Math 2.0

Statistics Bayesian inference Concept

Posterior probability

Builds on Bayes' theoremPrior probabilityConditional probability

Summary

The posterior probability of a hypothesis HH is its conditional probability P(H∣D)P(H \mid D) given the observed data DD. By Bayes' theorem, P(H∣D)=P(D∣H) P(H)/P(D)P(H \mid D) = P(D \mid H) \, P(H) / P(D). So the posterior is proportional to the likelihood P(D∣H)P(D \mid H) times the prior probability P(H)P(H). For an unknown parameter θ\theta with prior density π(θ)\pi(\theta) and likelihood L(θ∣x)L(\theta \mid x), the posterior density is π(θ∣x)=L(θ∣x) π(θ)/∫L(t∣x) π(t) dt\pi(\theta \mid x) = L(\theta \mid x) \, \pi(\theta) / \int L(t \mid x) \, \pi(t) \, dt.

Remarks

The posterior of one experiment can be the prior of the next experiment. So Bayesian updates can be done one observation at a time. A credible interval is an interval that has a given posterior probability, for example 95%95\%.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  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
  40. Prior probability concept
  41. Posterior probability concept

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Posterior_probability, Definition in the distributional case
About this page
Tag0KJ7ZS8AJA
URIm2://0KJ7ZS8AJA
Aliasesposterior, posterior-distribution
Kindconcept
WritingW0 stub
MSC62F15
UsesBayes' theorem, Prior probability, Conditional probability
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