Bayes rule and belief updates

Evidence changes belief through both how expected it was and how plausible the hypothesis was beforehand.

Bayes's rule relates conditional probabilities: P(H|E) = P(E|H) P(H) / P(E), when P(E) is positive. A striking observation can still have a modest posterior probability for a rare hypothesis if the observation is also common under alternatives.

For example: Suppose 1% of items are defective, a test catches 90% of defects, and 5% of good items fail. Among 10,000 items, about 90 defective and 495 good items fail: about 15.4% of failed items are defective.

Keep in mind: The arithmetic is only as useful as the model, prior, and likelihood estimates. A posterior probability does not by itself establish a causal mechanism.

Think about it

For a piece of evidence you find persuasive, how often would you expect to see it if your preferred explanation were false?

Connected ideas

The problem of induction · Correlation is not causation · Uncertainty is part of measurement

Sources and further reading

• Brown University Seeing Theory: Bayesian Inference

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