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Bayesian Updating

Why It Matters

Structured belief updating avoids both under-reaction (ignoring new evidence) and over-reaction (overweighting recent evidence). It's a discipline for incorporating information consistently.

The Key Move

Start with prior beliefs. When new evidence arrives, ask: how likely is this evidence if hypothesis A is true? How likely if hypothesis B is true? Update proportionally to this likelihood ratio. The posterior equals the prior times the likelihood ratio (normalized).

Classic Application

Medical diagnosis. A positive test result should update your belief based on the test's accuracy (sensitivity, specificity) and the base rate of the disease. False positives are common when testing for rare diseases.

Surprising Application

Evaluating expertise. When an expert makes a prediction that comes true, update based on: would they have made that prediction if they were just guessing? If the prediction was unusual and came true, update a lot (strong evidence of skill). If it was obvious, update little.

Failure Modes

  • Ignoring base rates - the prior matters; rare events stay rare even with positive tests
  • Updating on evidence you would have seen regardless - selection effects contaminate inference
  • Treating absence of evidence as evidence of absence - sometimes you just didn't look
  • Over-updating on vivid evidence - memorable events aren't necessarily diagnostic
  • Under-updating on statistical evidence - representative samples beat anecdotes

Go Deeper

Any Bayesian statistics text. Jaynes' "Probability Theory: The Logic of Science" for foundations.