the fastest, a market records the most chosen, a test records the ones who passed, a leaderboard records the model that won the item. In every case the outcome is an argmax over hidden qualities plus noise, and the task is to run the argmax backwards. Psychologists, economists, statisticians, sports raters and now AI evaluators have each formalized this, mostly without citing one another. The names differ; the object is the same.
This package computes that object — the distribution of the winner of many correlated competitors, and its inverse — and the timeline below is the lineage it sits in. Highlighted nodes are the ones where two fields turn out to have written the same equation.
Elements of Psychophysics turns a subjective comparison — which weight feels heavier — into a measurable quantity by modeling the sensation as a noisy magnitude. The idea that a latent intensity can be recovered from the frequency of a judgment is the seed of everything below.
The law of comparative judgment writes each stimulus as a Gaussian "discriminal process" and the probability that one is judged greater than another as the tail of their difference. This is the founding document: latent quality is normal, the observation is an argmax, and the model is inverted to place stimuli on a scale. Every later rating system is a variation on it.
Faced with an unfinished chess tournament, Zermelo derives maximum-likelihood strengths from win counts — the paired-comparison model that Bradley and Terry rediscover independently two decades later, and the direct ancestor of Elo.
Practical estimation of Thurstone's normal model (Case V), and a test of its fit. Psychometrics is now computing, not just theorizing, the inverse.
The logistic paired-comparison model, arrived at without reference to Zermelo. Swapping Thurstone's Gaussian noise for a logistic difference buys a closed form, and the trade — tractability for the exact normal model — is the one this package exists to undo.
The choice axiom: pick among many alternatives in proportion to strengths, independent of irrelevant alternatives. This is the softmax, stated as a behavioral law fifteen years before it becomes an econometric estimator and sixty before it is a neural-network layer.
Item response theory: a test taker's ability and an item's difficulty combine through a logistic to give the probability of a correct answer. The measurement machinery of standardized testing, and — sixty-five years on — the machinery being reinvented to score language models on benchmarks.
Assigning probabilities to the full order of finish of a multi-entry competition, motivated by horse racing. The winner is the argmin of running times; the place and show probabilities are the deeper order statistics. This is the problem this package solves exactly.
Conditional logit: random utility maximization as an econometric model of choice, which is Thurstone's 1927 model with Gumbel noise in place of Gaussian. The economists rediscover the psychologists' object, industrialize it, and win a Nobel for it (2000). Same argmax, new field.
A model for permutations built by drawing winners without replacement, the ranking companion to Luce's choice axiom. The likelihood of a whole finishing order, in the softmax family.
The unification. Yellott proves that Luce's choice axiom holds for a Thurstone model exactly when the noise is double-exponential (Gumbel) — that softmax is the Gumbel special case of the Gaussian race. The whole apparent divide between logit and probit is one distribution choice, a fact this package makes computational: one lattice, softmax and multinomial probit as two settings of it.
The chess rating system: a running estimate of each player's latent strength from game outcomes, the paired-comparison model turned into infrastructure. Glicko (1999) adds uncertainty, TrueSkill (2006) adds teams and a Gaussian belief; all are Thurstone kept online.
Permutation probabilities as models for horse races, treating the normal ranking probability directly — and finding it hard enough to propose an approximation. The exact normal race is now a recognized computational problem, not just a definition.
Numerical evaluation of multivariate normal probabilities by separation of variables and quasi–Monte Carlo — the workhorse, alongside the GHK simulator from econometrics, for the orthant integrals that the normal race demands. The standard, and still the right tool at high rank; this package's shared field is the linear-time complement for structured covariance.
A computer handicapping model — a multinomial logit over horses, combined with the public's own probabilities — run for profit to the tune of about a billion dollars at the Hong Kong track. The clearest proof that inverting the race is worth doing well.
Inferring relative ability from winning probability in multi-entrant
contests, in the SIAM Journal on Financial Mathematics: the lattice method
that represents each competitor's performance as a density and inverts the
winner map for arbitrary noise, shipped as
winning.classic.
Large language models are sampled many times per problem and scored by whether any attempt succeeds — pass@k, an order statistic of latent per-attempt abilities. The statistics that follow (difficulty priors, factor models of skill, adaptive item selection, leaderboard uncertainty) are item response theory and the theory of comparative judgment, rebuilt for a new examinee and rarely cited as such. The newest field to rediscover the oldest measurement.
Scalable inversion of contests with correlated performances: all the win probabilities of a factor-correlated race in one linear-time pass, their derivatives matrix-free, and the map from observed shares back to abilities inverted at a million alternatives. The Thurstone–Mosteller model, freed from the tractability compromise Bradley, Terry and Luce made, at modern scale.
The throughline
Fechner's noisy sensation, Thurstone's discriminal process, McFadden's random utility, the item response ability, the Elo rating and the latent per-attempt success rate of a language model are one variable seen from six doorways. In each, an argmax over hidden qualities is observed and the hidden qualities are recovered. Psychology measured it first; the other fields industrialize it, once a generation, under a new name. This package computes the argmax and its inverse for correlated competitors, and the list above is the company it keeps.