Math · Probability

Bayes’ Theorem

Bayes’ Theorem updates a probability after observing evidence: start with a prior, filter for the evidence, then compare what remains.

Concept

Before observing evidence, A has a base rate. Evidence B may occur when A is true, but it can also occur when A is false. Once B is observed, cases without B no longer matter; among the remaining B cases, find the fraction that belong to A.

Prior

P (A)

Evidence

P (B | A)

P (B | \neg A)

Posterior

P (A | B)

Equation

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

$P (B) = P (B | A) P (A) + P (B | \neg A) P (\neg A)$

$P (A | B) = \frac{P (B | A) P (A)}{P (B | A) P (A) + P (B | \neg A) P (\neg A)}$

Focus or select a term to trace it in the population.

Manipulate

Adjust the prior and the two evidence rates. The labels describe the relationship before you need to read the notation.

Prior probability

P (A)

Evidence if A is true

P (B | A)

Evidence if A is false

P (B | \neg A)

A without B¬A without BB among AB among ¬A (dashed)

Live calculation

Natural frequencies

This is why the denominator matters: $P (B | A)$ asks about B within A, while $P (A | B)$ asks about A within B. Reversing the condition changes the sample space.

A spam-filtering signal

Let A mean “a message is spam” and B mean “the message contains a particular signal.” A prior spam rate and the frequency of that signal among spam and non-spam messages combine to update $P (spam | signal)$ . Real filters use many signals and richer models; this is one clear slice of the reasoning.

Computing, data & machine learning

Bayes supports Bayesian inference, probabilistic classification, Naive Bayes methods, spam filtering, uncertainty-aware systems, and model updating. It provides a disciplined way to combine a base rate with new evidence.