optimization
Fill in the hidden variables
It is 1977, and a recurring frustration across statistics finally gets a unifying method: you want the maximum-likelihood fit of a model, but the likelihood is a tangled mess because some variables were never observed — which cluster a point came from, which component generated it. Directly maximizing over the missing structure is intractable. The trick is to alternate: given a current guess, compute the expected values of the hidden variables, then maximize as if those expectations were data, and repeat, provably never decreasing the likelihood. Frame this as optimization by iteratively lower-bounding the objective. Get it wrong and you wrestle the likelihood into a bad local answer, or miss that a family of latent-variable problems — mixtures, missing data, hidden Markov chains — all yield to one monotone ascent.
Who this problem belongs to
The two figures whose methods fit it best, out of 71 in contention.
Rubin's 1977 paper with Dempster and Laird, 'Maximum Likelihood from Incomplete Data via the EM Algorithm,' is this problem verbatim: it named and formalized exactly the alternate-and-maximize procedure the problem describes, unifying mixture models, missing data, and hidden Markov structure under one monotone-ascent framework. His broader career-long focus on missing data, multiple imputation, and the mechanics of inference when some variables are unobserved makes this his signature intellectual territory, not a detour. He proved the likelihood never decreases across iterations, precisely the 'provably never decreasing the likelihood' clause this problem specifies. His score is not 100 only because Dempster and Laird share direct co-authorship of the founding paper. His name is, without qualification, the R in the paper's title.
Jordan's career substantially deepened and extended EM's theoretical foundations, his work on variational methods, published widely in the 1990s, reframed EM itself as a special case of coordinate ascent on a variational lower bound, precisely the 'iteratively lower-bounding the objective' framing this problem specifies, and generalized it to intractable models where exact E-steps are impossible. His mentorship at Berkeley shaped how an entire generation understands EM's variational interpretation. He did not co-author the original 1977 Dempster-Laird-Rubin paper, but his theoretical reframing is arguably the modern lens through which EM is now taught and understood. His score reflects very strong, near-primary theoretical contribution. Michael I. Jordan's name would appear early in any serious literature review a team assembled before attempting to build and certify a solution to exactly this problem.
Fought here
In the mind map
The same ideas, as concepts rather than history — in John's ML knowledge map.
71 figures are scored on this problem. Draw it in a battle to see where you land.