Buzz
Bayes' theorem offers a framework to make sense of uncertainty and how probability influences various outcomes. MytourThomas Bayes was not only a mathematician and Presbyterian minister, but also a strong supporter of Sir Isaac Newton. Today, statisticians worldwide honor him, especially for a paper published two years after his passing.
Bayes passed away on April 7, 1761. As per his will, his friend and colleague, Richard Price, received his unpublished manuscripts, which contained a partial essay on the concept of probability—an idea that continues to captivate us today.
Fascinated and inspired, Price had an edited version of the work published in 1763, titled "An Essay Towards Solving a Problem in the Doctrine of Chances."
This work laid the groundwork for what we now refer to as Bayes' theorem (or "Bayes' rule"), one of the fundamental tools in contemporary statistics.
Odds and Ends
"Bayes' rule is applied in countless ways today. It provides a framework for understanding uncertainty (something we’re not particularly good at, according to decades of cognitive science research)," says Chris Wiggins, associate professor of applied mathematics at Columbia University, in an email interview.
The actual equation is shown above. Essentially, this formula is designed to determine the probability of "A" occurring given that "B" has already taken place or been observed.
To achieve this, we need to follow these steps:
- Change perspective: Determine the probability of "B" assuming "A" has already occurred or been observed.
- Multiply that by the total probability of "A."
- Divide the result by the overall probability of "B."
Conditional probability is the cornerstone of Bayes' theorem. The world is a complex system. In trying to calculate the likelihood of a specific event, we may need to adjust our predictions based on new data, changes, or existing information.
Enter the theorem. Whether you're an astrophysicist examining the age of the universe or a wildlife biologist estimating the population of a rarely spotted species, Bayes' theorem helps you refine your predictions by considering these conditional factors.
Now that we've covered some of the basics, let's put Mr. Bayes' formula into action and see it in practice.
True or False?
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 doctor for a test, we want to know the likelihood that I am sick if the test comes back positive."
"Paging Doctor Bayes!"
To illustrate how Thomas Bayes relates to the issue of false positives in medical tests, Waller offers a useful example. Take a look again at the printed formula. See the As and Bs? Now it’s time to substitute those abstract symbols with something more tangible.
"Imagine we use a test that has a 1 in 100 chance of producing a false positive for a healthy person, and a 99 in 100 chance of accurately detecting a sick person," explains Waller.
"If we tested 100 healthy people and 100 sick people, we would expect 1 false positive and 99 true positives. However, if we tested 100,000 healthy individuals and 100 sick ones, we'd anticipate 1,000 false positives and 99 true positives. Most of our positive test results would actually be false."
"Bayes' theorem," explains Waller, "shows how the ratios of sick and healthy people tested can shift the probability of a positive test for a healthy person into the probability of a healthy person having a positive test."
Outside the Laboratory
This concept led to the development of Bayesian statistics, a broader field of mathematics and probability theory.
While Bayesian thought has faced its share of critics, history reveals its lasting relevance. As Wiggins observes, modern mathematicians use different computational tools and seek new kinds of data compared to their predecessors.
"At times, we use data to describe the world scientifically; at other times, we predict a specific outcome; and occasionally, we prescribe treatments to optimize results," says Wiggins. "It’s no surprise, then, that the standards for what constitutes a strong model or good modeling practice have evolved."
In today's tech-driven world, Bayesian techniques are everywhere. Take email, for example. Some spam filters apply Bayes' Theorem to determine the likelihood that a message is unwanted spam based on its word choices.
Or consider how the U.S. Coast Guard made headlines in 2014 when one of its computer systems helped rescue a missing fisherman. As you might have guessed, that program used Bayes' theorem to accomplish the task.
"Performing 'a Bayesian analysis' doesn’t always guarantee a better outcome," notes Waller. "However, because Bayesian methods require precise mathematical definitions, they often offer more adaptability to a wider variety of situations than traditional methods."
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Like Thomas Bayes, Richard Price was also a minister, and a well-connected one at that. He had personal meetings with prominent figures like Benjamin Franklin, Thomas Jefferson, John Adams, and Thomas Paine. Additionally, Mary Wollstonecraft — the pioneering feminist and mother of "Frankenstein" author Mary Wollstonecraft Shelley — was one of his mentees.
