Law Of Likelihood

TL;DR

• Use new evidence to update how likely competing hypotheses are.
• A hypothesis with a higher prior probability and a better explanation of the data becomes relatively more likely.
• Belief shifts depend on both the prior probability and how well the hypothesis explains the observed data.

Definition

The law of likelihood is a principle in statistics that states that the likelihood of a hypothesis being true is directly proportional to its prior probability and the degree to which it explains the observed data.

Explanation

The principle requires two elements for comparing hypotheses: a prior probability for each hypothesis, and how well each hypothesis explains the observed data. When new evidence is observed, the law of likelihood implies that hypotheses with higher prior probabilities and stronger support from the data increase in likelihood, while those with lower priors or weaker support decrease in likelihood. In practice, this means updating beliefs about which hypothesis is more plausible in light of the evidence.

Examples

Detective example

You are a detective with two suspects, John and Mary. John has a prior probability of 0.3, and Mary has a prior probability of 0.7. After examining the evidence, the data strongly supports the hypothesis that the culprit is John. According to the law of likelihood, the likelihood of John being the culprit increases and the likelihood of Mary being the culprit decreases — beliefs are updated based on the new evidence.

Medical diagnosis example

You are a doctor with two hypotheses about a patient, Hypothesis A and Hypothesis B. Hypothesis A has a prior probability of 0.1, and Hypothesis B has a prior probability of 0.9. After examining the patient and conducting tests, the data strongly supports the hypothesis that the patient has the rare disease. Under the law of likelihood, the likelihood of Hypothesis A being true decreases, and the likelihood of Hypothesis B being true increases — beliefs are updated based on the new evidence.