Bayes, one square at a time

Observation: Positive

Round 1

H 1%¬H 99%

Width is belief. Height selects evidence. Area combines them.

Before the observation

Start with what you believe.

H and this outcome

¬H and this outcome

Total retained, P(outcome)

The same update, in numbers

P(H | E)=P(H) P(E | H)P(H) P(E | H) + P(¬H) P(E | ¬H)

15.38%Posterior probability of H

See this as expected counts

Play the transformation, then carry the posterior into the next observation.

One observation at a time.

The posterior from one observation is the prior for the next.

Update log · 0 observations committed

Round	Observed	Prior P(H)	P(outcome \| H)	P(outcome \| ¬H)	BF	Posterior P(H)
Your committed updates will appear here.

Why the picture works—and what sequential updating assumes

1. Width × height = joint probability

The full square represents all possibilities and has area 1. Its blue width is P(H); the remaining width is P(¬H). Within each region, the likelihood is the fraction consistent with the observed outcome.

Blue area = P(H) × P(E | H)
Orange area = P(¬H) × P(E | ¬H)

The posterior is the blue area divided by the total retained area. For the complementary outcome, replace each likelihood by its complement.

2. Make the retained area a new whole

Rearranging changes the shapes while preserving each retained area. The two pieces become a strip with shared height P(E). Their widths now equal the posterior probabilities.

Normalization stretches both pieces vertically by the same factor, 1/P(E), until the strip fills a unit square. Every area is divided by the same total, so the relative blue and orange proportions stay unchanged.

Blue and orange always identify the same hypotheses. The hatched region shows possibilities excluded by the observation.

3. Use each observation once

After an update, the posterior becomes the starting probability for the next observation. The general likelihood for the next step is conditional on both the hypothesis and all earlier observations.

P(H | E₁,…,Eₜ) ∝
P(H | E₁,…,Eₜ₋₁) P(Eₜ | H,E₁,…,Eₜ₋₁)

Reusing the same likelihoods assumes observations are independent given the hypothesis. If they are dependent, enter the appropriate conditional likelihoods for the next observation. Replaying the animation never adds evidence.

The evidence multiplier: BF = P(outcome | H) / P(outcome | ¬H). Posterior odds = prior odds × BF. Equal nonzero likelihoods leave beliefs unchanged. An outcome with zero total probability cannot be conditioned on. A hypothesis assigned exactly zero prior probability cannot gain probability through ordinary conditioning.