Existing in Mathlib · Probability

Bayes' Theorem

For two measurable events under a finite measure, the probability of t conditioned on s can be recovered from the reverse conditional probability, weighted by the measure of t and scaled by the inverse measure of s.

conditional probability
finite measures
measurable events
measure restriction
intersection symmetry
Bayes' theorem

Exact theorem

Exact Mathlib statement

theorem ProbabilityTheory.cond_eq_inv_mul_cond_mul {Ω : Type*} {m : MeasurableSpace Ω} {s t : Set Ω} (hms : MeasurableSet s) (hmt : MeasurableSet t) (μ : Measure Ω) [IsFiniteMeasure μ] : μ[t | s] = (μ s)⁻¹ * μ[s | t] * μ t

The theorem at a glance

Bayes' theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory Bayes theorem poster places the exact conditional-measure equation above two overlapping event contours, a shared gold intersection-mass core, reciprocal conditioning views, three checked source movements, and an exact-scope footer. — Both conditioning directions pass through the same intersection mass, yielding Mathlib's finite-measure form of Bayes' theorem.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Scientific figure illustrating: Overlapping blue and terracotta event chambers share a gold kernel, with opposing viewing paths and a balanced token comparison below. — Two conditioning directions isolate the same shared kernel and balance against their event-mass tokens.

Why it matters

A mathematical landmark

Bayes' theorem is one of the central identities of probability. Mathlib's selected declaration is especially instructive because it presents the identity at the level of measurable sets and finite measures: both conditioning directions reduce to the measure of the same intersection.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's finite-measure, measurable-set form of Bayes' theorem. For measurable s and t, μ[t | s] equals (μ s)⁻¹ times μ[s | t] times μ t. The selected declaration does not add a nonzero condition on μ s or μ t, require μ to be a probability measure, assert independence or causation, or specialize to a finite or discrete sample space.

The selected declaration does not say that s and t are independent or that either event causes the other.
It assumes only IsFiniteMeasure μ; it does not require μ to have total mass one or the sample space to be finite or discrete.
There is no explicit hypothesis μ s ≠ 0 or μ t ≠ 0 in this declaration; the statement uses Mathlib's ENNReal inverse convention.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: ProbabilityTheory.cond_eq_inv_mul_cond_mul in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.bayes-theorem.v001
Source: Open the pinned upstream reference