Likelihoodist statistics

Likelihoodist statistics or likelihoodism is an approach to statistics that exclusively or primarily uses the likelihood function. Likelihoodist statistics is a more minor school than the main approaches of Bayesian statistics and frequentist statistics, but has some adherents and applications. The central idea of likelihoodism is the likelihood principle: data are interpreted as evidence, and the strength of the evidence is measured by the likelihood function. Beyond this, there are significant differences within likelihood approaches: "orthodox" likelihoodists consider data only as evidence, and do not use it as the basis of statistical inference, while others make inferences based on likelihood, but without using Bayesian inference or frequentist inference. Likelihoodism is thus criticized for either not providing a basis for belief or action (if it fails to make inferences), or not satisfying the requirements of these other schools.

The likelihood function is also used in Bayesian statistics and frequentist statistics, but they differ in how it is used. Some likelihoodists consider their use of likelihood as an alternative to other approaches, while others consider it complementary and compatible with other approaches; see § Relation with other theories.

Relation with other theories

Criticism

History

Likelihoodism as a distinct school dates to Edwards (1972), which gives a systematic treatment of statistics, based on likelihood. This built on significant earlier work; see Dempster (1972) for a contemporary review.

While comparing ratios of probabilities dates to early statistics and probability, notably Bayesian inference as developed by Pierre-Simon Laplace from the late 1700s, likelihood as a distinct concept is due to Ronald Fisher in Fisher (1921). Likelihood played an important role in Fisher's statistics, but he developed and used many non-likelihood frequentist techniques as well. His late writings, notably Fisher (1955), emphasize likelihood more strongly, and can be considered an precursor to a systematic theory of likelihoodism.

The likelihood principle was proposed in 1962 by several authors, notably Barnard, Jenkins & Winsten (1962), Birnbaum (1962), and Savage (1962), and followed by the law of likelihood in Hacking (1965); these laid the foundation for likelihoodism. See Likelihood principle § History for early history.

While Edwards's version of likelihoodism considered likelihood as only evidence, which was followed by Royall (1997), others proposed inference based only on likelihood, notably as extensions of maximum likelihood estimation. Notable is John Nelder, who declared in Nelder (1999, p. 264):

At least once a year I hear someone at a meeting say that there are two modes of inference: frequentist and Bayesian. That this sort of nonsense should be so regularly propagated shows how much we have to do. To begin with there is a flourishing school of likelihood inference, to which I belong.

Textbooks that take a likelihoodist approach include the following: Kalbfleisch (1985), Azzalini (1996), Pawitan (2001), Rohde (2014), and Held & Sabanés Bové (2014). A collection of relevant papers is given by Taper & Lele (2004).

gollark: Python's arbitrarily large integers probably do *not* have constant time bitshifting, so I don't think this is actually the complexity you said.
gollark: Wait, is your estimate of the complexity assuming the bitshifts will take the same time regardless of how big the numbers are?
gollark: What's n meant to be?
gollark: Being Python, which uses bignums by default, an optimized C implementation which did multiplication too might be faster.
gollark: ... okay.

See also

References
    • Azzalini, Adelchi (1996), Statistical Inference—Based on the likelihood, Chapman & Hall
    • Barnard, G. A.; Jenkins, G. M.; Winsten, C. B. (1962), "Likelihood inference and time series", Journal of the Royal Statistical Society, Series A, 125 (3): 321–372, doi:10.2307/2982406, JSTOR 2982406
    • Birnbaum, Allan (1962), "On the foundations of statistical inference", Journal of the American Statistical Association, 57 (298): 269–326, doi:10.2307/2281640, JSTOR 2281640, MR 0138176 (With discussion.)
    • Dempster, A. P. (1972), "[Book Review] Likelihood. An Account of the Statistical Concept of Likelihood and Its Application to Scientific Inference. A. W. F. Edwards. Cambridge University Press, New York, 1972. xvi, 236 pp., illus. $13.50", Science, 177 (4052): 878–879, doi:10.1126/science.177.4052.878
    • Edwards, Anthony W. F. (1972), Likelihood (1st ed.), Cambridge University Press
    • Edwards, Anthony W. F. (1992), Likelihood (2nd ed.), Johns Hopkins University Press, ISBN 0-8018-4445-2
    • Fisher, R. A. (1921), "On the "probable error" of a coefficient of correlation deduced from a small sample", Metron, 1: 3–32
    • Fisher, Ronald (1955), "Statistical methods and scientific induction", Journal of the Royal Statistical Society, Series B, 17: 69–78
    • Hacking, Ian (1965), Logic of Statistical Inference, Cambridge University Press, ISBN 0-521-05165-7
    • Held, Leonhard; Sabanés Bové, Daniel (2014), Applied Statistical Inference—Likelihood and Bayes, Springer
    • Kalbfleisch, J. G. (1985), Probability and Statistical Inference, 2, Springer-Verlag
    • Nelder, John A. (1999), "From statistics to statistical science", Journal of the Royal Statistical Society. Series D (The Statistician), 48 (2): 257–269, JSTOR 2681191
    • Pawitan, Yudi (2001), In All Likelihood: Statistical Modelling And Inference Using Likelihood, Oxford University Press, ISBN 978-0-19967122-9
    • Rohde, Charles A. (2014), Introductory Statistical Inference with the Likelihood Function, Springer, ISBN 978-3-319-10460-7
    • Royall, Richard M. (1997), Statistical Evidence: A Likelihood Paradigm, Chapman & Hall, ISBN 0-412-04411-0
    • Savage, Leonard J.; et al. (1962), The Foundations of Statistical Inference, Methuen Publishing
    • Taper, M. L.; Lele, S. R., eds. (2004), The Nature of Scientific Evidence, University of Chicago Press

    Further reading
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