Bayes' Theorem

The mathematical framework for updating your beliefs in light of new evidence.

From Prior to Posterior

Bayes' Theorem is one of the most important concepts in probability and statistics. It provides a formal way to combine new evidence with existing beliefs (our "priors") to arrive at an updated, more accurate belief (a "posterior").

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}

P(A|B) — The Posterior Probability: The probability of event A being true, given that event B is true. This is what you are trying to calculate.
P(B|A) — The Likelihood: The probability of observing event B, given that event A is true. (This is often the accuracy of your test or signal).
P(A) — The Prior Probability: Your initial belief in the probability of event A before seeing any new evidence.
P(B) — The Marginal Probability: The total probability of observing event B under all circumstances.

Interactive Calculator: The Disease Test

A person tests positive for a rare disease. What's the real probability they have it? Adjust the sliders to see how the base rate and test accuracy dramatically change the outcome. This demonstrates the "base rate fallacy".

Disease Prevalence (Base Rate): 1.00%

How common is the disease in the general population? (P(Disease))

Test Sensitivity (True Positive Rate): 99.0%

If you have the disease, how likely is the test to be positive? (P(Positive|Disease))

Test Specificity (True Negative Rate): 95.0%

If you DON'T have the disease, how likely is the test to be negative? (P(Negative|No Disease))

Given a POSITIVE test result, the actual probability of having the disease is:

0.00%

(Posterior Probability P(Disease|Positive))