bayes-rule-posterior-proportional-likelihood-prior
IN premise — summaries/2026/08/24/wiki-Bayesian_inference-chunk-2.md
Created 2026-08-25T02:58:43+00:00
The posterior distribution p(θ|X,α) is proportional to the product of the likelihood p(X|θ,α) and the prior p(θ|α), with the marginal likelihood p(X|α) as the normalizing constant.
Summary
This is the core update rule: when new evidence arrives, your revised estimate of a parameter is shaped by combining what the data says about that parameter with your starting expectations, then scaling the result so all probabilities sum to one. It matters because this is the mechanism by which every subsequent inference in the system folds in fresh data rather than ignoring it.