- The hypothetical probability that an event that has already occurred would yield a specific outcome.
- Unlike probability, mathematical likelihood is used to evaluate past events.
Formerly, likelihood was a synonym for probability, as it still is in everyday English. In his paper “On the Mathematical Foundations of Theoretical Statistics” (Phil. Trans. Royal Soc. Ser. A. 222, (1922), p. 326). Fisher made clear for the first time the distinction between the mathematical properties of “likelihoods” and “probabilities”:
The solution of the problems of calculating from a sample the parameters of the hypothetical population, which we have put forward in the method of maximum likelihood, consists, then, simply of choosing such values of these parameters as have the maximum likelihood. Formally, therefore, it resembles the calculation of the mode of an inverse frequency distribution. This resemblance is quite superficial: if the scale of measurement of the hypothetical quantity be altered, the mode must change its position, and can be brought to have any value, by an appropriate change of scale; but the optimum, as the position of maximum likelihood may be called, is entirely unchanged by any such transformation. Likelihood also differs from probability in that it is not a differential element, and is incapable of being integrated: it is assigned to a particular point of the range of variation, not to a particular element of it.
Likelihood was first used in a Bayesian context by Harold Jeffreys in his “Probability and Scientific Method,” Proceedings of the Royal Society A, 146, (1934) p. 10. Jeffreys wrote “the theorem of Inverse Probability” in the form
Posterior Probability Prior Probability × Likelihood