In this post, we’ll unpack all you need to know about Bayes Theorem, defining exactly what it is, how it works, what the formula is, how to leverage it and more.
What Is Bayes Theorem?
Bayes Theorem (also known as Bayes’ Rule) is a mathematical framework that allows us to more accurately determine the probability of a hypothesis.
How Does It Work?
Bayes Theorem operates on conditional probability by focusing on how the likelihood of one event can change based on the occurrence of another. It works by updating the existing probability of a hypothesis based on new information.
What Is The Formula?
The formula for Bayes Theorem is: P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B): The probability of event A happening, given that event B has already happened.
- P(B|A): The probability of event B happening, given that event A has already happened.
- P(A): The prior probability of event A happening.
- P(B): The prior probability of event B happening
Bayes Theorem Example
Imagine a rare disease that affects only 1 in 1,000 people (0.1%). The test for this disease has the following characteristics:
- True Positive Rate: 99% — If you have the disease, the test will correctly detect it 99% of the time.
- False Positive Rate: 5% — If you don’t have the disease, the test will incorrectly indicate that you do 5% of the time.
Knowing this, if you test positive, what’s the actual chance that you have the disease?
Given the following probabilities:
- Probability of having the disease: P(Disease) = 0.001
- Probability of testing positive if you do have the disease (true positive): P(Positive | Disease) = 0.99
- Probability of testing positive if you don’t have the disease (false positive): P(Positive | No Disease) = 0.05
- Step 1: Calculate the probability of testing positive P(Positive)
P(Positive) = (0.99 * 0.001) + (0.05 * 0.99) = 0.05094
- Step 2: Calculate the probability of having the disease given a positive test result P(Disease | Positive)
Using Bayes’ Theorem: P(Disease | Positive) = (0.99 * 0.001) / 0.05094 = 0.0194
Therefore, despite a positive test result, the probability of actually having the disease is roughly 1.94%.
How To Leverage It
The key to leveraging Bayes Theorem is to continuously update the probability of a hypothesis occurring by combining existing information with new information as it becomes available.
In doing so, you improve the accuracy of your predictions by refining your understanding with each new piece of information.
Summary (TL;DR)
Bayes Theorem is a mathematical formula that allows us to more accurately determine the probability of a specific outcome.
It operates on conditional probability by focusing on how the likelihood of one event can change based on the occurrence of another.
The key to leveraging it is to continuously update the probability of a hypothesis occurring by combining existing information with new information as it becomes available.
