---
title: "Bayes' rule"
description: "Updates a prior belief about a parameter after observing evidence by reweighting parameter values according to how well they predict that evidence."
canonical_url: "https://fanout.sh/labs/math-decoder/formula/bayes-rule-posterior-update"
md_url: "https://fanout.sh/labs/math-decoder/formula/bayes-rule-posterior-update.md"
last_updated: "2026-08-09"
access: "public"
---

# Bayes' rule

Updates a prior belief about a parameter after observing evidence by reweighting parameter values according to how well they predict that evidence.

## Public overview

The probability of theta given x equals the probability of x given theta times the prior probability of theta, divided by the probability of x.

Start with what you believed before x, reward parameter values that predict x well, and renormalize the updated beliefs.

Conditioning is the conceptual verb: Bayes' rule reverses which variable is treated as known by reweighting and normalizing.

This is an exact identity. In large models the denominator or posterior may be computationally intractable, motivating variational or sampling approximations.

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