A likelihood is the probability of observed evidence conditional on a specified hypothesis or parameter value. In Bayes’ rule, P(B|A) is the likelihood, while P(A) is the prior and P(A|B) is the posterior. Likelihood measures how compatible the evidence is with the assumed condition; it is not itself the posterior probability. section 12.2
Height and professional basketball
Let A mean that a randomly chosen person is over six feet tall and B mean that the person plays professional basketball. The likelihood P(B|A) describes the probability of professional basketball membership among people over six feet tall. Observing B then raises the probability of A from its unconditional rate of about 0.0000013636 to about 0.0000193, illustrating Bayesian updating from prior and likelihood to posterior. section 12.2
The same structure extends from binary events to distributions. A parameter θ, such as an exponential rate or normal mean, is treated as uncertain; its prior describes beliefs before data, and the likelihood contributes the observed-data information used to form a posterior distribution. section 12.2
