Bayes' Theorem: Geometric Area Proportions, Likelihood Ratios & Posterior Updating
Replace rote formula memorization with 3Blue1Brown's geometric probability-square restriction view.
Bayes' Theorem updates the prior probability P(H) of a hypothesis H given observed evidence E by multiplying the prior by the likelihood P(E|H) and normalizing by the marginal probability of the evidence P(E).
Draw a 1x1 unit square where horizontal width is P(H) vs P(~H) and vertical height is the likelihood P(E|H) vs P(E|~H). Observing evidence E restricts the universe exclusively to the shaded 'E occurred' rectangles; the posterior probability P(H|E) is simply the area of the true-positive rectangle divided by the total shaded area.
Bayes' Theorem & Posterior Inference (BayesGeometricSquareScene)
Step-by-Step Symbolic LaTeX Derivation
Follow each algebraic transformation from first principles to computational implementation.
Log-Odds Form of Bayes' Rule
In log-odds space, Bayesian updating is pure addition: Posterior Log-Odds = Prior Log-Odds + Log Bayes Factor.
A rare condition affects 1 in 1,000 people. A test has 99% sensitivity and 1% false-positive rate. If a random person tests positive, what is P(Disease | Positive)?
What is Maximum A Posteriori (MAP) vs. Maximum Likelihood Estimation (MLE)?
MLE maximizes P(D | theta) alone; MAP maximizes P(D | theta) * P(theta), where a Gaussian prior P(theta) is mathematically equivalent to L2 weight decay.