Bayes theorem

Mathematical statement

Use exact rational masses in one complete finite atomic law. Both the two-event and partition forms require positive conditioning denominators. Partition hypotheses must be disjoint and exhaustive, and the evidence mass obtained by total probability must be positive. Bayes conditioning is not causation.

Representative formula (exact TeX)

B(E)

Assumptions

• The complete finite sample space has exact nonnegative rational masses summing to 1.
• Every denominator actually used for a conditional probability is strictly positive.

Symbol scope

• B — Bayes update
• E — evidence event

Intuition, definition, and misconceptions

The two-event formulas share the same intersection.

The partition form normalizes by the law of total probability.

Reversal needs the prior and likelihood and does not imply causation.

Do not ignore zero denominators, omit the prior or identify P(E|H) directly with P(H|E).

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Conditional probability in finite models — The definition requires a positive denominator.
• Event — Hypothesis and evidence are events.
• Probability on finite sample spaces — Joint, prior and evidence masses come from the law.

Verification boundary

Only complete finite laws are handled; no causal inference is asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add parameter models and posterior distributions.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.