Bayes’ rule is a way of representing rational updating - rational changes of credences in propositions - upon receiving new evidence. It can be used to calculate what the posterior credence of a proposition ought to be, given a certain prior credence.
- Let H be a proposition (e.g. that someone has cancer),
- let E be some piece of evidence of relevance for H (e.g. that a test has diagnosed them with cancer)
- let P(H) be their prior credence in the proposition H,
- let P(H|E) be their credence in H given that E is true, and
- let ¬H be the negation of H: “H is not true”.
Bayes’ rule then says that the rational credence in H given E (the posterior credence) should be calculated as follows:
Thus, the probability that they indeed have cancer conditional on this evidence - the posterior probability, P(H|E) - is determined by:
- The prior probability that they have cancer, P(H) - i.e. the probability that they would assign to that hypothesis prior to the test.
- The probability that they would be diagnosed with cancer, conditional on them having it - P(E|H).
- The probability that they would be diagnosed with cancer, conditional on them not having it - P(E|¬H).
Two important lessons follow from this analysis. The first is the importance of not ignoring prior probabilities, or base rates (Wikipedia 2016a). If the prior probability of cancer is low, even relatively strong evidence need not make the posterior probability very high. The second is the importance of not ignoring the chance of false positives - that the test could diagnose someone with cancer, even if they do not in fact have cancer (Wikipedia 2016b). If the number of false positives is large relative to the number of true positives - correct cancer diagnoses - then the test gives relatively weak evidence.
Arbital. 2016. Bayes’ rule.
A tailored, visual explanation.
Joyce, James. 2003. Bayes’ theorem. In Edward Zalta (ed.), Stanford Encyclopedia of Philosophy.
Wikipedia. 2016. Bayes’ theorem.
Wikipedia. 2016a. Base rate fallacy.
Wikipedia. 2016b. False positive paradox.