Likelihood¶
Summary¶
In Bayes' theorem, the likelihood is the probability of observing the evidence given that the hypothesis is true — written as P(E|H). It measures how well the hypothesis predicts the evidence.
Definition¶
Likelihood = P(Evidence | Hypothesis is true)
Role in Bayes' Theorem¶
Posterior = Prior × Likelihood / Total Evidence
The likelihood determines how much the evidence should change your belief: - High likelihood (evidence is common if hypothesis is true) → belief increases - Low likelihood (evidence is rare if hypothesis is true) → belief decreases - Equal likelihood (evidence equally likely either way) → belief doesn't change (irrelevant evidence)
Example: Steve the Librarian¶
- Likelihood if librarian: 40% would fit "meek and tidy" description
- Likelihood if farmer: 10% would fit the description
- The 4× difference in likelihoods updates the prior, but not enough to overcome the 20:1 base rate
Likelihood vs. Probability¶
- Probability = P(Evidence) — how likely the evidence is overall
- Likelihood = P(Evidence | Hypothesis) — how likely the evidence is if the hypothesis is true