Changes
m
Reverted edits by 47.247.206.28 (talk) to last version by GreenC bot
{{distinguish|Statistical interference}}
'''Statistical inference''' is the process of using [[data analysis]] to deduce properties of an underlying [[probability distribution]].<ref name="Oxford">Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. {{isbn|978-0-19-954145-4}}</ref> Inferential statistical analysis infers properties of a [[Statistical population|population]], for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is [[Sampling (statistics)|sampled]] from a larger population.
Inferential statistics can be contrasted with [[descriptive statistics]]. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.
== Introduction ==
Statistical inference makes propositions about a population, using data drawn from the population with some form of [[Sampling (statistics)|sampling]]. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) [[Model selection|selecting]] a [[statistical model]] of the process that generates the data and (second) deducing propositions from the model.{{Citation needed|date=December 2016}}
Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling".<ref>Konishi & Kitagawa (2008), p. 75.</ref> Relatedly, [[David Cox (statistician)|Sir David Cox]] has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".<ref>Cox (2006), p. 197.</ref>
The [[Logical consequence|conclusion]] of a statistical inference is a statistical [[proposition]].{{Citation needed|date=February 2012}} Some common forms of statistical proposition are the following:
* a [[point estimate]], i.e. a particular value that best approximates some parameter of interest;
* an [[interval estimate]], e.g. a [[confidence interval]] (or set estimate), i.e. an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the [[frequency probability|probability]] at the stated [[confidence level]];
* a [[credible intervals|credible interval]], i.e. a set of values containing, for example, 95% of posterior belief;
* rejection of a [[statistical hypothesis testing|hypothesis]];{{efn|According to Peirce, acceptance means that inquiry on this question ceases for the time being. In science, all scientific theories are revisable.}}
* [[Cluster analysis|clustering]] or [[Statistical classification|classification]] of data points into groups.
==Models and assumptions==
{{Main|Statistical model|Statistical assumptions}}
Any statistical inference requires some assumptions. A '''statistical model''' is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference.<ref name=Cox2006>Cox (2006) page 2</ref> Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.<ref>{{cite book| last=Evans|first=Michael|title=Probability and Statistics: The Science of Uncertainty|year=2004|publisher=Freeman and Company|page=267|display-authors=etal|url=https://books.google.com/books?id=hkWK8kFzXWIC&printsec=frontcover#v=onepage&q=%22descriptive%20statistics%22&f=false|isbn=9780716747420}}</ref>
===Degree of models/assumptions===
Statisticians distinguish between three levels of modeling assumptions;
* '''[[Parametric model|Fully parametric]]''': The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters.<ref name=Cox2006/> For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by [[Simple random sample|'simple' random sampling]]. The family of [[Generalized linear model#Model components|generalized linear models]] is a widely used and flexible class of parametric models.
* '''[[Nonparametric statistics#Non-parametric models|Non-parametric]]''': The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal.<ref>van der Vaart, A.W. (1998) ''Asymptotic Statistics'' Cambridge University Press. {{isbn|0-521-78450-6}} (page 341)</ref> For example, every continuous probability distribution has a median, which may be estimated using the sample median or the [[Hodges–Lehmann estimator|Hodges–Lehmann–Sen estimator]], which has good properties when the data arise from simple random sampling.
* '''[[Semiparametric model|Semi-parametric]]''': This term typically implies assumptions 'in between' fully and non-parametric approaches. For example, one may assume that a population distribution has a finite mean. Furthermore, one may assume that the mean response level in the population depends in a truly linear manner on some covariate (a parametric assumption) but not make any parametric assumption describing the variance around that mean (i.e. about the presence or possible form of any [[heteroscedasticity]]). More generally, semi-parametric models can often be separated into 'structural' and 'random variation' components. One component is treated parametrically and the other non-parametrically. The well-known [[Cox model]] is a set of semi-parametric assumptions.
===Importance of valid models/assumptions===
Whatever level of assumption is made, correctly calibrated inference in general requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified.
Incorrect assumptions of [[Simple random sample|'simple' random sampling]] can invalidate statistical inference.<ref>Kruskal 1988</ref> More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions.<ref>
[[David A. Freedman|Freedman, D.A.]] (2008) "Survival analysis: An Epidemiological hazard?". ''The American Statistician'' (2008) 62: 110-119. (Reprinted as Chapter 11 (pages 169–192) of Freedman (2010)).</ref> Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference.<ref>Berk, R. (2003) ''Regression Analysis: A Constructive Critique (Advanced Quantitative Techniques in the Social Sciences) (v. 11)'' Sage Publications. {{isbn|0-7619-2904-5}}</ref> The use of '''any''' parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal."<ref name=Brewer>{{cite book|first=Ken |last=Brewer| title=Combined Survey Sampling Inference: Weighing of Basu's Elephants| publisher=Hodder Arnold|page=6|year= 2002|isbn=978-0340692295}}</ref> In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population."<ref name=Brewer/> Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy tailed.
====Approximate distributions====
{{Main|Statistical distance|Asymptotic theory (statistics)|Approximation theory}}
Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.
With finite samples, [[approximation theory|approximation results]] measure how close a limiting distribution approaches the statistic's [[sample distribution]]: For example, with 10,000 independent samples the [[normal distribution]] approximates (to two digits of accuracy) the distribution of the [[sample mean]] for many population distributions, by the [[Berry–Esseen theorem]].<ref name=JHJ>
Jörgen Hoffman-Jörgensen's ''Probability With a View Towards Statistics'', Volume I. Page 399 {{full citation needed|date=November 2012}}
</ref>
Yet for many practical purposes, the normal approximation provides a good approximation to the sample-mean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience.<ref name=JHJ/> Following Kolmogorov's work in the 1950s, advanced statistics uses [[approximation theory]] and [[functional analysis]] to quantify the error of approximation. In this approach, the [[metric geometry]] of [[probability distribution]]s is studied; this approach quantifies approximation error with, for example, the [[Kullback–Leibler divergence]], [[Bregman divergence]], and the [[Hellinger distance]].<ref>Le Cam (1986) {{page needed|date=June 2011}}</ref><ref>Erik Torgerson (1991) ''Comparison of Statistical Experiments'', volume 36 of Encyclopedia of Mathematics. Cambridge University Press. {{full citation needed|date=November 2012}}
</ref><ref>{{cite book
|author1=Liese, Friedrich |author2=Miescke, Klaus-J.
|lastauthoramp=yes | title = Statistical Decision Theory: Estimation, Testing, and Selection
| year = 2008
| publisher = Springer
| isbn = 0-387-73193-8
}}
</ref>
With indefinitely large samples, [[asymptotic theory (statistics)|limiting results]] like the [[central limit theorem]] describe the sample statistic's limiting distribution, if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.<ref>Kolmogorov (1963, p.369): "The frequency concept, <!-- comma missing in original --> based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials".</ref><ref>"Indeed, limit theorems 'as <math>n</math> tends to infinity' are logically devoid of content about what happens at any particular <math>n</math>. All they can do is suggest certain approaches whose performance must then be checked on the case at hand." — Le Cam (1986) (page xiv)</ref><ref>Pfanzagl (1994): "The crucial drawback of asymptotic theory: What we expect from asymptotic theory are results which hold approximately . . . . What asymptotic theory has to offer are limit theorems."(page ix) "What counts for applications are approximations, not limits." (page 188)
</ref> However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the [[generalized method of moments]] and the use of [[generalized estimating equation]]s, which are popular in [[econometrics]] and [[biostatistics]]. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation<!-- and approximation results -->.<ref>Pfanzagl (1994) : "By taking a limit theorem as being approximately true for large sample sizes, we commit an error the size of which is unknown. [. . .] Realistic information about the remaining errors may be obtained by simulations." (page ix)
</ref> The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional [[statistical model|models]] with [[logarithmically concave function|log-concave]] [[likelihood function|likelihood]]s (such as with one-parameter [[exponential families]]).
===Randomization-based models===
{{Main|Randomization}}
{{See also|Random sample|Random assignment}}
For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the null-hypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments.<ref>[[Jerzy Neyman|Neyman, J.]](1934) "On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection", ''[[Journal of the Royal Statistical Society]]'', 97 (4), 557–625 {{jstor|2342192}}</ref><ref name="Hinkelmann and Kempthorne">Hinkelmann and Kempthorne(2008) {{page needed|date=June 2011}}</ref> Statistical inference from randomized studies is also more straightforward than many other situations.<ref>ASA Guidelines for a first course in statistics for non-statisticians. (available at the ASA website)</ref><ref>[[David A. Freedman]] et alia's ''Statistics''.</ref><ref>Moore et al. (2015).</ref> In [[Bayesian inference]], randomization is also of importance: in [[survey sampling]], use of [[sampling without replacement]] ensures the [[exchangeability]] of the sample with the population; in randomized experiments, randomization warrants a [[missing at random]] assumption for [[covariate]] information.<ref>[[Andrew Gelman|Gelman A.]] et al. (2013). ''Bayesian Data Analysis'' ([[Chapman & Hall]]).</ref>
Objective randomization allows properly inductive procedures.<ref>Peirce (1877-1878)</ref><ref>Peirce (1883)</ref><ref>
[[David A. Freedman|David Freedman]] et alia ''Statistics'' and [[David A. Freedman]] ''Statistical Models''.</ref><ref>
[[C. R. Rao|Rao, C.R.]] (1997) ''Statistics and Truth: Putting Chance to Work'', World Scientific. {{isbn|981-02-3111-3}}</ref>
Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures.<ref>Peirce; Freedman; Moore et al. (2015).{{Citation needed|date=March 2010}}</ref> (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.<ref>Box, G.E.P. and Friends (2006) ''Improving Almost Anything: Ideas and Essays, Revised Edition'', Wiley. {{isbn|978-0-471-72755-2}}</ref><ref>
Cox (2006), page 196</ref>)
Similarly, results from [[randomized experiment]]s are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena.<ref>ASA Guidelines for a first course in statistics for non-statisticians. (available at the ASA website)
* David A. Freedman et alia's ''Statistics''.
* Moore et al. (2015).
</ref>
However, a good observational study may be better than a bad randomized experiment.
The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.<ref>Neyman, Jerzy. 1923 [1990]. "On the Application of Probability Theory to AgriculturalExperiments. Essay on Principles. Section 9." ''Statistical Science'' 5 (4): 465–472. Trans. [[Dorota Dabrowska|Dorota M. Dabrowska]] and Terence P. Speed.</ref><ref>Hinkelmann & Kempthorne (2008) {{page needed|date=June 2011}}</ref>
However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.
====Model-based analysis of randomized experiments====
It is standard practice to refer to a statistical model, often a linear model, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme.<ref name="Hinkelmann and Kempthorne"/> Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.<ref>Hinkelmann and Kempthorne (2008) Chapter 6.</ref>
==Paradigms for inference==
Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.
Bandyopadhyay & Forster<ref>Bandyopadhyay & Forster (2011). The quote is taken from the book's Introduction (p.3). See also "Section III: Four Paradigms of Statistics".</ref> describe four paradigms: "(i) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics". The classical (or [[Frequentist inference|frequentist]]) paradigm, the [[Bayesian inference|Bayesian]] paradigm, and the [[Akaike information criterion|AIC]]-based paradigm are summarized below. The likelihood-based paradigm is essentially a sub-paradigm of the AIC-based paradigm.
===Frequentist inference===
{{See also|Frequentist inference}}
This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging.
====Examples of frequentist inference====
* [[p-value|''p''-value]]
* [[Confidence interval]]
====Frequentist inference, objectivity, and decision theory====
One interpretation of [[frequentist inference]] (or classical inference) is that it is applicable only in terms of [[frequency probability]]; that is, in terms of repeated sampling from a population. However, the approach of Neyman<ref>{{cite journal | last1 = Neyman | first1 = J. | authorlink = Jerzy Neyman | year = 1937 | title = Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability | jstor = 91337 | journal = Philosophical Transactions of the Royal Society of London A | volume = 236 | issue = 767| pages = 333–380 | doi=10.1098/rsta.1937.0005}}</ref> develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e. probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach.
The frequentist procedures of significance testing and confidence intervals can be constructed without regard to [[utility function]]s. However, some elements of frequentist statistics, such as [[statistical decision theory]], do incorporate [[utility function]]s.{{Citation needed|date=April 2012}} In particular, frequentist developments of optimal inference (such as [[minimum-variance unbiased estimator]]s, or [[uniformly most powerful test]]ing) make use of [[loss function]]s, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property.<ref>Preface to Pfanzagl.</ref> However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under [[absolute value]] loss functions, in that they minimize expected loss, and [[least squares]] estimators are optimal under squared error loss functions, in that they minimize expected loss.
While statisticians using frequentist inference must choose for themselves the parameters of interest, and the [[estimators]]/[[Test statistic#Common test statistics|test statistic]] to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.{{Citation needed|date=April 2012}}
===Bayesian inference===
{{See also|Bayesian Inference}}
The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate to one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are [[Bayesian probability#Justification of Bayesian probabilities|several different justifications]] for using the Bayesian approach.
====Examples of Bayesian inference====
* [[Credible interval]] for [[interval estimation]]
* [[Bayes factor]]s for model comparison
====Bayesian inference, subjectivity and decision theory====
Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's [[utility function]] need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been [[Bayesian probability#Personal probabilities and objective methods for constructing priors|proposed]] but not yet fully developed.)
Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides [[optimal decision]]s in a [[decision theory|decision theoretic]] sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) [[Coherence (statistics)|incoherent]]; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be [[Coherence (statistics)|coherent]]. Some advocates of [[Bayesian inference]] assert that inference ''must'' take place in this decision-theoretic framework, and that [[Bayesian inference]] should not conclude with the evaluation and summarization of posterior beliefs.
===AIC-based inference===
{{Main|Akaike information criterion}}
{{expand section|date=November 2017}}
The ''[[Akaike information criterion]]'' (AIC) is an [[estimator]] of the relative quality of [[statistical model]]s for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for [[model selection]].
AIC is founded on [[information theory]]: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the [[goodness of fit]] of the model and the simplicity of the model.)
===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.
==Inference topics==
The topics below are usually included in the area of '''statistical inference'''.
#[[Statistical assumptions]]
#[[Statistical decision theory]]
#[[Estimation theory]]
#[[Statistical hypothesis testing]]
#[[Revising opinions in statistics]]
#[[Design of experiments]], the [[analysis of variance]], and [[Regression analysis|regression]]
#[[Survey sampling]]
#[[Summarizing statistical data]]
==See also==
* [[Algorithmic inference]]
* [[Induction (philosophy)]]
* [[Informal inferential reasoning]]
* [[Population proportion]]
* [[Philosophy of statistics]]
* [[Predictive inference]]
* [[Information field theory]]
== Notes ==
{{Notelist}}
==Citations==
{{Reflist}}
==References==
* {{Citation |editor1-first= P. S. |editor1-last = Bandyopadhyay |editor2-first= M. R. |editor2-last = Forster | title= Philosophy of Statistics |year= 2011 |publisher = [[Elsevier]] }}.
* {{cite book |last1=Bickel |first1=Peter J. |last2 = Doksum |first2 = Kjell A. |title = Mathematical statistics: Basic and selected topics |volume = 1 |edition = Second (updated printing 2007) |year = 2001 |publisher = [[Prentice Hall]] |mr = 443141 |isbn = 0-13-850363-X }}
* [[David R. Cox|Cox, D. R.]] (2006). ''Principles of Statistical Inference'', [[Cambridge University Press]]. {{ISBN|0-521-68567-2}}.
* [[Ronald A. Fisher|Fisher, R. A.]] (1955), "Statistical methods and scientific induction", ''[[Journal of the Royal Statistical Society]], Series B'', 17, 69—78. (criticism of statistical theories of [[Jerzy Neyman]] and [[Abraham Wald]])
* {{cite book |authorlink=David A. Freedman |last=Freedman |first=D. A. |title = Statistical Models: Theory and practice |edition=revised |year=2009 |publisher=[[Cambridge University Press]] |pages = xiv+442 pp.
|isbn = 978-0-521-74385-3 |mr = 2489600 }}
* [[David A. Freedman|Freedman, D. A.]] (2010). ''Statistical Models and Causal Inferences: A Dialogue with the Social Sciences'' (Edited by David Collier, Jasjeet S. Sekhon, and Philip B. Stark), [[Cambridge University Press]].
* {{cite journal |last1=Hampel |first1=Frank |title = The proper fiducial argument |date=Feb 2003 |issue=Research Report No. 114 |url = http://e-collection.library.ethz.ch/eserv/eth:26403/eth-26403-01.pdf |accessdate=29 March 2016 |ref = harv }}
* {{cite journal |doi=10.1198/016214501753168398 |title=Model Selection and the Principle of Minimum Description Length: Review paper |first1=Mark H. |last1=Hansen |first2=Bin |last2=Yu |author2-link=Bin Yu |journal=[[Journal of the American Statistical Association]] |volume=96 |issue=454 |date=June 2001 |pages=746–774 |jstor=2670311 |mr=1939352 |url=http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps |archive-url=https://web.archive.org/web/20041116080440/http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps |dead-url=yes |archive-date=2004-11-16 }}
* {{cite book
|last1 = Hinkelmann |first1=Klaus |last2=Kempthorne |first2=Oscar |authorlink2=Oscar Kempthorne
|year = 2008
|title = Introduction to Experimental Design
|url = https://books.google.com/?id=T3wWj2kVYZgC&printsec=frontcover
|edition = Second
|publisher = Wiley
|isbn = 978-0-471-72756-9
}}
* {{cite journal |authorlink=Andrei N. Kolmogorov|first=Andrei N.|last=Kolmogorov |year=1963 |title = On tables of random numbers |journal=[[Sankhya (journal)|Sankhyā]] Ser. A.|volume=25 |pages=369–375 |mr=178484 }} Reprinted as {{cite journal |first=Andrei N. |last=Kolmogorov |year=1998 |title=On tables of random numbers |journal = [[Theoretical Computer Science (journal) |Theoretical Computer Science]] |volume=207 |issue=2 |pages = 387–395 |doi = 10.1016/S0304-3975(98)00075-9 |mr = 1643414 }}
* Konishi S., Kitagawa G. (2008), ''Information Criteria and Statistical Modeling'', Springer.
* {{cite journal
|title = Miracles and statistics: the casual assumption of independence (ASA Presidential Address)
|authorlink = William Kruskal
|first = William
|last = Kruskal
|journal = [[Journal of the American Statistical Association]]
|volume = 83
|issue = 404
|date = December 1988
|pages = 929–940
|jstor = 2290117
|doi = 10.2307/2290117
}}
* [[Lucien Le Cam|Le Cam, Lucian]]. (1986) ''Asymptotic Methods of Statistical Decision Theory'', Springer. {{isbn|0-387-96307-3}}
* [[David S. Moore|Moore, D. S.]]; McCabe, G. P.; Craig, B. A. (2015), ''Introduction to the Practice of Statistics'', Eighth Edition, Macmillan.
* {{cite journal
|title = Note on an article by Sir Ronald Fisher
|authorlink=Jerzy Neyman
|first=Jerzy
|last=Neyman
|journal=[[Journal of the Royal Statistical Society, Series B]]
|volume=18
|issue=2
|year=1956
|pages=288–294
|jstor=2983716 }} (reply to Fisher 1955)
* [[Charles Sanders Peirce|Peirce, C. S.]] (1877–1878), "Illustrations of the logic of science" (series), ''[[Popular Science Monthly]]'', vols. 12-13. Relevant individual papers:
** (1878 March), "The Doctrine of Chances", ''Popular Science Monthly'', v. 12, March issue, pp. [https://books.google.com/books?id=ZKMVAAAAYAAJ&jtp=604 604]–615. ''Internet Archive'' [https://archive.org/stream/popscimonthly12yoummiss#page/612/mode/1up Eprint].
** (1878 April), "The Probability of Induction", ''Popular Science Monthly'', v. 12, pp. [https://books.google.com/books?id=ZKMVAAAAYAAJ&jtp=705 705]–718. ''Internet Archive'' [https://archive.org/stream/popscimonthly12yoummiss#page/715/mode/1up Eprint].
** (1878 June), "The Order of Nature", ''Popular Science Monthly'', v. 13, pp. [https://books.google.com/books?id=u8sWAQAAIAAJ&jtp=203 203]–217.''Internet Archive'' [https://archive.org/stream/popularsciencemo13newy#page/203/mode/1up Eprint].
** (1878 August), "Deduction, Induction, and Hypothesis", ''Popular Science Monthly'', v. 13, pp. [https://books.google.com/books?id=u8sWAQAAIAAJ&jtp=470 470]–482. ''Internet Archive'' [https://archive.org/stream/popularsciencemo13newy#page/470/mode/1up Eprint].
*[[Charles Sanders Peirce|Peirce, C. S.]] (1883), "A Theory of probable inference", ''Studies in Logic'', pp. [https://books.google.com/books?id=V7oIAAAAQAAJ&pg=PA126 126-181], Little, Brown, and Company. (Reprinted 1983, [[John Benjamins Publishing Company]], {{ISBN|90-272-3271-7}})
* {{cite book |title=Parametric Statistical Theory |last1=Pfanzagl |first1=Johann |authorlink= <!-- Johann Pfanzagl --> |last2=with the assistance of R. Hamböker |year=1994 |publisher=[[Walter de Gruyter]] |location=Berlin |isbn = 3-11-013863-8 |mr = 1291393 }}
* {{cite book |authorlink=Jorma Rissanen |first=Jorma |last=Rissanen |year=1989 |title = Stochastic Complexity in Statistical Inquiry |location=Singapore |publisher=[[World Scientific]] |series = Series in Computer Science |volume=15 |mr=1082556 |isbn = 9971-5-0859-1 }}
* {{cite journal |title = Principal information-theoretic approaches (Vignettes for the Year 2000: Theory and Methods, ed. by George Casella) | last=Soofi |first=Ehsan S.|journal=[[Journal of the American Statistical Association]] |volume=95 |issue=452 |date=December 2000 |pages=1349–1353 |jstor=2669786 |mr=1825292 |doi = 10.1080/01621459.2000.10474346 }}
* {{cite book
|first1 = Joseph F.|last1= Traub|first2=G. W.|last2= Wasilkowski|first3=H. |last3=Wozniakowski
|authorlink1 = Joseph F. Traub
|year = 1988
|title = Information-Based Complexity
|publisher=Academic Press
|isbn = 0-12-697545-0 }}
* {{cite journal |doi = 10.1214/ss/1177011233 |title = R. A. Fisher and Fiducial Argument
|first=S. L.
|last=Zabell <!-- |authorlink=Sandy L. Zabell -->
|journal=Statistical Science
|volume=7
|issue=3
|date=Aug 1992
|pages=369–387
|jstor=2246073
}}
==Further reading==
<!-- should be additional reading, not cited in text -->
*[[George Casella|Casella, G.]], Berger, R.L. (2001). ''Statistical Inference''. Duxbury Press. {{isbn|0-534-24312-6}}
* {{cite journal | last1 = Freedman | first1 = D.A. | authorlink = David A. Freedman | year = 1991 | title = Statistical models and shoe leather | url = | journal = Sociological Methodology | volume = 21 | issue = | pages = 291–313 | doi=10.2307/270939| jstor = 270939 }}
* Held L., Bové D.S. (2014). ''Applied Statistical Inference—Likelihood and Bayes'' (Springer).
*{{cite journal | last1 = Lenhard | first1 = Johannes | year = 2006 | title = Models and Statistical Inference: the controversy between Fisher and Neyman–Pearson | url = http://www.stats.org.uk/statistical-inference/Lenhard2006.pdf| journal = [[British Journal for the Philosophy of Science]] | volume = 57 | issue = | pages = 69–91 | doi=10.1093/bjps/axi152}}
* {{cite journal | last1 = Lindley | first1 = D | year = 1958 | title = Fiducial distribution and Bayes' theorem | url = | journal = [[Journal of the Royal Statistical Society]], Series B | volume = 20 | issue = | pages = 102–7 }}
*Rahlf, Thomas (2014). "Statistical Inference", in Claude Diebolt, and Michael Haupert (eds.), "Handbook of Cliometrics ( Springer Reference Series)", Berlin/Heidelberg: Springer. http://www.springerreference.com/docs/html/chapterdbid/372458.html
*{{Cite journal | doi = 10.1111/insr.12067| title = On Some Principles of Statistical Inference| journal = International Statistical Review| volume = 83| issue = 2| pages = n/a| year = 2014| last1 = Reid | first1 = N. | last2 = Cox | first2 = D. R. }}
*Young, G.A., Smith, R.L. (2005). ''Essentials of Statistical Inference'', CUP. {{isbn|0-521-83971-8}}
== External links ==
{{Commons category|Statistical inference}}
{{Wikiversity}}
* MIT [http://dspace.mit.edu/handle/1721.1/45587 OpenCourseWare]: Statistical Inference
* [http://www.nptel.ac.in/courses/111105043/ NPTEL Statistical Inference], [https://www.youtube.com/playlist?list=PLbMVogVj5nJRkNUH5v9qNEJvW7r2A7rEY youtube link]
* [https://www.academia.edu/3247833/ Statistical induction and prediction]
{{-}}
{{Statistics|inference}}
{{Portal bar|Statistics}}
{{Authority control}}
{{DEFAULTSORT:Statistical Inference}}
[[Category:Statistical inference| ]]
[[Category:Inductive reasoning]]
[[Category:Deductive reasoning]]
<!-- [[Category:Abduction]] -->
[[Category:Logic and statistics]]
[[Category:Philosophy of science]]
[[Category:Psychometrics]]
[[de:Mathematische Statistik]]
'''Statistical inference''' is the process of using [[data analysis]] to deduce properties of an underlying [[probability distribution]].<ref name="Oxford">Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. {{isbn|978-0-19-954145-4}}</ref> Inferential statistical analysis infers properties of a [[Statistical population|population]], for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is [[Sampling (statistics)|sampled]] from a larger population.
Inferential statistics can be contrasted with [[descriptive statistics]]. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.
== Introduction ==
Statistical inference makes propositions about a population, using data drawn from the population with some form of [[Sampling (statistics)|sampling]]. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) [[Model selection|selecting]] a [[statistical model]] of the process that generates the data and (second) deducing propositions from the model.{{Citation needed|date=December 2016}}
Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling".<ref>Konishi & Kitagawa (2008), p. 75.</ref> Relatedly, [[David Cox (statistician)|Sir David Cox]] has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".<ref>Cox (2006), p. 197.</ref>
The [[Logical consequence|conclusion]] of a statistical inference is a statistical [[proposition]].{{Citation needed|date=February 2012}} Some common forms of statistical proposition are the following:
* a [[point estimate]], i.e. a particular value that best approximates some parameter of interest;
* an [[interval estimate]], e.g. a [[confidence interval]] (or set estimate), i.e. an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the [[frequency probability|probability]] at the stated [[confidence level]];
* a [[credible intervals|credible interval]], i.e. a set of values containing, for example, 95% of posterior belief;
* rejection of a [[statistical hypothesis testing|hypothesis]];{{efn|According to Peirce, acceptance means that inquiry on this question ceases for the time being. In science, all scientific theories are revisable.}}
* [[Cluster analysis|clustering]] or [[Statistical classification|classification]] of data points into groups.
==Models and assumptions==
{{Main|Statistical model|Statistical assumptions}}
Any statistical inference requires some assumptions. A '''statistical model''' is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference.<ref name=Cox2006>Cox (2006) page 2</ref> Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.<ref>{{cite book| last=Evans|first=Michael|title=Probability and Statistics: The Science of Uncertainty|year=2004|publisher=Freeman and Company|page=267|display-authors=etal|url=https://books.google.com/books?id=hkWK8kFzXWIC&printsec=frontcover#v=onepage&q=%22descriptive%20statistics%22&f=false|isbn=9780716747420}}</ref>
===Degree of models/assumptions===
Statisticians distinguish between three levels of modeling assumptions;
* '''[[Parametric model|Fully parametric]]''': The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters.<ref name=Cox2006/> For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by [[Simple random sample|'simple' random sampling]]. The family of [[Generalized linear model#Model components|generalized linear models]] is a widely used and flexible class of parametric models.
* '''[[Nonparametric statistics#Non-parametric models|Non-parametric]]''': The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal.<ref>van der Vaart, A.W. (1998) ''Asymptotic Statistics'' Cambridge University Press. {{isbn|0-521-78450-6}} (page 341)</ref> For example, every continuous probability distribution has a median, which may be estimated using the sample median or the [[Hodges–Lehmann estimator|Hodges–Lehmann–Sen estimator]], which has good properties when the data arise from simple random sampling.
* '''[[Semiparametric model|Semi-parametric]]''': This term typically implies assumptions 'in between' fully and non-parametric approaches. For example, one may assume that a population distribution has a finite mean. Furthermore, one may assume that the mean response level in the population depends in a truly linear manner on some covariate (a parametric assumption) but not make any parametric assumption describing the variance around that mean (i.e. about the presence or possible form of any [[heteroscedasticity]]). More generally, semi-parametric models can often be separated into 'structural' and 'random variation' components. One component is treated parametrically and the other non-parametrically. The well-known [[Cox model]] is a set of semi-parametric assumptions.
===Importance of valid models/assumptions===
Whatever level of assumption is made, correctly calibrated inference in general requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified.
Incorrect assumptions of [[Simple random sample|'simple' random sampling]] can invalidate statistical inference.<ref>Kruskal 1988</ref> More complex semi- and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions.<ref>
[[David A. Freedman|Freedman, D.A.]] (2008) "Survival analysis: An Epidemiological hazard?". ''The American Statistician'' (2008) 62: 110-119. (Reprinted as Chapter 11 (pages 169–192) of Freedman (2010)).</ref> Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference.<ref>Berk, R. (2003) ''Regression Analysis: A Constructive Critique (Advanced Quantitative Techniques in the Social Sciences) (v. 11)'' Sage Publications. {{isbn|0-7619-2904-5}}</ref> The use of '''any''' parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal."<ref name=Brewer>{{cite book|first=Ken |last=Brewer| title=Combined Survey Sampling Inference: Weighing of Basu's Elephants| publisher=Hodder Arnold|page=6|year= 2002|isbn=978-0340692295}}</ref> In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population."<ref name=Brewer/> Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy tailed.
====Approximate distributions====
{{Main|Statistical distance|Asymptotic theory (statistics)|Approximation theory}}
Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.
With finite samples, [[approximation theory|approximation results]] measure how close a limiting distribution approaches the statistic's [[sample distribution]]: For example, with 10,000 independent samples the [[normal distribution]] approximates (to two digits of accuracy) the distribution of the [[sample mean]] for many population distributions, by the [[Berry–Esseen theorem]].<ref name=JHJ>
Jörgen Hoffman-Jörgensen's ''Probability With a View Towards Statistics'', Volume I. Page 399 {{full citation needed|date=November 2012}}
</ref>
Yet for many practical purposes, the normal approximation provides a good approximation to the sample-mean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience.<ref name=JHJ/> Following Kolmogorov's work in the 1950s, advanced statistics uses [[approximation theory]] and [[functional analysis]] to quantify the error of approximation. In this approach, the [[metric geometry]] of [[probability distribution]]s is studied; this approach quantifies approximation error with, for example, the [[Kullback–Leibler divergence]], [[Bregman divergence]], and the [[Hellinger distance]].<ref>Le Cam (1986) {{page needed|date=June 2011}}</ref><ref>Erik Torgerson (1991) ''Comparison of Statistical Experiments'', volume 36 of Encyclopedia of Mathematics. Cambridge University Press. {{full citation needed|date=November 2012}}
</ref><ref>{{cite book
|author1=Liese, Friedrich |author2=Miescke, Klaus-J.
|lastauthoramp=yes | title = Statistical Decision Theory: Estimation, Testing, and Selection
| year = 2008
| publisher = Springer
| isbn = 0-387-73193-8
}}
</ref>
With indefinitely large samples, [[asymptotic theory (statistics)|limiting results]] like the [[central limit theorem]] describe the sample statistic's limiting distribution, if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.<ref>Kolmogorov (1963, p.369): "The frequency concept, <!-- comma missing in original --> based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials".</ref><ref>"Indeed, limit theorems 'as <math>n</math> tends to infinity' are logically devoid of content about what happens at any particular <math>n</math>. All they can do is suggest certain approaches whose performance must then be checked on the case at hand." — Le Cam (1986) (page xiv)</ref><ref>Pfanzagl (1994): "The crucial drawback of asymptotic theory: What we expect from asymptotic theory are results which hold approximately . . . . What asymptotic theory has to offer are limit theorems."(page ix) "What counts for applications are approximations, not limits." (page 188)
</ref> However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the [[generalized method of moments]] and the use of [[generalized estimating equation]]s, which are popular in [[econometrics]] and [[biostatistics]]. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation<!-- and approximation results -->.<ref>Pfanzagl (1994) : "By taking a limit theorem as being approximately true for large sample sizes, we commit an error the size of which is unknown. [. . .] Realistic information about the remaining errors may be obtained by simulations." (page ix)
</ref> The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional [[statistical model|models]] with [[logarithmically concave function|log-concave]] [[likelihood function|likelihood]]s (such as with one-parameter [[exponential families]]).
===Randomization-based models===
{{Main|Randomization}}
{{See also|Random sample|Random assignment}}
For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the null-hypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments.<ref>[[Jerzy Neyman|Neyman, J.]](1934) "On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection", ''[[Journal of the Royal Statistical Society]]'', 97 (4), 557–625 {{jstor|2342192}}</ref><ref name="Hinkelmann and Kempthorne">Hinkelmann and Kempthorne(2008) {{page needed|date=June 2011}}</ref> Statistical inference from randomized studies is also more straightforward than many other situations.<ref>ASA Guidelines for a first course in statistics for non-statisticians. (available at the ASA website)</ref><ref>[[David A. Freedman]] et alia's ''Statistics''.</ref><ref>Moore et al. (2015).</ref> In [[Bayesian inference]], randomization is also of importance: in [[survey sampling]], use of [[sampling without replacement]] ensures the [[exchangeability]] of the sample with the population; in randomized experiments, randomization warrants a [[missing at random]] assumption for [[covariate]] information.<ref>[[Andrew Gelman|Gelman A.]] et al. (2013). ''Bayesian Data Analysis'' ([[Chapman & Hall]]).</ref>
Objective randomization allows properly inductive procedures.<ref>Peirce (1877-1878)</ref><ref>Peirce (1883)</ref><ref>
[[David A. Freedman|David Freedman]] et alia ''Statistics'' and [[David A. Freedman]] ''Statistical Models''.</ref><ref>
[[C. R. Rao|Rao, C.R.]] (1997) ''Statistics and Truth: Putting Chance to Work'', World Scientific. {{isbn|981-02-3111-3}}</ref>
Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures.<ref>Peirce; Freedman; Moore et al. (2015).{{Citation needed|date=March 2010}}</ref> (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.<ref>Box, G.E.P. and Friends (2006) ''Improving Almost Anything: Ideas and Essays, Revised Edition'', Wiley. {{isbn|978-0-471-72755-2}}</ref><ref>
Cox (2006), page 196</ref>)
Similarly, results from [[randomized experiment]]s are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena.<ref>ASA Guidelines for a first course in statistics for non-statisticians. (available at the ASA website)
* David A. Freedman et alia's ''Statistics''.
* Moore et al. (2015).
</ref>
However, a good observational study may be better than a bad randomized experiment.
The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.<ref>Neyman, Jerzy. 1923 [1990]. "On the Application of Probability Theory to AgriculturalExperiments. Essay on Principles. Section 9." ''Statistical Science'' 5 (4): 465–472. Trans. [[Dorota Dabrowska|Dorota M. Dabrowska]] and Terence P. Speed.</ref><ref>Hinkelmann & Kempthorne (2008) {{page needed|date=June 2011}}</ref>
However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.
====Model-based analysis of randomized experiments====
It is standard practice to refer to a statistical model, often a linear model, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme.<ref name="Hinkelmann and Kempthorne"/> Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.<ref>Hinkelmann and Kempthorne (2008) Chapter 6.</ref>
==Paradigms for inference==
Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.
Bandyopadhyay & Forster<ref>Bandyopadhyay & Forster (2011). The quote is taken from the book's Introduction (p.3). See also "Section III: Four Paradigms of Statistics".</ref> describe four paradigms: "(i) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics". The classical (or [[Frequentist inference|frequentist]]) paradigm, the [[Bayesian inference|Bayesian]] paradigm, and the [[Akaike information criterion|AIC]]-based paradigm are summarized below. The likelihood-based paradigm is essentially a sub-paradigm of the AIC-based paradigm.
===Frequentist inference===
{{See also|Frequentist inference}}
This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging.
====Examples of frequentist inference====
* [[p-value|''p''-value]]
* [[Confidence interval]]
====Frequentist inference, objectivity, and decision theory====
One interpretation of [[frequentist inference]] (or classical inference) is that it is applicable only in terms of [[frequency probability]]; that is, in terms of repeated sampling from a population. However, the approach of Neyman<ref>{{cite journal | last1 = Neyman | first1 = J. | authorlink = Jerzy Neyman | year = 1937 | title = Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability | jstor = 91337 | journal = Philosophical Transactions of the Royal Society of London A | volume = 236 | issue = 767| pages = 333–380 | doi=10.1098/rsta.1937.0005}}</ref> develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e. probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach.
The frequentist procedures of significance testing and confidence intervals can be constructed without regard to [[utility function]]s. However, some elements of frequentist statistics, such as [[statistical decision theory]], do incorporate [[utility function]]s.{{Citation needed|date=April 2012}} In particular, frequentist developments of optimal inference (such as [[minimum-variance unbiased estimator]]s, or [[uniformly most powerful test]]ing) make use of [[loss function]]s, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property.<ref>Preface to Pfanzagl.</ref> However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under [[absolute value]] loss functions, in that they minimize expected loss, and [[least squares]] estimators are optimal under squared error loss functions, in that they minimize expected loss.
While statisticians using frequentist inference must choose for themselves the parameters of interest, and the [[estimators]]/[[Test statistic#Common test statistics|test statistic]] to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.{{Citation needed|date=April 2012}}
===Bayesian inference===
{{See also|Bayesian Inference}}
The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate to one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are [[Bayesian probability#Justification of Bayesian probabilities|several different justifications]] for using the Bayesian approach.
====Examples of Bayesian inference====
* [[Credible interval]] for [[interval estimation]]
* [[Bayes factor]]s for model comparison
====Bayesian inference, subjectivity and decision theory====
Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's [[utility function]] need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been [[Bayesian probability#Personal probabilities and objective methods for constructing priors|proposed]] but not yet fully developed.)
Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides [[optimal decision]]s in a [[decision theory|decision theoretic]] sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) [[Coherence (statistics)|incoherent]]; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be [[Coherence (statistics)|coherent]]. Some advocates of [[Bayesian inference]] assert that inference ''must'' take place in this decision-theoretic framework, and that [[Bayesian inference]] should not conclude with the evaluation and summarization of posterior beliefs.
===AIC-based inference===
{{Main|Akaike information criterion}}
{{expand section|date=November 2017}}
The ''[[Akaike information criterion]]'' (AIC) is an [[estimator]] of the relative quality of [[statistical model]]s for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for [[model selection]].
AIC is founded on [[information theory]]: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the [[goodness of fit]] of the model and the simplicity of the model.)
===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.
==Inference topics==
The topics below are usually included in the area of '''statistical inference'''.
#[[Statistical assumptions]]
#[[Statistical decision theory]]
#[[Estimation theory]]
#[[Statistical hypothesis testing]]
#[[Revising opinions in statistics]]
#[[Design of experiments]], the [[analysis of variance]], and [[Regression analysis|regression]]
#[[Survey sampling]]
#[[Summarizing statistical data]]
==See also==
* [[Algorithmic inference]]
* [[Induction (philosophy)]]
* [[Informal inferential reasoning]]
* [[Population proportion]]
* [[Philosophy of statistics]]
* [[Predictive inference]]
* [[Information field theory]]
== Notes ==
{{Notelist}}
==Citations==
{{Reflist}}
==References==
* {{Citation |editor1-first= P. S. |editor1-last = Bandyopadhyay |editor2-first= M. R. |editor2-last = Forster | title= Philosophy of Statistics |year= 2011 |publisher = [[Elsevier]] }}.
* {{cite book |last1=Bickel |first1=Peter J. |last2 = Doksum |first2 = Kjell A. |title = Mathematical statistics: Basic and selected topics |volume = 1 |edition = Second (updated printing 2007) |year = 2001 |publisher = [[Prentice Hall]] |mr = 443141 |isbn = 0-13-850363-X }}
* [[David R. Cox|Cox, D. R.]] (2006). ''Principles of Statistical Inference'', [[Cambridge University Press]]. {{ISBN|0-521-68567-2}}.
* [[Ronald A. Fisher|Fisher, R. A.]] (1955), "Statistical methods and scientific induction", ''[[Journal of the Royal Statistical Society]], Series B'', 17, 69—78. (criticism of statistical theories of [[Jerzy Neyman]] and [[Abraham Wald]])
* {{cite book |authorlink=David A. Freedman |last=Freedman |first=D. A. |title = Statistical Models: Theory and practice |edition=revised |year=2009 |publisher=[[Cambridge University Press]] |pages = xiv+442 pp.
|isbn = 978-0-521-74385-3 |mr = 2489600 }}
* [[David A. Freedman|Freedman, D. A.]] (2010). ''Statistical Models and Causal Inferences: A Dialogue with the Social Sciences'' (Edited by David Collier, Jasjeet S. Sekhon, and Philip B. Stark), [[Cambridge University Press]].
* {{cite journal |last1=Hampel |first1=Frank |title = The proper fiducial argument |date=Feb 2003 |issue=Research Report No. 114 |url = http://e-collection.library.ethz.ch/eserv/eth:26403/eth-26403-01.pdf |accessdate=29 March 2016 |ref = harv }}
* {{cite journal |doi=10.1198/016214501753168398 |title=Model Selection and the Principle of Minimum Description Length: Review paper |first1=Mark H. |last1=Hansen |first2=Bin |last2=Yu |author2-link=Bin Yu |journal=[[Journal of the American Statistical Association]] |volume=96 |issue=454 |date=June 2001 |pages=746–774 |jstor=2670311 |mr=1939352 |url=http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps |archive-url=https://web.archive.org/web/20041116080440/http://www.stat.berkeley.edu/webmastr/users/binyu/ps/mdl.ps |dead-url=yes |archive-date=2004-11-16 }}
* {{cite book
|last1 = Hinkelmann |first1=Klaus |last2=Kempthorne |first2=Oscar |authorlink2=Oscar Kempthorne
|year = 2008
|title = Introduction to Experimental Design
|url = https://books.google.com/?id=T3wWj2kVYZgC&printsec=frontcover
|edition = Second
|publisher = Wiley
|isbn = 978-0-471-72756-9
}}
* {{cite journal |authorlink=Andrei N. Kolmogorov|first=Andrei N.|last=Kolmogorov |year=1963 |title = On tables of random numbers |journal=[[Sankhya (journal)|Sankhyā]] Ser. A.|volume=25 |pages=369–375 |mr=178484 }} Reprinted as {{cite journal |first=Andrei N. |last=Kolmogorov |year=1998 |title=On tables of random numbers |journal = [[Theoretical Computer Science (journal) |Theoretical Computer Science]] |volume=207 |issue=2 |pages = 387–395 |doi = 10.1016/S0304-3975(98)00075-9 |mr = 1643414 }}
* Konishi S., Kitagawa G. (2008), ''Information Criteria and Statistical Modeling'', Springer.
* {{cite journal
|title = Miracles and statistics: the casual assumption of independence (ASA Presidential Address)
|authorlink = William Kruskal
|first = William
|last = Kruskal
|journal = [[Journal of the American Statistical Association]]
|volume = 83
|issue = 404
|date = December 1988
|pages = 929–940
|jstor = 2290117
|doi = 10.2307/2290117
}}
* [[Lucien Le Cam|Le Cam, Lucian]]. (1986) ''Asymptotic Methods of Statistical Decision Theory'', Springer. {{isbn|0-387-96307-3}}
* [[David S. Moore|Moore, D. S.]]; McCabe, G. P.; Craig, B. A. (2015), ''Introduction to the Practice of Statistics'', Eighth Edition, Macmillan.
* {{cite journal
|title = Note on an article by Sir Ronald Fisher
|authorlink=Jerzy Neyman
|first=Jerzy
|last=Neyman
|journal=[[Journal of the Royal Statistical Society, Series B]]
|volume=18
|issue=2
|year=1956
|pages=288–294
|jstor=2983716 }} (reply to Fisher 1955)
* [[Charles Sanders Peirce|Peirce, C. S.]] (1877–1878), "Illustrations of the logic of science" (series), ''[[Popular Science Monthly]]'', vols. 12-13. Relevant individual papers:
** (1878 March), "The Doctrine of Chances", ''Popular Science Monthly'', v. 12, March issue, pp. [https://books.google.com/books?id=ZKMVAAAAYAAJ&jtp=604 604]–615. ''Internet Archive'' [https://archive.org/stream/popscimonthly12yoummiss#page/612/mode/1up Eprint].
** (1878 April), "The Probability of Induction", ''Popular Science Monthly'', v. 12, pp. [https://books.google.com/books?id=ZKMVAAAAYAAJ&jtp=705 705]–718. ''Internet Archive'' [https://archive.org/stream/popscimonthly12yoummiss#page/715/mode/1up Eprint].
** (1878 June), "The Order of Nature", ''Popular Science Monthly'', v. 13, pp. [https://books.google.com/books?id=u8sWAQAAIAAJ&jtp=203 203]–217.''Internet Archive'' [https://archive.org/stream/popularsciencemo13newy#page/203/mode/1up Eprint].
** (1878 August), "Deduction, Induction, and Hypothesis", ''Popular Science Monthly'', v. 13, pp. [https://books.google.com/books?id=u8sWAQAAIAAJ&jtp=470 470]–482. ''Internet Archive'' [https://archive.org/stream/popularsciencemo13newy#page/470/mode/1up Eprint].
*[[Charles Sanders Peirce|Peirce, C. S.]] (1883), "A Theory of probable inference", ''Studies in Logic'', pp. [https://books.google.com/books?id=V7oIAAAAQAAJ&pg=PA126 126-181], Little, Brown, and Company. (Reprinted 1983, [[John Benjamins Publishing Company]], {{ISBN|90-272-3271-7}})
* {{cite book |title=Parametric Statistical Theory |last1=Pfanzagl |first1=Johann |authorlink= <!-- Johann Pfanzagl --> |last2=with the assistance of R. Hamböker |year=1994 |publisher=[[Walter de Gruyter]] |location=Berlin |isbn = 3-11-013863-8 |mr = 1291393 }}
* {{cite book |authorlink=Jorma Rissanen |first=Jorma |last=Rissanen |year=1989 |title = Stochastic Complexity in Statistical Inquiry |location=Singapore |publisher=[[World Scientific]] |series = Series in Computer Science |volume=15 |mr=1082556 |isbn = 9971-5-0859-1 }}
* {{cite journal |title = Principal information-theoretic approaches (Vignettes for the Year 2000: Theory and Methods, ed. by George Casella) | last=Soofi |first=Ehsan S.|journal=[[Journal of the American Statistical Association]] |volume=95 |issue=452 |date=December 2000 |pages=1349–1353 |jstor=2669786 |mr=1825292 |doi = 10.1080/01621459.2000.10474346 }}
* {{cite book
|first1 = Joseph F.|last1= Traub|first2=G. W.|last2= Wasilkowski|first3=H. |last3=Wozniakowski
|authorlink1 = Joseph F. Traub
|year = 1988
|title = Information-Based Complexity
|publisher=Academic Press
|isbn = 0-12-697545-0 }}
* {{cite journal |doi = 10.1214/ss/1177011233 |title = R. A. Fisher and Fiducial Argument
|first=S. L.
|last=Zabell <!-- |authorlink=Sandy L. Zabell -->
|journal=Statistical Science
|volume=7
|issue=3
|date=Aug 1992
|pages=369–387
|jstor=2246073
}}
==Further reading==
<!-- should be additional reading, not cited in text -->
*[[George Casella|Casella, G.]], Berger, R.L. (2001). ''Statistical Inference''. Duxbury Press. {{isbn|0-534-24312-6}}
* {{cite journal | last1 = Freedman | first1 = D.A. | authorlink = David A. Freedman | year = 1991 | title = Statistical models and shoe leather | url = | journal = Sociological Methodology | volume = 21 | issue = | pages = 291–313 | doi=10.2307/270939| jstor = 270939 }}
* Held L., Bové D.S. (2014). ''Applied Statistical Inference—Likelihood and Bayes'' (Springer).
*{{cite journal | last1 = Lenhard | first1 = Johannes | year = 2006 | title = Models and Statistical Inference: the controversy between Fisher and Neyman–Pearson | url = http://www.stats.org.uk/statistical-inference/Lenhard2006.pdf| journal = [[British Journal for the Philosophy of Science]] | volume = 57 | issue = | pages = 69–91 | doi=10.1093/bjps/axi152}}
* {{cite journal | last1 = Lindley | first1 = D | year = 1958 | title = Fiducial distribution and Bayes' theorem | url = | journal = [[Journal of the Royal Statistical Society]], Series B | volume = 20 | issue = | pages = 102–7 }}
*Rahlf, Thomas (2014). "Statistical Inference", in Claude Diebolt, and Michael Haupert (eds.), "Handbook of Cliometrics ( Springer Reference Series)", Berlin/Heidelberg: Springer. http://www.springerreference.com/docs/html/chapterdbid/372458.html
*{{Cite journal | doi = 10.1111/insr.12067| title = On Some Principles of Statistical Inference| journal = International Statistical Review| volume = 83| issue = 2| pages = n/a| year = 2014| last1 = Reid | first1 = N. | last2 = Cox | first2 = D. R. }}
*Young, G.A., Smith, R.L. (2005). ''Essentials of Statistical Inference'', CUP. {{isbn|0-521-83971-8}}
== External links ==
{{Commons category|Statistical inference}}
{{Wikiversity}}
* MIT [http://dspace.mit.edu/handle/1721.1/45587 OpenCourseWare]: Statistical Inference
* [http://www.nptel.ac.in/courses/111105043/ NPTEL Statistical Inference], [https://www.youtube.com/playlist?list=PLbMVogVj5nJRkNUH5v9qNEJvW7r2A7rEY youtube link]
* [https://www.academia.edu/3247833/ Statistical induction and prediction]
{{-}}
{{Statistics|inference}}
{{Portal bar|Statistics}}
{{Authority control}}
{{DEFAULTSORT:Statistical Inference}}
[[Category:Statistical inference| ]]
[[Category:Inductive reasoning]]
[[Category:Deductive reasoning]]
<!-- [[Category:Abduction]] -->
[[Category:Logic and statistics]]
[[Category:Philosophy of science]]
[[Category:Psychometrics]]
[[de:Mathematische Statistik]]
