Math equation could help find missing Malaysian plane

Bayes' Theorem helped researchers locate Air France Flight 447's black box in 2011

Indonesian Air Force officers examine a map of the Malacca Strait during a briefing following a search operation for the missing Malaysia Airlines Boeing 777, at Suwondo air base in North Sumatra, Indonesia, on Wednesday.

Days after a Malaysian airliner with 239 people aboard went missing en route to Beijing, searchers are still struggling to find any confirmed sign of the plane. Authorities have acknowledged that they didn't even know what direction it was heading when it disappeared.

As frustrations mount over the failures of the latest technology in the hunt for Malaysia Airlines Flight MH370, some scientists say an 18th-century mathematical equation – used in a previous search for an Air France jetliner's black box recorder – could help pinpoint the location of the Malaysian plane.

In 2009, Air France Flight 447 en route to Paris from Rio de Janeiro vanished over the Atlantic Ocean, triggering the most expensive and exhaustive search effort ever conducted for a plane. After two years, officials could only narrow the location of the plane's black box down to an area the size of Switzerland.

But Flight 447’s black box was found in just five days after authorities contacted scientific consultants who applied a centuries-old equation called Bayes’ Theorem.

“It’s a very short, simple equation that says you can start out with hypothesis about something — and it doesn’t matter how good the hypothesis is,” said Sharon Bertsch McGrayne, author of “The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy.”

That’s because the hypothesis can keep changing and improve, and still be used with the theorem, McGrayne told Al Jazeera.

“You are committed to modify that hypothesis every time a new piece of information arises,” McGrayne said.

Bayes’ Theorem, which is also used in Google’s driverless cars and predictions in stock markets, is based on probability. Because the theorem starts with a hypothesis – something McGrayne said “can be very subjective” – it had been seen as controversial until the 1960s. But because it forces researchers to change their hypothesis with each new piece of information, the probability becomes more accurate.

The theorem was used in World War II to locate German U-boats and the lost nuclear submarine U.S.S. Scorpion. It was also used during the Cold War to spot Soviet submarines.

AF 447
Air France Flight 447's path
Oona Raisanen/Wikimedia Commons

Applying all available information – including wind and water currents, previous flight patterns and underwater drift – Bayes’ Theorem helped French authorities determine where Flight 447 would most likely be. The flight’s black box was then found under more than 12,000 feet of water.

Despite assistance from Australia, China, Thailand, Indonesia, Singapore, Vietnam, Philippines and the United States, Malaysian search efforts are even further from locating Flight MH370. The search area has been expanded to almost 27,000 square nautical miles – an area roughly equivalent to the state of Indiana – authorities said. That’s more than 10,000 nautical square miles larger than the search for Air France Flight 447, before Bayes’ Theorem was applied.

“The AF 447 search is rooted in Bayesian inference,” Lawrence D. Stone, chief scientist at Virginia-based scientific consultancy Metron – which was contacted to apply Bayes’ Theorem in the search for the Air France plane – wrote in ORMS Today magazine in 2011. Bayes’ Theorem “allows the organization of available data with associated uncertainties and computation of the PDF (probability distribution function) for target location given these data,” he said.

Stone told Al Jazeera the company is not currently involved in search efforts for Flight MH370, and that at this point it is “highly unlikely” that Bayes’ Theorem is being applied in the search.

But Professor Bradley Efron, statistician at Stanford University, told Al Jazeera that though Bayes’ Theorem is not being applied formally, “the people who are doing the searches are Bayesian.”

“Their search process has a certain Bayesian flavor, but it then got upset when their prior calculations were incorrect.”

Efron was referring to Malaysian authorities coming to a false calculation that the plane had crashed over the Malacca Straits, where they spotted an oil slick thought to have originated from the plane.

Though Bayes’ rule allows for constant modification to a hypothesis, it only works if the situation is similar enough to evidence applied from past incidents.

Searchers' focus on the oil slick in the Malaysia Airlines case would have guided Bayes’ Theorem in the wrong direction because the slick was not related to the missing plane.

A “weakness of Bayes’ theorem is that you have to have reasonably accurate past experiences in order for the theorem to calculate accurately,” Efron said. 

Find Al Jazeera America on your TV

Get email updates from Al Jazeera America

Sign up for our weekly newsletter

Get email updates from Al Jazeera America

Sign up for our weekly newsletter