Bayes Theorem is the handiwork of an 18th-century minister and statistician named Thomas Bayes, first released in a paper Bayes wrote entitled "An Essay Towards Solving a Problem in the Doctrine of Chances."
There is some debate among economists on whether credit should also be given to another economist, Richard Price, who edited and corrected Bayes' paper in 1763, after his death in 1761.
Another wrinkle on Bayes Theorem stems from a 1774 paper by French mathematician Pierre-Simon Laplace, who was apparently unaware of Bayes original thesis. Laplace formalized the Bayes concept and is now viewed by economists as the individual who should share the credit for developing what's known as the "Bayesian probability."
Historically, the Bayes Theorem has led to significant breakthroughs.
The theorem was used to crack to infamous Nazi Enigma code in World War II. Alan Turing, a British mathematician, used Bayes Theorem to assess the translations culled from the Enigma encryption machine used to crack the German messaging code. Applying probability models, Turing and his staff were able to break down the almost infinite number of possible translations based on the messages that were most likely to be translatable, and ultimately crack the German Enigma code.
Interestingly, there is no known portrait of Bayes in existence, and nobody really knows what he looked like (there is a sketched image floating around the internet, but it's never been officially confirmed as the "real Bayes.")
But exist he did, and his theory on conditional probability remains widely praised today by mathematicians, businesses and even poker players, all over the world.
What Is Bayes Theorem?
Bayes Theorem is a mathematic model, based in statistics and probability, that aims to calculate the probability of one scenario based on its relationship with another scenario.
Largely defined, conditional probability is the likelihood of an event transpiring, due to its association with another scenario. For instance, your likelihood of playing a round of golf within four hours depends on the time of previous rounds played, the time of day, the course you're playing, how many other people you're golfing with, and where and how often you hit your golf ball.
Or, consider this scenario: your son is coming home from college for a long weekend and tells that he's bringing a friend with him. All you have to go is that there is a 50% chance the friend is female. Then your son texts you and says "Oh, you remember my friend with the long blond hair." With this additional information, under Bayes Theorem, the probability is more likely the friend is female.
In Bayes' line of thinking, events are actually tests that indicate the probability of something happening. Bayes saw tests as a way to measure the probability of an event occurring, even though tests really are not events, and results from tests are invariably flawed.
Using testing models and equations, Bayes plugged in previous information plus data formulations to predict multiple probabilities in a given situation.
Bayes Theorem Formula
The most popular and pervasive formula used for Bayes' Theorem is as follows:
P(A ∣ B) = P(B ∣ A)P(A) / P(B)
Broken down, A and B are two events and P(B) ≠ 0
P(A ∣ B) is the conditional probability of event A occurring given that B is true.
P(B ∣ A) is the conditional probability of event B occurring given that A is true.
P(A) and P(B) are the probabilities of A and B occurring independently of one another (the marginal probability).
Here, probability is the basis of the Bayes Theorem. More specifically, what is the probability that a scenario will take place, given existing knowledge about similar scenarios?
Consider this example, which is commonly used when citing Bayes Theorem, using the mathematician's testing model:
- Approximately 1% of women aged 40-50 have breast cancer.
- A woman with breast cancer has a 90% chance of a positive test from a mammogram.
- A woman without cancer has a 10% chance of a false positive result.
Applying, Bayes' Theorem, can identify the probability that a woman is suffering from breast cancer, even from the application of just one breast cancer test.
Or, consider an advertising campaign that is selling products and service from a major investment firm. Existing data from previous campaigns estimate that 4% of those high-net-worth individuals targeted by the ad campaign will respond favorably and buy shares of mutual funds from the Wall Street giant.
Yet once you factor in other probabilities, like the ad recipient's age, his or her retirement needs, his or her total portfolio assets, and appetite for risk, the likelihood of that individual buying shares of the investment firm's mutual funds grows beyond that original 4% figure.
Quantity is almost as important as quality when applying those conditional variables that are so important to Bayes Theorem.
Basically, the more an individual can compare conditional probabilities to find likely facts using a mathematic equation, the closer to the truth that individual gets. Or, as the theory itself presupposes, the more variables that are in play, and the more certain an individual becomes about those variables, the more certain an accurate conclusion can be drawn, using conditional probabilities.
That's why Bayes Theorem is big on the number of tests applied and the more data used in moving from assumptions to conclusions.
How to Use Bayes Theorem for Business and Finance
In finance and business circles, corporate financial specialists have been applying Bayes' Theorem for centuries.
Consider these applications:
- In evaluating interest rates. Companies rely on interest rates for multiple reasons - borrowing money, investing in the fixed income market, and trading in currencies overseas. Any unexpected shifts in interest rate values can hit a company hard in the pocketbook, and can negatively impact profits and revenues. With Bayes Theorem and estimated probabilities, companies can better evaluate systematic changes in interest rates, and steer their financial resources to take maximum advantage.
- With net income. Businesses are keen to be on top of their net income streams, or the profit a business earns after subtracting expenses out of the equation. Net income is highly vulnerable to external events, like legal proceedings, weather, the cost of necessary equipment and materials, and geopolitical events, for starters. Plugging probability scenarios into the net income equation when these scenarios arise gives financial decision makers a stronger platform when managing resources and making critical decisions.
- For extending credit. Under the Bayes Theorem conditional probability model, financial companies can make better decisions and better evaluate the risk of lending cash to unfamiliar or even existing borrowers. For example, an existing client may have had a good previous track record of repaying loans, but lately the client has been slow in playing This additional information, based on probability theory, can lead the company to treat the slow payment history as a red flag, and either hike interest rates on the loan or reject it altogether.
The Takeaway on Bayes Theorem
Bayes Theorem has come back into vogue as data analysis has grown in stature in mathematics and economic disciplines. Everything from climate change to cancer screenings has gone under the Bayes microscope, and data scientists say the possibilities are endless in matching data technology with probability analysis.
Real world applications abound.
Insurance companies are already using the probability theorem to assess the risk of flooding in coastal areas, and developers of driverless vehicles see Bayes Theorem as a way to improve decision making using various probabilities on roadways.
The theorem is so prevalent and widely used, that scientists openly wonder, as physicist John Mather has stated, that Bayesian concepts, coupled with the advancements in machine learning and artificial intelligence, may make human-based thinking "obsolete."
Bayes may not have grasped the long-term ramifications of his now legendary theorem. But in all probability, generations of probability theorists will continue to do so for centuries to come.