Bayes’ Theorem Helps Us Nail Down Probabilities
By: Mark Mancini
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Thomas Bayes was a mathematician, a Presbyterian minister and a defender of Sir Isaac Newton. Today he’s celebrated by statisticians all over the world because of a document released two years after his death.
Bayes died April 7, 1761. As stipulated in the Englishman’s will, a friend and colleague named Richard Price was given his unpublished notes. These included a partial essay about a topic that’s always weighing on our minds: probability.
Impressed and intrigued, Price had an edited version published in 1763 under the title, “An Essay Towards Solving a Problem in the Doctrine of Chances.”
Here, the foundations were laid for what we now call Bayes’ theorem (or “Bayes’ rule”), one of the most well-worn tools in modern statistics.
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“Bayes’ rule is used today in innumerable ways. It gives you a tool for thinking clearly about uncertainty (which decades of cognitive science research have shown we’re not particularly good at),” says Chris Wiggins, a Columbia University associate professor of applied mathematics, in an email interview.
The actual equation is displayed above. In a nutshell, the aim of this formula is to determine what the probability of “A” is given that “B” has already happened or been observed.
To do this, we must take the following steps:
Conditional probability lies at the heart of Bayes’ theorem. The world is an intricate place. When we try to determine the chances that a specific thing will happen, sometimes we need to revise our calculations because of new information, new developments and preexisting data.
Enter the theorem. Whether you’re an astrophysicist studying the age of the universe or a wildlife biologist coming up with population estimates for a rarely seen species, Bayes’ theorem can help you update your outlook and worldview along these conditional lines.
Now that we know some of the basics, let’s take Mr. Bayes’ formula out for a spin.
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Medical professionals know to watch out for false positives.
If a test tells you that something is present when it’s actually absent, that’s a false positive, amigo. The shepherd boy cried wolf, but he didn’t really see one.
True positives are test results that align with reality. They’re what you get when a test exposes a condition that genuinely exists. So, in this scenario, the wolf is real and the shepherd boy was telling the truth.
“Bayes’ theorem can provide insight into the performance of diagnostic tests,” explains Emory University biostatistician Lance Waller in a recent email exchange.
“When we go to the clinic and get tested, we want to know the probability that I am sick given the test is positive.“
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To explain how Thomas Bayes fits into the conversation about false positives in medical tests, Waller’s got a helpful hypothetical. Take another look at our printed formula. See the As and the Bs? Now it’s time to replace those letters with something less abstract.
“Suppose we apply a test that has a 1 in 100 chance of giving a false positive result to a healthy person, and that same test has a 99 in 100 chance of giving a true positive result to a sick person,” says Waller.
“If we apply this test to 100 healthy people and 100 sick people, we would expect 1 false positive and 99 true positives. If we gave the same test to 100,000 healthy people and 100 sick people, we would expect 1,000 false positives and 99 true positives. Most of our positive test results would be false.”
“Bayes’ theorem,” Waller tells us, “defines how the proportions of people tested who are sick and healthy change the probability of a positive test given a healthy person to the probability of a healthy person given a positive test.”
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The theorem gave rise to Bayesian statistics, a wider approach to mathematics and probability.
This school of thought has had its share of critics over the years. Yet history’s shown there’s a place for Bayesian thought. As Wiggins points out, mathematicians now use different computing tools — and look for different kinds of data — than prior generations did.
“Sometimes we use data to describe, scientifically, the world as it is; other times to make predictions of a particular outcome; and other times to prescribe the treatment which will optimize an outcome,” says Wiggins. “It’s no surprise, then, that the norms as to what constitutes a good model or a good modeling practice have also advanced.”
In our computer-driven culture, Bayesian methods are all around us. Consider electronic mail. Some email filters use Bayes’ Theorem to calculate the odds that an individual message is unwanted spam given its word choices.
Or look at how the U.S. Coast Guard made waves in 2014 when one of its computer programs led to the rescue of a fisherman who’d gone missing. As you may have guessed, that program got the job done with Bayes’ theorem.
“Doing ‘a Bayesian analysis’ doesn’t always mean a better analysis,” observes Waller. “[But] since Bayesian methods require detailed mathematical definitions, a Bayesian analysis often provides the flexibility to adapt to a broader range of applications than traditional approaches.”
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Like Thomas Bayes, Richard Price was a practicing minister, and a well-connected one at that. He personally met with the likes of Benjamin Franklin, Thomas Jefferson, John Adams and Thomas Paine. Also, Mary Wollstonecraft — a groundbreaking feminist and the mother of “Frankenstein” creator Mary Wollstonecraft Shelley — was one of his mentees.
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