Updated 12 September 2026

Bayes’ theorem

Bayes’ theorem tells you how to update a belief when new evidence arrives. You start with a prior probability $P(H)$ that some hypothesis H is true. You observe evidence E. The posterior is:

The denominator is the total probability of seeing the evidence at all:

Reading the area diagram

The rectangle represents the entire sample space. Width encodes the prior split: left = P(H), right = P(¬H). Height encodes the likelihoods: shaded top fraction of each column = probability of evidence E given that column’s hypothesis.

The posterior P(H|E) is the proportion of the total shaded area that falls in the left (H) column. Drag the sliders and watch the ratio shift.

What to notice

• Strong evidence isn’t enough. Set P(H) very low (rare hypothesis) and P(E|¬H) only a little lower than P(E|H). Even 95% sensitivity gives a low posterior when prevalence is 1%. This is base rate neglect.
• False positives matter when the hypothesis is rare. Reducing P(E|¬H) can substantially raise the posterior because the large ¬H population may contribute many of the positive results. The size of the change depends on all three input probabilities.
• The prior can be overwhelmed. With strong enough evidence (high P(E|H), low P(E|¬H)), the posterior can approach 1 for a fixed positive prior as the likelihood ratio grows. A zero prior stays zero when the evidence has positive marginal probability. Convergence also depends on the model and how the evidence is generated.

Make the denominator visible

At the default prior 0.3, likelihood P(E|H)=0.8 and alternative likelihood P(E|not H)=0.2, a hypothetical 10,000 cases contain 3,000 with H and 7,000 without it. Expected E counts are 2,400 and 1,400. Conditioning on E leaves 3,800 cases, of which 2,400 have H: 63.1579%. The likelihood 80% answers a different conditional question from this posterior.

Keep the likelihoods fixed and reduce the prior to 0.01. Expected E counts become 80 and 1,980, giving posterior 3.8835%. These synthetic numbers illustrate sensitivity to the prior; they are not estimates for a real screening system.

Sequential evidence needs a dependence model

Posterior odds equal prior odds times the likelihood ratio. With the default likelihoods, each positive observation multiplies odds by four and each negative by one quarter. Two positives give odds (3/7)×16=48/7 and posterior 87.2727%. One positive and one negative cancel their likelihood ratios, returning the posterior to 30%.

This multiplication requires observations that are independent conditional on the same H, with the specified likelihoods. Two copies of the same result are perfectly dependent, so counting both as independent would exaggerate the evidence. The order of observations does not matter in this particular fixed-likelihood model; other models can require additional state.

Reference

ProbabilityCourse: Bayes’ rule derives the reversed conditional probability and the total-probability denominator. The repeated-evidence control applies that identity under the additional conditional-independence assumption stated above.