Bayes' Theorem

A precise treatment of Bayes' theorem, base rates, partitions, diagnostic tests, and interpretation.

Events are subsets of a sample space, and conditional probability changes the reference set. A sample-space rectangle contains two overlapping event regions and a small two-stage probability tree. The overlap marks joint occurrence, while branch labels show how later probabilities can depend on an earlier outcome.

Precise definition

For events $A,B$ with $P(B)>0$, Bayes' theorem is $P(A\mid B)=P(B\mid A)P(A)/P(B)$. If $A_1,\ldots,A_k$ partition the sample space, then $P(A_i\mid B)=P(B\mid A_i)P(A_i)/\sum_jP(B\mid A_j)P(A_j)$.

Notation and mathematical language

$P(A)$ is a prior probability, $P(B\mid A)$ a likelihood for event data, $P(A\mid B)$ a posterior, and $P(B)$ the evidence or normalizing probability. In diagnostic testing, sensitivity is $P(+\mid D)$ and specificity is $P(-\mid D^c)$; neither is the positive predictive value $P(D\mid+)$.

Conceptual picture

Bayes reverses a conditional probability by reweighting prior possibilities according to how compatible each is with the evidence. A probability tree or frequency table makes the denominator visible and prevents ignoring the many false positives possible when a condition is rare.

Fully worked example

Interpretation and application

Bayesian updating supports diagnosis, quality control, classification, and scientific inference. The posterior is conditional on the prior, likelihood model, data quality, and population; it is not an assumption-free statement of certainty or causal proof.