regression
The probability of default
It is the late 1950s, and you must model a yes/no outcome — will this loan default, will this patient survive — from a set of predictors, but fitting a straight line to a zero/one response is absurd: it predicts probabilities below zero and above one and assumes constant-variance errors that binary data never have. Model the log-odds of the outcome as linear instead, fit it by maximum likelihood, and interpret each coefficient as a change in odds. Get it wrong and a lender or a hospital acts on predicted "probabilities" that aren't probabilities at all — negative risks, certainties above one — and the entire apparatus of scoring binary outcomes rests on choosing the right link between a linear predictor and a bounded chance.
Who this problem belongs to
The two figures whose methods fit it best, out of 62 in contention.
This is Cox's own contribution: his 1958 paper 'The Regression Analysis of Binary Sequences' is the founding modern treatment of exactly this problem, modeling the log-odds of a binary outcome as a linear function of predictors and fitting it by maximum likelihood rather than forcing a straight line onto a zero-one response. He worked out precisely why ordinary least squares fails for binary data — predicted probabilities outside the valid range, non-constant error variance — and supplied the logistic link function as the principled fix, interpreting each coefficient as a change in log-odds. Set in the late 1950s, this is not Cox applying an inherited toolkit; he is, in that very moment, building the toolkit this problem is asking to be built.
Fisher's development of maximum likelihood estimation in the 1920s is the essential statistical engine Cox's logistic regression runs on — without a rigorous, general method for fitting a model's parameters by maximizing the likelihood of observed data, there would be no principled way to estimate the log-odds coefficients this problem requires. His work on classification and discriminant analysis for binary and categorical outcomes also engages closely related territory. But Fisher's own major contributions predate the specific logistic link function by decades and centered on linear discriminant analysis as his preferred tool for binary classification, a related but mathematically distinct approach from Cox's log-odds regression, leaving him foundational rather than the direct originator here.
In the mind map
The same ideas, as concepts rather than history — in John's ML knowledge map.
62 figures are scored on this problem. Draw it in a battle to see where you land.