Foundations of statistics
Foundations of statistics is the usual name for the epistemological debate in statistics over how one should conduct inductive inference from data. Among the issues considered in statistical inference are the question of Bayesian inference versus frequentist inference, the distinction between Fisher's "significance testing" and Neyman-Pearson "hypothesis testing", and whether the likelihood principle should be followed. Some of these issues have been debated for up to 200 years without resolution [Efron, 1978].
Bandyopadhyay & Forster  describe four statistical paradigms: "(1) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics".
Other reading 
For a short introduction to the foundations of statistics, see ch. 8 ("Probability and statistical inference") of Kendall's Advanced Theory of Statistics (6th edition, 1994).
In his book Statistics As Principled Argument, Robert P. Abelson articulates the position that statistics serves as a standardized means of settling disputes between scientists who could otherwise each argue the merits of their own positions ad infinitum. From this point of view, statistics is a form of rhetoric; as with any means of settling disputes, statistical methods can succeed only as long as all parties agree on the approach used.
See also 
- Abelson, Robert P. (1995), Statistics as Principled Argument, Lawrence Erlbaum Associates, ISBN 0-8058-0528-1, "... the purpose of statistics is to organize a useful argument from quantitative evidence, using a form of principled rhetoric."
- Bandyopadhyay P.S., Forster M.R. (2011), Philosophy of Statistics (Elsevier).
- Efron, B. (1978), "Controversies in the foundations of statistics", American Mathematical Monthly 85 (4): 231–246, doi:10.2307/2321163.
- Stuart A., Ord J.K. (1994). Kendall's Advanced Theory of Statistics, volume I: Distribution Theory (Edward Arnold).
- Savage, L. J. (1972). The Foundations of Statistics (2nd revised edition). New York: Dover Publications.
Further reading 
- Cox, David R. (2006), Principles of Statistical Inference, Cambridge University Press, ISBN 978-0-521-68567-2
- Efron, Bradley (1986), "Why Isn't Everyone a Bayesian? (with discussion)", The American Statistician 40 (1): 1–11, doi:10.2307/2683105, JSTOR 2683105
- Good, I. J. (1988), "The Interface Between Statistics and Philosophy of Science", Statistical Science 3 (4): 386–397, doi:10.1214/ss/1177012754, JSTOR 2245388
- Kadane J.B., Schervish M.J., Seidenfeld T. (1999), Rethinking the Foundations of Statistics (Cambridge University Press). [Bayesian.]
- Lindley, D.V. (2000), "The philosophy of statistics", Journal of the Royal Statistical Society, Series D 49 (3): 293–337, doi:10.1111/1467-9884.00238
- Mayo, Deborah G. (1992), "Did Pearson reject the Neyman-Pearson philosophy of statistics?", Synthese 90 (2): 233–262, doi:10.1007/BF00485352
- Royall, Richard M. (1997), Statistical Evidence: A Likelihood Paradigm, CRC Press, ISBN 0-412-04411-0
- Citations of Savage (1972) at Google Scholar. [Over 10000 citations.]
- Stanford Encyclopedia of Philosophy entry on probability interpretations.
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