In 1907, Francis Galton examined 787 tickets from a weight-guessing contest at an English livestock fair and found something that’s been quoted ever since: the median guess for the ox’s dressed weight, 1,197 pounds, came within a single pound of the true value, 1,198, despite most contestants having no particular expertise in cattle. It’s become one of the founding anecdotes of a broader mathematical idea, formalized more than a century earlier by the Marquis de Condorcet, that averaging many independent, only-slightly-better-than-random judgments can produce something close to the truth. Modern machine learning has an entire family of techniques, ensemble methods, that get casually described as this same principle implemented in code. That’s true for about half of them. The other half works through a mechanism that’s arguably the mathematical opposite of what Galton and Condorcet actually require.
Scientific Foundation
Condorcet’s Jury Theorem, formalized in 1785, makes a precise claim about group decision-making: if a group of voters each have some probability just slightly better than a coin flip of reaching the correct answer on a binary choice, and their judgments are statistically independent of one another, then the probability that a majority vote reaches the correct decision increases as the group grows, approaching certainty as the group size goes to infinity. Galton’s ox-weighing analysis, a related but technically distinct result built around averaging continuous numerical estimates rather than majority-voting on a binary choice, provided striking empirical support for the same broad intuition roughly 120 years later. The mathematical engine behind both results is the same: independent errors, when combined, partially cancel each other out rather than reinforcing one another, which is precisely why the aggregate can end up more accurate than any single contributor. That independence requirement isn’t a minor technicality — philosophers and social-choice theorists studying these jury theorems describe it as one of the framework’s genuinely restrictive assumptions, and there’s an active research tradition specifically examining what happens once voters’ errors become correlated, including formal models documenting real conditions under which the wisdom of crowds actually fails, once shared biases or social influence break the independence the theorem depends on.
Cross-Domain Connection
One major branch of ensemble machine learning is a direct, deliberate engineering implementation of exactly this mathematical requirement. Bagging, short for bootstrap aggregating, trains many individual models, typically decision trees, each on a different randomly resampled subset of the training data, then averages or votes across their predictions. Random forests, a refinement of bagging, go a step further: at every single decision point within every tree, only a random subset of the available features is even considered as a candidate for splitting. This isn’t an arbitrary design flourish — it exists specifically because researchers found that plain bagging alone still left individual trees too similar to each other, since a handful of strongly predictive variables would tend to dominate the earliest, most influential splits across nearly every tree in the ensemble regardless of which data subset it was trained on. Randomly restricting feature access at each split is an explicit, engineered attempt to decorrelate the trees further, pushing the ensemble closer to the genuine statistical independence Condorcet’s mathematics actually requires for averaging to help rather than merely repeat the same errors in unison.
What Remains Undemonstrated
It would be a real overreach to describe “ensemble methods” broadly as the computational form of the wisdom of crowds, because boosting, an equally prominent and often higher-performing ensemble family that includes AdaBoost and gradient boosting, works through a mechanism that runs in essentially the opposite direction. Rather than training many models independently and combining them, boosting builds its ensemble sequentially: each new weak learner is deliberately constructed to focus specifically on the examples the ensemble built so far has gotten wrong, with those mistaken examples up-weighted so the next model pays disproportionate attention to fixing them. That’s a design that actively correlates each new model with the current ensemble’s specific error pattern, rather than seeking independence from it — nearly the reverse of what bagging is engineered to do. The mathematical justification for why boosting works comes from a completely different theoretical tradition than Condorcet-style independent-error cancellation, and the structural difference has real, documented practical consequences: boosting algorithms are known to be considerably more sensitive to noisy or mislabeled data than bagging, precisely because a noisy data point that gets misclassified tends to accumulate increasing weight across successive rounds rather than simply being diluted away by independent resampling the way it would be in a bagged ensemble.
So the honest, precise version of this comparison is narrower than the original framing suggests. Bagging and random forests are the genuine computational descendants of Galton’s ox and Condorcet’s jury — built, explicitly, around the same independence requirement, with real engineering effort spent enforcing it. Boosting belongs to the same broad “ensemble learning” umbrella term, and often outperforms bagging in practice, but it’s solving the aggregation problem through a mathematically unrelated route, one that Condorcet’s original proof gives no reason to expect would work at all.
Why It Matters
Getting this distinction right matters for anyone reasoning by analogy from crowd wisdom to machine learning, or vice versa. If someone wants to understand why an ensemble technique works by pointing to Galton’s ox, that argument genuinely applies to bagging and random forests, where decorrelating the individual models is a stated, deliberate design goal serving exactly the function Condorcet’s math describes. Pointing to the same anecdote to explain boosting would be reaching for the wrong theory entirely — boosting’s success has its own, separate mathematical story, built on sequential error-correction rather than independent averaging, and conflating the two risks importing false intuitions about noise sensitivity, diversity requirements, or failure modes from one family into the other.
Human Dimension
There’s something worth appreciating in discovering that a phrase as broad and comfortable as “ensemble learning” is quietly covering two genuinely different mathematical bets. One bet, bagging’s, is Galton’s bet: gather enough independent, decorrelated opinions and let their disagreements cancel each other out. The other, boosting’s, is a different wager entirely: build a sequence of specialists, each one deliberately obsessed with fixing whatever the group got wrong last time. Both bets pay off, often spectacularly. But only one of them was actually placed at a county fair in 1906, and it’s worth knowing which one you’re actually relying on before reaching for the anecdote to explain either.
Sources:
1. Springer Nature Link, Synthese — “A skewed jury theorem: more theorems in search of the truth” — https://link.springer.com/article/10.1007/s11229-025-05125-0
2. arXiv — “(Failure of the) Wisdom of the crowds in an endogenous opinion dynamics model with multiply biased agents” — https://arxiv.org/pdf/1309.3660
3. arXiv — “WoCE: a framework for clustering ensemble by exploiting the wisdom of Crowds theory” — https://arxiv.org/pdf/1612.06598
4. arXiv — “Delegation and Participation in Decentralized Governance: An Epistemic View” — https://arxiv.org/pdf/2505.04136
5. Stanford Encyclopedia of Philosophy — “Jury Theorems” — https://plato.stanford.edu/entries/jury-theorems/
6. arXiv — “Wisdom of crowds: much ado about nothing” — https://arxiv.org/pdf/2008.01485
7. arXiv — Blier-Wong, C. et al., “Boosting insights in insurance tariff plans with tree-based machine learning methods” — https://arxiv.org/pdf/1904.10890
8. arXiv — “Improved Weighted Random Forest for Classification Problems” — https://arxiv.org/pdf/2009.00534
9. arXiv — “Probabilistic Random Forest: A machine learning algorithm for noisy datasets” — https://arxiv.org/pdf/1811.05994
10. MachineLearningMastery.com — “Bagging and Random Forest Ensemble Algorithms for Machine Learning” — https://machinelearningmastery.com/bagging-and-random-forest-ensemble-algorithms-for-machine-learning/
Idea originated at artificialideas.org. Article researched and written by Claude Sonnet 5. Published at artificialideas.org.
Artificial Ideas 