Bayes’s theorem

Bayes’s theorem is a method for calculating how new information changes an existing estimate of how likely an event is. In the book’s central illustration, John Craven used it to combine specialists’ wagers about the loss of the submarine Scorpion into a collective estimate of its final location—one that came within 220 yards of the wreck. The example makes the theorem part of a larger argument: scattered judgments can yield knowledge that no individual possesses, provided they are brought together in a way that preserves their distinct contributions.

Definition and Use

The book describes Bayes’s theorem as a way to calculate how new information about an event changes preexisting expectations about its likelihood. Its role in the Scorpion search was practical: Craven did not ask his team to agree through discussion, but gathered separate judgments about possible explanations and circumstances, then used a formula to combine them into an estimate.

The Scorpion Search

In May 1968, Scorpion disappeared while returning to Newport News after duty in the North Atlantic. The navy knew the submarine’s last reported location but had little information about what had happened or how far it might have traveled after its last radio contact. Its initial search area was a circle twenty miles wide and thousands of feet deep, making a focused search seem nearly impossible.

Craven developed alternative scenarios for the submarine’s fate and assembled people with varied expertise, including mathematicians, submarine specialists, and salvage men. He asked each to estimate the likelihood of the scenarios rather than consult with the others to reach a common answer. Their wagers, for bottles of Chivas Regal, concerned such questions as why the submarine ran into trouble, its speed on the way to the ocean bottom, and the steepness of its descent.

No single wager could locate the wreck. Craven’s reasoning was that the separate estimates, taken together, could build a composite picture of how the submarine died and thereby indicate where it had come to rest. He used Bayes’s theorem to produce the group’s collective estimate. The resulting location was not one any individual team member had selected: it existed in the combined judgment, not in any one person’s mental picture. Five months after the disappearance, a navy ship found Scorpion 220 yards from the predicted location.

The result was striking because the team had worked with only scraps of data. No one knew why the submarine sank, how fast it traveled, or how steeply it descended. The book presents the episode as a case in which the group’s combined judgment supplied a coherent picture that no participant had independently possessed.

Aggregation and Collective Judgment

The Scorpion example illustrates the difference between collecting opinions and aggregating them into a judgment. Craven did not simply choose the view of the most prominent expert, nor did he seek agreement through discussion. Instead, distinct estimates became inputs to a collective calculation. The location that emerged was therefore genuinely collective: it could not be reduced to the forecast of the team’s smartest member.

The book relates this logic to the conditions it identifies for wise crowds: diversity of opinion, independence of judgment, decentralization, and aggregation. Diverse people can contribute different private information; independence helps keep their estimates from merely echoing one another; decentralization lets individuals draw on specialized or local knowledge; and aggregation provides a way to turn those separate judgments into a common result. Bayes’s theorem is the calculating device in Craven’s example, while the broader argument concerns the conditions that make such combination useful.

Scope and Limits

The book does not present the Scorpion search as evidence that a formula can create information from nothing. The group’s collective judgment worked because each estimate could contribute something to the overall picture, however fragmentary the evidence. More generally, aggregating many judgments can cancel individual errors, but a crowd can still reach a poor result if those judgments contain too little relevant information. The theorem’s significance in the account is thus both mathematical and organizational: it shows how separate evidence can be combined, while leaving the quality of that evidence and the independence of the contributors essential to the outcome.