Searching for Lost Nuclear Bombs: Bayes’ Theorem in Action

 

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What does searching for lost nuclear weapons, hunting German U-boats submarines, breaking NAZI codes, savaging the high seas for Soviet submarines, and searching for 747s that go missing in the Pacific Ocean have in common? No, not a diabolical death wish, but a 200-year-old algorithm, deceivingly simple, yet computationally robust. This algorithm and its myriad offshoots is Bayes Theorem. The book The Theory That Would Not Die, by Sharon Bertsch McGrayne, opens the reader to the 200-year controversial history of the theorem, beginning in the late 1700s with Reverand Thomas Bayes’ discovery, through the application by Pierre Laplace, to its contentious struggle against the statistical frequentist approach in the early 1900s.

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It wasn’t until the dawn of the computer age that Bayes made an impact on predictive analytics. The narrative that McGrayne portrays is one of the misunderstood analyst, so desperately clinging to a belief that will offer clearer predictive power, trying to convey a simple algorithm that offers a more powerful means of testing phenomenon.

What is Bayes?

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A: Hypothesis

B: Data

The idea is simple: We learn new information each day. In essence, we update the knowledge that we already have on a daily basis from our past experiences. Each new day that passes we update our prior beliefs. We assign a probability of events occurring in the future based on these prior beliefs. This prior belief system is at the core of Bayes theorem. Simply put, Bayes is a way of updating our beliefs with new information to arrive at a more exact prediction based on probability. Another way of looking at the rule is below from www.pyschologyinaction.org.

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What does Bayes look like in action? 

Caveat: This is an extremely simplified version of the model used and is by no means represents the sheer volume of calculations involved by highly skilled statisticians.

A popular case documented by McGrayne in The Theory that Would Not Die is an incident that happened in the 1960s. The US military loses things, more often than we like to think. Many nuclear bombs have been accidentally dropped or lost in transit. While not activated, these bombs are a national security threat that most presidents want to get a hold of as quickly as possible. One specific example of this is in 1966 when a B-52G bomber crashed mid-air with a refueling KC-135 tanker over the Mediterranean Sea. The plane subsequently jettisoned four nuclear warheads. Three of these were found on land while the third lay in the waters off the Spanish coast.

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The Navy employed probabilistic Bayes experts to find the warheads. Specifically, they used Bayes’ Rule to find the probability of the warhead being in a given area given a positive signal from their sonar devices. Since going 2,550 feet below the ocean in a submersible is expensive and dangerous, the Navy wanted to ensure it was making the trip with a purpose and not to find a cylindrical rock formation.

The Navy searched for 80 days without results. However, Bayes tells us that an inconclusive result is a constructive result nonetheless. As McGrayne states,  “Once a large search area was divided into small cells, Bayes’ rule said that the failure to find something in one cell enhances the probability of finding it in the others. Bayes described in mathematical terms an everyday hunt for a missing sock: an exhaustive but fruitless search of the bedroom and a cursory look in the bath would suggest the sock is more likely to be found in the laundry. Thus, Bayes could provide useful information even if the search was unsuccessful.” (McGrayne, 189)

The simplistic model for this search in a single square would be…

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  • P(A|B): Probability the square contains the warhead given a positive signal from Navy instruments
  • P(A): Probability the square contains the warhead
  • P(B|A): Probability of a positive signal from the instrument given the warhead is present
  • P(B): Probability of a positive signal from instrument

We would then add these cells to derive a more general picture of the search area. The search area would be updated as new data flowed in, creating a new prior and posterior hypothesis that stemmed directly from the new data. Remember, this is one component of a much larger model, but it should help you get the picture. To get a more robust model, we would need to create similar calculations for the last known position of the aircraft when it went down, the currents of the ocean, the climate over the past months, etc. Computationally, this can get very heavy very quickly, but the underlying principles remain the same: base a hypothesis on prior knowledge and update to form a posterior hypothesis.

As you can see, as Navy vessels complete searches in one area, Bayes’ model is updated. While the above pictures are from a study involving the search for Air France Flight 447 in the Pacific Ocean, the story remains the same. We create a hypothesis, we test it, we gain new data, and we use that data to update our hypothesis. If you are curious about how Bayes was used for the successful recovery of France Flight 447, I highly recommend the 2015 METRON slides.

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Posterior hypothesis after initial search

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Similar methodologies in large-scale search endeavors have been well documented. these include the current search for the missing Malaysia Flight 370. Below are links to interesting use-cases of Bayes in action.

Bayesian Methods in the Search for MH370

How Statisticians Found Air France Flight 447 Two Years After It Crashed Into Atlantic

Missing Flight Found Using Bayes’ Theorem?

Operations Analysis During the Underwater Search for Scorpion

Can A 250-Year-Old Mathematical Theorem Find A Missing Plane?

And yes, they found the bomb!

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Background reference:

The Theory That Would Not Die, Sharon Bertsch McGrayne

https://en.wikipedia.org/wiki/1966_Palomares_B-52_crash

 

Image references:

https://sinews.siam.org/Details-Page/Bayesian-Search-for-Missing-Aircraft-Ships-and-People

https://www.motherjones.com/politics/2015/08/nuclear-weapon-obama-most-expensive-ever/

https://www.military.com/daily-news/2017/10/17/surge-wake-ship-collisions-tests-navy-new-deployment-plan.html

https://www.rogerebert.com/reviews/the-hunt-for-red-october-1990

http://thediplomatinspain.com/ecologistas-en-accion-lleva-al-supremo-la-radiactividad-de-palomares/

https://www.psychologyinaction.org/psychology-in-action-1/2012/10/22/bayes-rule-and-bomb-threats

https://navaltoday.com/#newsitem-123777

 

 

 

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