Two New Properties of Mathematical Likelihood

By R. Fisher. 1934

To Thomas Bayes must be given the credit of broaching the problem of using the concepts of mathematical probability in discussing problems of inductive inference, in which we argue from the particular to the general; or, in statistical phraselogy, argue form the sample to the population, from which, ex hypothesi , the sample was drawn. Bayes put forward, with considerable caution, a method by which such problems could be reduced to the form of problems of probability. His method of doing this depended essentially on postualting a priori knowledge, not of the particular population of which our observations from a sample, but of an imaginary population of populations from which this population was regarded as having been drawn at random. Clearly, if we have possession of such a priori knowledge, our problem is not properly an inductive one at all, for the population under discussion is then regarded merely as a particular case of a general type, of which we already possess exact knowledge, and are therefore in a positioin to draw exact deductive inferences. To the merit of broaching a fundamentally important problem, Bayes added that of perceiving, much more clearly than some of his followers have done, the logical weakness of the form of solutiion he put forward. Indeed we are told that it was his doubts respecting the validity of the postulate needed for establishing the method of inverse probability that led to this withholding his entire treatise from publication. Actually it was not published until after his death.

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Citations in the corpus

The hardness of conditional independence testing and the generalised covariance measure (2018, background): “Conditional independences lie at the heart of several fundamental concepts such as sufficiency [21] and ancillarity [22, 23]; see also Jensen and Sørensen [28].”
R.A. Fisher and the making of maximum likelihood 1912-1922 (1997, background): “By 1930 Fisher 1930, p. 531 knew the Ž .original and by 1934 he knew Fisher, 1934a p. 616 the ‘‘Gauss]Markov’’ justification of 1821; in 1912]1922 he seems to have known only bowdlerized Gauss.”
ANCILLARY STATISTICS: A REVIEW (2010, methodology): “The concept of ancillary statistics is one of R. A. Fisher’s fundamental contributions to statistical inference.”
Applying Generalized Linear Models (2000, background): “…a binomial distribution with the complementary log log link (Fisher, 1922); • exponential family — a class of distributions with sufficient statistics for the parameters (Fisher, 1934); • probit analysis — a binomial distribution with the probit link (Bliss, 1935); 1.1 Statistical Modelling 5 •…”
Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity (2019, methodology): “Therefore, with unknown assignment mechanisms, the assumptions for the RI approach would not be satisfied, while our method would still provide valid inference. We argue now that, even under uniform assignment probability, RI would usually provide an unsatisfactory solution to the inference problem ”
Asymptotics and the theory of inference (2003, methodology): “These may be computed for a fixed data value and a range of values for ψ0, leading to confidence limits or more generally a confidence distribution function or significance function [Fraser (1990); Efron (1997)].”
Sufficiency and Conditional Estimation of Person Parameters in the Polytomous Rasch Model (2010, methodology): “Sufficiency was defined in terms of separability of parameters (Fisher, 1934) and considered specifically with respect to the separation of person and item parameters in Rasch models (Masters & Wright, 1984).”
Minimax Estimation of Functionals of Discrete Distributions (2014, unknown): “Maximum likelihood is the most widely used statistical estimation technique, which emerged in modern form 90 years ago in a series of remarkable papers by Fisher [146]–[148].”
Fisher and Regression (2005, methodology): “Bartlett thus made explicit the scheme implicit in Fisher (1934)—or rather half explicit; the reader has to define quasi-sufficiency in general.”
Assessing syntactic sophistication in L2 writing: A usage-based approach (2017, methodology): “Collostructional strength is calculated using the Fisher-Yates exact test (Fisher, 1934; Yates, 1934), which is calculated as: pobserved distribution = a c a b d b N a b + + + + * ∑ pall more extreme distributions.”
Ancillaries and Conditional Inference (2004, background): “Cox initially considered the appropriate sample space for statistical inference, but then developed it in terms of conditioning on an ancillary statistic (Fisher, 1925, 1934, 1935).”
Conditioning, Likelihood, and Coherence: A Review of Some Foundational Concepts (2000, background): “Fisher (1921, 1925, 1934) defined the likelihood function as i= I Of course, the likelihood function appeared implicitly much earlier in the work of Bayes and Laplace, who used what we now call Bayesian inference to solve statistical problems.”
Most Likely Transformations (2015, methodology): “This model (with arbitrary FZ) has also been presented in Foresi and Peracchi (1995) and is called distribution regression in Chernozhukov et al.”
Covariate powered cross‐weighted multiple testing (2017, methodology): “dea to use weights with the first stage p-value as the covariate. Conditional inference is a classical topic in statistics, aiming at the relevance and tractability of conducted inference (Cox, 1958; Fisher, 1934; Lee et al., 2016). However, for single hypothesis testing, it has also been considered”