In addition to its use, or misuse, in court cases, Bayesian inference methods lie behind a number of new products on the market. For example, the paperclip advisor that pops up on the screen of users of Microsoft Office — the system monitors the user’s actions and uses Bayesian inference to predict likely future actions and provide appropriate advice accordingly. For another example, chemists can take advantage of a software system that uses Bayesian methods to improve the resolution of nuclear magnetic resonance (NMR) spectrum data. Chemists use such data to work out the molecular structure of substances they wish to analyze. The system uses Bayes’ formula to combine the new data from the NMR device with existing NMR data, a procedure that can improve the resolution of the data by several orders of magnitude.
Other recent uses of Bayesian inference are in the evaluation of new drugs and medical treatments, the analysis of human DNA to identify particular genes, and in analyzing police arrest data to see if any officers have been targeting one particular ethnic group.
CHAPTER VI BOMB BAYES OPEN
Another particularly fine example of Bayes’ theorem in action is displayed in the following example. On January 17, 1966, while attempting a mid-air refuel at 30,000 feet off the coast if Palmares Spain, a Strategic Air Command B-52 Statofortress Bomber (See illustration 1), collided with an air borne KC-135 Stratotanker fuel tanker aircraft. The collision killed all four of the crewmembers on the KC-135 and three of the seven crewmembers on the B-52.
When the collision occurred, the aircraft was carrying four Hydrogen Bombs. As result of the collision, all four bombs departed the aircraft in the air. Three of the four bombs were recovered almost immediately along the shore of Spain (See illustration 2). However, search teams could not locate the fourth bomb. It was lost and had presumably fallen to the bottom of the Mediterranean ocean.
1966 was a time of tremendous strain in the relations between the then Soviet Union and the United States of America. Tensions were running high partially because the on-going conflict in Vietnam and failed Bay-of-Pigs invasion of Cuba. The cold war was at its peak and competition for superior nuclear weapons design between the United States and the Soviet Union was fierce. When the United States government notified the government of Spain of the incident, its government, fearing a radiation leak, demanded a clean up and assurances from the United States that they would recover the bomb. The United States military knew the Soviets were aware of the accident and in an effort to retrieve the bomb and glean secrets from its design and construction, were looking for the bomb too. United States president Lyndon Johnson refused to believe his military’s assertions that there was a good probability the bomb could not and would not ever be recovered, by ether side, because of its presumed depth and condition. The president demanded its recovery.
A team of was assembled to try to pinpoint the location for a search and to attempt to retrieve the weapon once it was (if ever) located. The group was attempting to use Bayes’ Theorem. A group of mathematicians were assembled to construct a map of the sea bottom outside Palomares, Spain. Once the map was completed, the U.S. Navy assembled a group of submarine and salvage experts to place probabilities that Sontag describes as”?Las Vegas-style bets?” [pg. 63] of each of the different scenarios (outcomes), that might describe how the bomb was lost and what happened to it once it departed the aircraft. Each scenario left the bomb in not just a different place, but in a wide variety of places. Then, each possible location (all inclusive), was considered using the formula that was based on the probabilities created in the initial phase of the equation, that is the phase that set the initial probabilities.
The theory created by the 18th century theologian and amateur mathematician provided a way to apply quantitative reasoning to what we normally think of as a scientific method. That is, when several alternative theories, such as the case of the missing H-bomb, about an outcome exist, you can test them conducting experimental tests to see, whether or not those consequences actually occur. Put another way, if an idea predicts that something should happen and it does actually happen, it strengthens the belief in the validity of the idea. It acts as a spoiler too, if an actual outcome contradicts the idea, it may weaken the belief in the idea.
After a betting round to assign probabilities of the location of the bomb, the locations were then plotted again, sometimes great distances from where logic and acoustic science would have place them. The bomb, according to S.D. Bono, the Historian of the National Atomic Museum in Albuquerque NM, had been connected to two parachutes designed by the Sandia Corporation. “Sandia was the general contractor of the weapon system itself as well as the parachutes” (S.D. Bono, personal communication, February 14, 2001). The parachutes complicated the issue further because no one knew if they functioned or not. Whether they did or not, and what condition they were in, could have had dramatic effect on the bomb’s ultimate resting place. As part of applying Bayes’s theorem, the researchers asked the experts individually how they expected the event unfolded. Going over each part of the event with each participant. The team of mathematicians wrote out possible endings to the crash story and took bets (set probabilities), on which ending the believed to be most likely. After the betting was over, they used the odds created to assign probabilities to several locations identified by the betting. The site was again mapped, but this time, the most probable locations were marked according to relevancy.
The team used the Bayes theorem to map out where they believed the bomb was. According to their calculations, the most probable location of the bomb was a tremendous distance from where the other three bombs were located and a good distance from where most of the aircraft’s debris had hit the water. The team provided their data to the search team who immediately began the search.
The search was complicated by the fact that they had pinpointed a location that was at the bottom of a deep undersea ravine, making the search very difficult. After just two weeks of searching, president Johnson called the man responsible for the search to Washington to be briefed on the search. Upon learning the technique that had been used to determine the location of bomb, Johnson was furious. He could not believe the hope of finding the bomb before the Russian’s was tied to what appeared to outsiders, to such a plan of betting on where experts thought it would be. He called for another review of the data and circumstances by yet another group of mathematicians. The second group of mathematicians gathered, looked at the method used to determine the bombs location and could develop nothing better, and reported to president Johnson that there was no better way. Within days of the second group reporting to Johnson and after weeks of searching and revising the probabilities based on actual results, the bomb was located and eventually retrieved. The bomb was exactly where the team’s latest Bayesian calculations said it would be. The theory developed by an 18th century minister had found America’s lost Bomb.
CONCLUSION
As demonstrated by the examples shown in this paper, Bayesian thought has found its place in statistical thinking. It provides a mathematical approach to what is often called a hunch, and takes advantage of information an expert may have, but it unable to put into words. It allows one to refine ones theories about outcomes based on what has actually occurred, in practical use, it works well.
There are some that say its ineffective for the very reason it has gained much favor, but for many it serves a useful purpose in mathematical thinking. A amateur mathematician changed the was we see and calculate probabilities, and had it not been for his friend, Richard Price, who sent his essays to the Royal Society after his death, we would be without this very useful theory.
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APPENDIX
Equation 1
Equation 2
Equation 3
Equation 4
ILLUSTRATIONS
Illustration 1
A B52 Stratofortress owned by the National Atomic Museum, similar to the one that crash off the coast of Spain in 1966.
Illustration 2
One of the three Atomic bombs found on the coast of Palmares Spain, in 1966.