Editing Statistical inference (section)

===Other paradigms for inference===

====Minimum description length====
{{Main|Minimum description length}}

The minimum description length (MDL) principle has been developed from ideas in [[information theory]]<ref name="Soofi 2000 1349–1353">Soofi (2000)</ref> and the theory of [[Kolmogorov complexity]].<ref name=HY>Hansen & Yu (2001)</ref> The (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or non-falsifiable "data-generating mechanisms" or [[probability models]] for the data, as might be done in frequentist or Bayesian approaches.

However, if a "data generating mechanism" does exist in reality, then according to [[Claude Shannon|Shannon]]'s [[source coding theorem]] it provides the MDL description of the data, on average and asymptotically.<ref name=HY747>Hansen and Yu (2001), page 747.</ref> In minimizing description length (or descriptive complexity), MDL estimation is similar to [[maximum likelihood estimation]] and [[maximum a posteriori estimation]] (using [[Maximum entropy probability distribution|maximum-entropy]] [[Bayesian probability|Bayesian priors]]). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.<ref name=HY747/><ref name=JR>Rissanen (1989), page 84</ref>

The MDL principle has been applied in communication-[[coding theory]] in [[information theory]], in [[linear regression]],<ref name=JR/> and in [[data mining]].<ref name=HY/>

The evaluation of MDL-based inferential procedures often uses techniques or criteria from [[computational complexity theory]].<ref>Joseph F. Traub, G. W. Wasilkowski, and H. Wozniakowski. (1988) {{page needed|date=June 2011}}</ref>

====Fiducial inference====
{{Main|Fiducial inference}}
[[Fiducial inference]] was an approach to statistical inference based on [[fiducial probability]], also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious.<ref>Neyman (1956)</ref><ref>Zabell (1992)</ref> However this argument is the same as that which shows<ref>Cox (2006) page 66</ref> that a so-called [[confidence distribution]] is not a valid [[probability distribution]] and, since this has not invalidated the application of [[confidence interval]]s, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's [[Fiducial probability|fiducial argument]] as a special case of an inference theory using [[upper and lower probabilities]].{{sfn|Hampel|2003}}

====Structural inference====

Developing ideas of Fisher and of Pitman from 1938 to 1939,<ref>Davison, page 12. {{full citation needed|date=November 2012}}</ref> [[George A. Barnard]] developed "structural inference" or "pivotal inference",<ref>Barnard, G.A. (1995) "Pivotal Models and the Fiducial Argument", International Statistical Review, 63 (3), 309–323. {{JSTOR|1403482}}</ref> an approach using [[Haar measure|invariant probabilities]] on [[group family|group families]]. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be well-defined and useful. [[Donald A. S. Fraser]] developed a general theory for structural inference<ref>{{Cite book|last=Fraser|first=D. A. S.|url=https://www.worldcat.org/oclc/440926|title=The structure of inference|date=1968|publisher=Wiley|isbn=0-471-27548-4|location=New York|oclc=440926}}</ref> based on [[group theory]] and applied this to linear models.<ref>{{Cite book|last=Fraser|first=D. A. S.|url=https://www.worldcat.org/oclc/3559629|title=Inference and linear models|date=1979|publisher=McGraw-Hill|isbn=0-07-021910-9|location=London|oclc=3559629}}</ref> The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.<ref>{{Cite journal|last1=Taraldsen|first1=Gunnar|last2=Lindqvist|first2=Bo Henry|date=2013-02-01|title=Fiducial theory and optimal inference|url=https://projecteuclid.org/journals/annals-of-statistics/volume-41/issue-1/Fiducial-theory-and-optimal-inference/10.1214/13-AOS1083.full|journal=The Annals of Statistics|volume=41|issue=1|doi=10.1214/13-AOS1083|arxiv=1301.1717|s2cid=88520957|issn=0090-5364}}</ref>