Math 2.0

Statistics Theorem

Bayes' theorem

Builds on Conditional probabilityLaw of total probability

Statement

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space and let A,B∈ℱA, B \in \mathcal{F} with P(A)>0P(A) > 0 and P(B)>0P(B) > 0. Then P(A∣B)=P(B∣A) P(A)P(B).P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}.

Remarks

Let A1,A2,…A_1, A_2, \dots be a finite or countable partition of Ω\Omega into events of positive probability. With the law of total probability, P(Aj∣B)=P(B∣Aj) P(Aj)∑iP(B∣Ai) P(Ai)P(A_j \mid B) = \dfrac{P(B \mid A_j) \, P(A_j)}{\sum_i P(B \mid A_i) \, P(A_i)}. In Bayesian language, P(A)P(A) is the prior probability, P(B∣A)P(B \mid A) is the likelihood and P(A∣B)P(A \mid B) is the posterior probability. The proof: both P(A∣B) P(B)P(A \mid B) \, P(B) and P(B∣A) P(A)P(B \mid A) \, P(A) are equal to P(A∩B)P(A \cap B).

History

The theorem is named after Thomas Bayes. Pierre-Simon Laplace found it independently and developed it further.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementUniversal set, definition, no evidence levelUniversal setdefinitionPartition of a set, definition, no evidence levelPartition of a setdefinitionPower set, definition, no evidence levelPower setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionConditional probability, definition, no evidence levelConditional probabili…definitionLaw of total probability, statement, no evidence levelLaw of total probabil…statementBayes' theorem, statement, no evidence levelBayes' theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Empty set object
  15. Family of sets definition
  16. Intersection definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Disjoint sets definition
  23. Subset definition
  24. Union of a family of sets definition
  25. The rationals form an ordered field statement
  26. Universal set definition
  27. Partition of a set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Interval definition
  32. Sigma-algebra definition
  33. Event definition
  34. Probability axioms concept
  35. Probability measure definition
  36. Probability space definition
  37. Conditional probability definition
  38. Law of total probability statement
  39. Bayes' theorem statement

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bayes'_theorem, Statement of theorem
About this page
Tag0NM6E9VW04
URIm2://0NM6E9VW04
Aliasesbayes-rule, bayes-law, bayes-formula
Kindstatement, theorem
WritingW1 stated
MSC62F15
StatusThis copy of the site has no build data. Run m2 status 0NM6E9VW04.
UsesConditional probability, Law of total probability
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