Bayes Theorem and some of the mysteries it has solved

This video was sponsored by Curiosity Stream and features over 2,000 documentaries and non-fiction titles for curious minds. In 2014, a man named John Aldridge was working on a lobster boat, he was 40 miles off the tip of Long Island and it was midnight, but while moving a handle he was using as a lever broke, sending him flying backwards into the ocean. , he shouted to the other two people in the boat, but they were sleeping and the engine was too loud, so the boat that was on autopilot vanished into the night. It was 3:30 in the morning and John was stranded in the Atlantic Ocean trying to stay alive.

It wasn't until 6 a.m. when the other two men on board woke up and immediately called the Coast Guard saying they had a Man Overboard now they had no idea if he went down five minutes ago or eight hours ago all they knew was his current location and the last time who saw him, based on water temperature and weather, the Coast Guard determined that John had a survival of approximately 19 hours. window before hypothermia took hold and his muscles failed, so now it was a race against the clock to find out where he was and this is what they did based on the boat's current position and two autopilot speeds, they were able to determine the maximum distance John could be.

From that location we can draw a circle that tells us that if John stood still, he has to be

some

where on that circle, but the ship was on a heading away from the coast, so we were actually able to narrow that region down and then the Coast Guard enters everything. The information provided in Something Knows optimal search and rescue planning system that operates based on mathematical statistics and real data about the weather, ocean currents, etc., the system overlays a grid over the ocean and assigns each square a probability of John being there given the information since there was a period of several hours where he could have fallen, we expect to see a long stretch of high probabilities and therefore started a search now, if we first search here, say , and we didn't find it, does that mean it's not there? there and therefore the new probability is zero, the answer is no, it's a small head in a huge ocean, we could have missed them, so after searching a grid we need to update its probability, not just set it to zero and This brings us to Bayesian. search theory an application of Bayesian statistics used to find lost objects the basis: this is obviously Bayes'

theorem

, but I'm just going to use conditional probability to keep things simple anyway, the goal here is to find the probability that

some

event is a being. true given that another event B is true, for example, if I roll the die and hide the result from you, then I ask what are the chances of five, obviously you would say one in six, but if I give you more information like I told you, the rule came out odd, this is a relevant piece of information that reveals more about the situation, so the probability that it is a five given that it is odd is one-third.

New information can really change things. Now the base

theorem

for the probability of a given B is written like this, which can be derived from this formula that I will use soon, and by the way, this numerator just means the probability of both events being true. now going back, let's say we looked for this square and we didn't find it, now we want a new probability, the probability. that he is there since we search and we don't find him well in those two squares, either he is there or he is not, maybe this square has a 5% chance of him being there according to the software, if he is in that square he will be found or no, maybe the sky is clear, so there's a 70% chance you'll find it if you look, but still a 30% chance you'll miss it if it's not in the square, so of course no they will find him with 100% certainty from the formula we saw earlier, first we are going to find the probability that he is in the square and also not found, since those are the two events that I listed here, the probability that he is in the square and is not found can be determined simply by going down these branches and multiplying those probabilities together, then we need to find the probability that he is not in that square period which can be determined by going down these two branches that end and, not being found , we add the product of those probabilities and we get a value of about 1.5 5%, which again is the probability that it is still in that Square given that we couldn't find it there if we go back to our grid, that Square that we just searched for now it was updated to its new probability of 1.5 5% since then. the square lowered the probabilities of the other squares increase by a small amount, obviously it is better to continue our search probably with this square, but after a few hours of searching we can see that it might be more beneficial to search for that original square again instead of one of these lower. probability of green squares here now through several studies a while ago it was determined that the probability of detecting an object is inversely related to the cube of the distance or Alisa serves as a solid estimate, but more general detection equations are not typically used , the constant is usually determined experimentally based on things like the depth of the water, whether the object is submerged, or the size of the object you are looking for.

This is one way that the probability value we saw earlier could be estimated for those who are curious, so if John is here, say, and a helicopter flies this straight line path looking for him at any given point, there is some probability of locate John based on distance, this probability changes at each point and if we integrate or add the values ​​along the line we get a total probability of detection for that line, but we don't know where John is, so we just have to make sweeps that optimize our probabilities, meaning we should apply optimized search patterns now if there is a high probability that John is in a certain location like If we knew exactly where he landed, then a search in an expanding square might be ideal for cover higher probability areas.

Sometimes a progressive online search is better, but in this case it was better to use a search parallel to the high probability strip and Yes, that software we saw before was calculating all of this using mathematics and statistics based on given information. Now, after hours in the water, John managed to find something to keep him afloat, but he had lost so much energy that he was preparing me for the real possibility of dying. He soon decided to tie his body to the object so that when it was found, the body could be given to his parents so they would have something to bury, but finally at three in the afternoon in one of the final sweeps, the search team saw John and brought him to safety. to the hospital where he was treated for hypothermia and made a full recovery to be able to tell this story without John's ability to stay calm at sea and a formula that dates back a few hundred years, it is very likely that this man was not alive next day.

In 1966, a United States Air Force bomber collided with another aircraft during a mid-air refueling over the Mediterranean Sea near Spain. On board the bomber there were four hydrogen bombs that fell to the ground, three were recovered within 24 hours but no one knew where they were. The fourth was, and therefore analysts assumed it was resting somewhere on the ocean floor to narrow the search. Some experts reapplied Bayesian search theory. The factors here were different from the previous story, although as if the lost object was submerged in the ocean. This time, however, in this case a local fisherman witnessed something parachute into the ocean on the day of the accident, giving the government an idea of ​​where the bomb was likely located, which really helped shape the case. initial probability map now that two months had passed.

There are no results, but remember that failed searches lead to higher chances of finding other squares as things update. After approximately 80 days, the bomb was found almost 3,000 feet underwater and within 1.5 kilometers of where the fisherman predicted. The testimony of the fishermen is supposed to have saved the government for more than a year. work below On June 1, 2009, Air France Flight 447 mysteriously disappeared over the Atlantic Ocean while traveling from Brazil to France and no one knew what caused it. After a few days, floating debris was found on some bodies quite close to the last one. Known location of the plane, but to determine what caused the crash they first need to find the black box or flight recorders.

They determine the maximum distance the plane could have flown from its last known position. They hoped to find the remains somewhere in this. Then, based on ocean currents and bodies found, experts performed backward drift to produce trajectories that could locate the crash site. They then found 95% confidence in a sone or rectangle that likely contained the location of the wreck after several fruitless searches or investigative consulting was performed. They were tasked with creating a probability map using the information available, this was the initial distribution but given the failed searches they were constantly creating more updated maps and by the way I can't really do justice to all the analysis that was done but shortly after resuming the search.

Using statistical analysis, sonar scanners discovered large portions of debris and a few weeks later flight recorders around the high probability areas were discovered and discovered that the crash was caused by inconsistencies with airspeed measurements, probably because the tubes were clogged by ice. glass the plane ended up stalling and was not recovered and 228 people died on this flight many bodies were recovered and returned to Barry's families however 74 were never found two years after this in 1968 because the submarines mysteriously disappeared all of which occurred in a few months period the french submarine has never been found the israeli submarine was not found for 30 years and the soviet submarine was partially recovered six years later but the US navy USS scorpion was found only a few months after missing and although I'm not going to go into detail about this, it was the same Bayesian search theory that helped locate the remains.

Now, to be fair to acoustics, the expert used sonar technology to help determine where to look here, which was a huge factor, of course, so I'm not saying. Statistics and probability were the only reasons these

mysteries

were

solved

but they played a big role, unfortunately when the submarine sank it took 99 people to kill them all and the cause of what sank the USS Scorpion still It is unknown to this day. In 1857, the SS Central America sank the Turing. A hurricane took 425 of the 578 passengers and 14,000 kilograms of gold to the bottom of the ocean. Its current equivalent is 292 million dollars.

By the way, the location of the gold remains a mystery for over 100 years, but in 1988 a man named Tommy Gregory Thompson used route information and documents from the time of the disaster to reconstruct the ship's trajectory and then created a probability map of the likely locations where the gold would be located, and despite the skeptics, the hard work paid off. when he and his team recovered over 100 million dollars worth of gold, so for anyone who doesn't see the use of statistics, this guy became a millionaire, but then he sold millions of dollars worth of gold before paying his investors and his team.

He was then sued by several of those people, went into hiding in 2012 and was arrested three years later and in November 2018, a jury returned several million dollars to investors and team members. Well, this guy is so stupid. I'll wait, can we recognize everything? the dates I just said here because this gold was lost in 1857 before the American Civil War, it was discovered in 1988, just a few years before I was born and the central case I just mentioned occurred in November 2018, which was four months ago as of posting this video, this is very recent and it seems crazy to me that a storm that occurred over 150 years ago is still having an effect to this day.

Well, let's go back to the next story and move on to a new category in 1787 and 1788. Alexander. Hamilton James Madison and John J anonymously wrote 85 essays known as the Federalist Papers that were intended to persuade people to ratify the Constitution. The specific author of each of these essays was mostlyknown and Hamilton and Madison wrote most; However, 12 were left up for debate because one did not know whether Hamilton or Madison wrote them. Aaron Burr shot Hamilton and died in 1804, taking the secret to the grave. This remained a mystery for about one hundred and seventy-five years until two statisticians decided to give it a try and took papers. in which they knew if Hamilton or Madison was the author and analyzed certain words that one used more than the other, the writing styles were extremely similar but words like while upon and so on were some words of interest since for example Madison rarely used the word pond, while Hamilton actually did so across 49 of Hamilton's articles, they found that the word pond appeared 3.2 4 times per 1,000 words on average after analyzing all known articles by Madison and Hamilton , a comparison could be made that shows a great difference.

When faced with a mystery article, statisticians first assumed 50/50 probabilities that it had been written by any of the authors, then analyzed the frequency of 30 words of interest and updated the probability using Bayesian statistics and given information such as the frequency with which the word allowed would appear. a more up-to-date and accurate probability of who wrote it now, they first ran this test on well-known Hamilton and Madison articles and it predicted the author correctly every time and not just by a little. The least conclusive result occurred when running the numbers on an article. written by Hamilton and still said that there was a 95 percent chance that he was the author, so the tests seemed accurate and when it finally came time to test the 12 mystery items, the results indicated that each of them was written by Madison, the weakest case was the number. 55 which still said that there was a ninety-nine point six percent chance that Madison would write that this is a guarantee, no, even if the word frequencies were accurate, it is possible that the two may have edited each other's articles or something like that, but we are definitely more sure than ever that Madison is the real author of those twelve articles.

So I'm sure many of you recognize that this is a scene from the movie The Imitation Game. about how mathematicians deciphered encrypted German messages during World War II in the film you see this electromechanical machine that tests possible real solutions of an enigma that the Nazis used to encrypt messages due to the large number of possible configurations Alan Turing one of the inventors The machine we just saw needed to drastically reduce the number of tests it had to perform. Turing developed a manual method of doing this in which he guessed strings of letters in an unencrypted message, which he could do because things like the weather report or phrases like Heil Hitler were repetitive and predictable.

He then measured the validity of his guesses using Bayesian methods, updating the probabilities as more information arrived. In fact, while he was doing the analysis, one of his co-workers asked him: Aren't you using essentially which one? he responded: I guess it wasn't exactly Bayes' theorem as we know it, but most people say this analysis involved Beijing and business. Now of course there is a lot more to this and in fact Numberphile has some excellent videos on how code breakers would slip the expected. word along a string of coded letters that help narrow down the possibilities and I'll link it below, but as we now know, the analysis developed by Alan Turing and his co-workers is supposed to have ended the war several years earlier and saved millions of lives.

Again, probability and statistics weren't the only factors, of course, but without them, who knows how things would have turned out, and by the way, Alan Turing wasn't the only person involved in code-breaking during the Second World War. World War, in fact, a mathematician named Bill Tut, who You've probably never heard of what some described as one of the greatest intellectual feats of World War II and which also helped save millions of lives. Unfortunately, I don't have time to go over it, but if you want to learn more than me, I highly recommend heading over to Curiosity Stream, who I would like to thank for sponsoring this video.

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