{ "artifactId": "artifact.library.mathlib.bayes-theorem.v001", "candidateOnly": false, "category": "Probability", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Conditional probability", "id": "library.mathlib.bayes-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/bayes-theorem/", "entryData": "/data/library-theorems/bayes-theorem.json", "evidence": "/proofs/artifact.library.mathlib.bayes-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.bayes-theorem.v001.evidence.json", "source": "/sources/upstream/bayes-theorem/" }, "landmark": { "completeFormalType": "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", "concepts": [ "conditional probability", "finite measures", "measurable events", "measure restriction", "intersection symmetry", "Bayes' theorem" ], "plainLanguage": "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.", "poster": { "alt": "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.", "byteSize": 3010777, "caption": "Both conditioning directions pass through the same intersection mass, yielding Mathlib's finite-measure form of Bayes' theorem.", "description": "The poster foregrounds μ[t | s] = (μ s)⁻¹ · μ[s | t] · μ t for measurable s and t under a finite measure. Cobalt s and terracotta t contours overlap around one gold measure-mass core; paired measure-weight chambers emphasize the two conditioning directions without identifying measure with geometric area. The lower route converts reverse conditional mass to μ(t ∩ s), commutes the intersection, and unfolds conditioning on s.", "derivatives": [ { "byteSize": 139776, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:0b4dc99e772fd00c1089614ce0442675e3085a317d79ee3b55d1acb43dfb237d", "width": 640 }, { "byteSize": 333036, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:e559706014cf8d74d0d8f6196034c1bc503c097f30046210763333c6cd14dabc", "width": 1024 } ], "generatedAt": "2026-07-27T10:26:36-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-poster-v1.png", "sha256": "sha256:2755865c9ecf9079764d46609e118bdaca4847ca78d17b7778908bf2d29828a1", "title": "Bayes' theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "PROBABILITY · CONDITIONAL MEASURE\nBAYES' THEOREM\ns, t measurable · μ finite\nμ[t | s] = (μ s)⁻¹ · μ[s | t] · μ t\nONE INTERSECTION, TWO CONDITIONING DIRECTIONS\nHOW THE CHECKED ROUTE MOVES\n1 · Convert μ[s | t] · μ t to μ(t ∩ s)\n2 · Commute the intersection to s ∩ t\n3 · Unfold conditioning on s\nEXACT SCOPE\nFinite measure. No independence or causation claim. No nonzero hypothesis on μ s." }, "visual": { "alt": "Two large engraved event fields narrow toward one shared gold intersection-mass kernel, while opposing conditioning arcs and cobalt, terracotta, and central measure weights meet in a balanced lower mechanism.", "byteSize": 3183067, "caption": "Conditioning in either direction isolates the same intersection mass; rescaling by the conditioning event relates the two views.", "description": "A dark forest-green engraved field uses asymmetric cobalt and terracotta event chambers with varied measure tokens, a single shared gold intersection kernel, opposing curved viewing arcs, and a lower three-weight balance. Shape, hatch, and token size distinguish the two event masses and their shared core without treating geometric area as the measure.", "derivatives": [ { "byteSize": 52920, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:5d912b71ed033f24ac338ecf2f87c131637c4beb30758a2f2e798e59602d246a", "width": 640 }, { "byteSize": 215360, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:2cc1d75224e48aa06c55155629a8bcc03d67764345795691d97b461bf623f93c", "width": 1200 } ], "generatedAt": "2026-07-27T10:26:36-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/bayes-theorem/theorem-schematic-v1.png", "sha256": "sha256:9133d5f400909f28332314819c5d108d0257883182e120c9f89f22546805b9d7", "title": "One intersection, two conditioning directions", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.bayes-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "The two conditional directions are related through the same measurable intersection mass.", "targetId": "target.library.mathlib.bayes-theorem.v001", "title": "Bayes' Theorem", "upstreamOrigin": { "declarationName": "ProbabilityTheory.cond_eq_inv_mul_cond_mul", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Probability/ConditionalProbability.lean#L273", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:c3e6b20df733dd6e50267cd5c6b84ad0d0df362e2ea19de6e8efa9b2e18c90f3", "sourceByteLength": 13118, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Probability/ConditionalProbability.lean", "sourceLine": 273, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }