Mathematics
Bayesian Probability
Quick fact
Bayes' theorem, the foundation of Bayesian probability, was discovered by Reverend Thomas Bayes in the 18th century but only gained widespread use with modern computing.
Why this is interesting
You've probably made a snap judgment about someone, then changed your mind after learning more. But how can you systematically update your beliefs when new information arrives?
Read the full explanation
Understanding Bayesian Probability
Imagine you have a hunch that it might rain tomorrow. That's your initial belief, or prior probability. When you check the weather forecast and see dark clouds, you update your belief. The new, revised belief is the posterior probability. Bayesian probability is a mathematical framework for exactly this process: starting with a prior belief, combining it with the likelihood of new evidence, and producing an updated belief. The key formula is Bayes' theorem, which calculates the posterior probability as proportional to the prior times the likelihood. This is not about frequencies of events but about degrees of belief that can be assigned probabilities.
A deeper explanation
Bayesian probability rests on Bayes' theorem: P(A|B) = P(B|A) P(A) / P(B). Here, P(A) is your prior belief about hypothesis A, P(B|A) is the likelihood of observing evidence B if A is true, and P(B) is the total probability of B. The result, P(A|B), is the posterior belief after considering B. This framework's power lies in its iterative nature: today's posterior becomes tomorrow's prior, allowing continuous learning. It matters because it provides a principled way to incorporate prior knowledge, handle small data sets, and quantify uncertainty. Applications range from spam filtering and medical diagnosis to A/B testing and artificial intelligence, where Bayesian methods help models update from data without overfitting.