EQ-BAYES · Probability
Bayes' Theorem
-P_A*P_B_given_A + P_A_given_B*P_B = 0
Variables
variable
P_A
prior probability of hypothesis A
- Object
- bayesian_hypothesis
- Property
- PriorProbability
- Context
- bayesian
variable
P_A_given_B
posterior probability of hypothesis A given evidence B
- Object
- bayesian_hypothesis
- Property
- PosteriorProbability
- Context
- bayesian
variable
P_B
marginal probability of evidence B (normalising constant)
- Object
- bayesian_hypothesis
- Property
- Probability
- Context
- bayesian
- Constraint
- marginal_evidence
variable
P_B_given_A
likelihood: probability of evidence B given hypothesis A
- Object
- bayesian_hypothesis
- Property
- ConditionalProbability
- Context
- bayesian
- Constraint
- likelihood
Axioms
algebraic commutative_factors deterministic dimensionless linear
Assumptions
- P(B) > 0 (the evidence has nonzero prior probability, so conditioning is well-defined)
- All probabilities lie in [0,1] and form a coherent probability measure
- A and B are events in a shared sample space
Derivation
- Definition of conditional probability: P(A∩B) = P(A|B)P(B) = P(B|A)P(A)
- Bayes, An Essay towards solving a Problem in the Doctrine of Chances, Phil. Trans. 53 (1763)
- Laplace rediscovered and generalised to continuous densities
References
- Jaynes, Probability Theory: The Logic of Science, §1.8
- Bishop, Pattern Recognition and Machine Learning, §1.2.3