MathLiveBayes' theorem with total probability expansion
Gives the posterior probability of one event in a partition given evidence B, by combining the likelihoods with the priors and normalising using the law of total probability. Used whenever the evidence probability must be expanded over a partition of mutually exclusive, exhaustive events.
P(A_i \mid B)=\frac{P(B \mid A_i)\,P(A_i)}{\sum_{j=1}^{n} P(B \mid A_j)\,P(A_j)}What the symbols mean
| posterior probability of event A_i given that B has occurred | |
| likelihood of observing B given A_j | |
| prior probability of event A_j | |
| a partition of the sample space into mutually exclusive and exhaustive events | |
| the observed evidence event, with P(B) > 0 |
About this formula
- FieldMathematics
- TopicProbability theory
- LevelUniversity
- Also calledBayes theorem total probability, Bayes rule with law of total probability, posterior probability formula, Bayes formula denominator expansion, conditional probability partition formula, Bayes theorem partition of sample space
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