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Prior-Likelihood-Posterior Decomposition

Why It Matters

We chronically muddle these together. Someone says "the evidence strongly supports X" - but do they mean the evidence is very probable if X is true (high likelihood), or that X is very probable given the evidence (high posterior)? These are different claims. Strong evidence for an improbable hypothesis still leaves it improbable. This confusion generates systematic errors in medical diagnosis, legal reasoning, and scientific interpretation. The tool forces analytical separation where intuition conflates.

The Key Move

When evaluating any claim under uncertainty, explicitly identify three distinct questions:

  1. How plausible was the hypothesis before this evidence? (prior)
  2. How likely is this evidence if the hypothesis is true versus if it's false? (likelihood ratio)
  3. How plausible is the hypothesis after considering the evidence? (posterior). Refuse to answer question 3 without addressing questions 1 and 2. Write them down separately if the stakes are high.

Classic Application

Medical diagnosis. A test is "95% accurate" (high likelihood of positive result given disease). Patient tests positive. But if the disease has 1% base rate (low prior), the posterior probability of actually having the disease is only about 16%. Doctors who skip the prior-posterior distinction massively overdiagnose. The tool forces: What's the base rate? What's the test accuracy in both directions? Only then: What's the probability of disease given this positive test?

Surprising Application

Evaluating surprising claims from credible sources. A scientist you trust makes a startling claim (low prior plausibility, high source credibility). Naive reasoning: trusted source implies accept claim (conflating likelihood with posterior). The tool reveals: Yes, a trusted scientist is more likely to make this claim if it's true (high likelihood ratio), but extraordinary claims need extraordinary evidence. One credible source on an extraordinary claim might shift your posterior from 0.1% to 5% - meaningful evidence that still leaves you skeptical. The framework prevents both uncritical acceptance and dismissing all contrary evidence.

Failure Modes

False precision - treating qualitative priors as if you've done the math. Over-formalization - sometimes you just need more data, not more careful probability parsing. Prior sensitivity - when evidence is weak, different reasonable priors yield wildly different posteriors, and the formalism obscures that this is fundamentally uncertain. Ignoring model uncertainty - the framework assumes your hypotheses exhaust the possibilities.

Go Deeper

Gelman et al., Bayesian Data Analysis (3rd ed.), Chapter 1; Pearl, Causality, Chapter 1.4 on probabilistic reasoning; McGrayne, The Theory That Would Not Die for historical context.