Historical Remarks
In English, "likelihood" has been distinguished as being related to but weaker than "probability" since its earliest uses. The comparison of hypotheses by evaluating likelihoods has been used for centuries, for example by John Milton in Aeropagitica: "when greatest likelihoods are brought that such things are truly and really in those persons to whom they are ascribed".
In Danish, "likelihood" was used by Thorvald N. Thiele in 1889.
In English, "likelihood" appears in many writings by Charles Sanders Peirce, where model-based inference (usually abduction but sometimes including induction) is distinguished from statistical procedures based on objective randomization. Peirce's preference for randomization-based inference is discussed in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883)".
"probabilities that are strictly objective and at the same time very great, although they can never be absolutely conclusive, ought nevertheless to influence our preference for one hypothesis over another; but slight probabilities, even if objective, are not worth consideration; and merely subjective likelihoods should be disregarded altogether. For they are merely expressions of our preconceived notions" (7.227 in his Collected Papers).
"But experience must be our chart in economical navigation; and experience shows that likelihoods are treacherous guides. Nothing has caused so much waste of time and means, in all sorts of researchers, as inquirers' becoming so wedded to certain likelihoods as to forget all the other factors of the economy of research; so that, unless it be very solidly grounded, likelihood is far better disregarded, or nearly so; and even when it seems solidly grounded, it should be proceeded upon with a cautious tread, with an eye to other considerations, and recollection of the disasters caused." (Essential Peirce, volume 2, pages 108–109)
Like Thiele, Peirce considers the likelihood for a binomial distribution. Peirce uses the logarithm of the odds-ratio throughout his career. Peirce's propensity for using the log odds is discussed by Stephen Stigler.
In Great Britain, "likelihood" was popularized in mathematical statistics by R.A. Fisher in 1922: "On the mathematical foundations of theoretical statistics". In that paper, Fisher also uses the term "method of maximum likelihood". Fisher argues against inverse probability as a basis for statistical inferences, and instead proposes inferences based on likelihood functions. Fisher's use of "likelihood" fixed the terminology that is used by statisticians throughout the world.
Read more about this topic: Likelihood Function
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