Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is 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.
Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.
Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, 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".
- 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 probability at the stated confidence level;
- a credible interval, i.e. a set of values containing, for example, 95% of posterior belief;
- rejection of a hypothesis;[note 1]
- clustering or classification of data points into groups.
Models and assumptionsEdit
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. Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.
Degree of models/assumptionsEdit
Statisticians distinguish between three levels of modeling assumptions;
- 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. 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 sampling. The family of generalized linear models is a widely used and flexible class of parametric models.
- Non-parametric: The assumptions made about the process generating the data are much less than in parametric statistics and may be minimal. For example, every continuous probability distribution has a median, which may be estimated using the sample median or the Hodges–Lehmann–Sen estimator, which has good properties when the data arise from simple random sampling.
- 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/assumptionsEdit
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 sampling can invalidate statistical inference. 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. Incorrect assumptions of Normality in the population also invalidates some forms of regression-based inference. 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." 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." 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.
Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.
With finite samples, 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. 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. 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 distributions is studied; this approach quantifies approximation error with, for example, the Kullback–Leibler divergence, Bregman divergence, and the Hellinger distance.
With indefinitely large samples, 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. 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 equations, 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. The heuristic application of limiting results to finite samples is common practice in many applications, especially with low-dimensional models with log-concave likelihoods (such as with one-parameter exponential families).
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. Statistical inference from randomized studies is also more straightforward than many other situations. 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.
Objective randomization allows properly inductive procedures. Many statisticians prefer randomization-based analysis of data that was generated by well-defined randomization procedures. (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.) Similarly, results from randomized experiments are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena. However, a good observational study may be better than a bad randomized experiment.
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 experimentsEdit
It is standard practice to refer to a statistical model, e.g., a linear or logistic models, 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. 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.
Model-free randomization inferenceEdit
Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of reality-simplification. The former combine, evolve, ensemble and train algorithms dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations. 
For example, model-free simple linear regression is based either on
- a random design, where the pairs of observations are independent and identically distributed (iid), or
- a deterministic design, where the variables are deterministic, but the corresponding response variables are random and independent with a common conditional distribution, i.e., , which is independent of the index .
In either case, the model-free randomization inference for features of the common conditional distribution relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature conditional mean, , can be consistently estimated via local averaging or local polynomial fitting, under the assumption that is smooth. The conditional mean can be consistently estimated as a local average and/or by fitting a local polynomial. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the conditional mean, . 
Paradigms for inferenceEdit
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 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) paradigm, the Bayesian paradigm, the likelihoodist paradigm, and the AIC-based paradigm are summarized below.
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 inferenceEdit
Frequentist inference, objectivity, and decision theoryEdit
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 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 functions. However, some elements of frequentist statistics, such as statistical decision theory, do incorporate utility functions. In particular, frequentist developments of optimal inference (such as minimum-variance unbiased estimators, or uniformly most powerful testing) make use of loss functions, 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. 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 to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.
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 several different justifications for using the Bayesian approach.
Examples of Bayesian inferenceEdit
Bayesian inference, subjectivity and decision theoryEdit
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 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 decisions in a 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) incoherent; a feature of Bayesian procedures which use proper priors (i.e. those integrable to one) is that they are guaranteed to be 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.
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Likelihoodism approaches statistics by using the likelihood function. Some likelihoodists reject inference, considering statistics as only computing support from evidence. Others, however, propose inference based on the likelihood function, of which the best-known is maximum likelihood estimation.
This section needs expansion. You can help by adding to it. (November 2017)
The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models 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 inferenceEdit
Minimum description lengthEdit
The minimum description length (MDL) principle has been developed from ideas in information theory and the theory of Kolmogorov complexity. 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 Shannon's source coding theorem it provides the MDL description of the data, on average and asymptotically. In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy 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.
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. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments. An attempt was made to reinterpret the early work of Fisher's fiducial argument as a special case of an inference theory using Upper and lower probabilities.
Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. Barnard developed "structural inference" or "pivotal inference", an approach using invariant probabilities on 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.
The topics below are usually included in the area of statistical inference.
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- Upton, G., Cook, I. (2008) Oxford Dictionary of Statistics, OUP. ISBN 978-0-19-954145-4.
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- Cox (2006) page 2
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- van der Vaart, A.W. (1998) Asymptotic Statistics Cambridge University Press. ISBN 0-521-78450-6 (page 341)
- Kruskal 1988
- 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)).
- Berk, R. (2003) Regression Analysis: A Constructive Critique (Advanced Quantitative Techniques in the Social Sciences) (v. 11) Sage Publications. ISBN 0-7619-2904-5
- Brewer, Ken (2002). Combined Survey Sampling Inference: Weighing of Basu's Elephants. Hodder Arnold. p. 6. ISBN 978-0340692295.
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- "Indeed, limit theorems 'as tends to infinity' are logically devoid of content about what happens at any particular . All they can do is suggest certain approaches whose performance must then be checked on the case at hand." — Le Cam (1986) (page xiv)
- 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)
- 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)
- 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
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- Peirce (1877-1878)
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- Cox (2006), p. 196.
- 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).
- Neyman, Jerzy. 1923 . "On the Application of Probability Theory to AgriculturalExperiments. Essay on Principles. Section 9." Statistical Science 5 (4): 465–472. Trans. Dorota M. Dabrowska and Terence P. Speed.
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- Hinkelmann and Kempthorne (2008) Chapter 6.
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- Politis, D.N. (2019). "Model-free inference in statistics: how and why". IMS Bulletin. 48.
- Bandyopadhyay & Forster (2011). The quote is taken from the book's Introduction (p.3). See also "Section III: Four Paradigms of Statistics".
- Neyman, J. (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society of London A. 236 (767): 333–380. doi:10.1098/rsta.1937.0005. JSTOR 91337.
- Preface to Pfanzagl.
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- Soofi (2000)
- Hansen & Yu (2001)
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- Rissanen (1989), page 84
- Joseph F. Traub, G. W. Wasilkowski, and H. Wozniakowski. (1988)[page needed]
- Neyman (1956)
- Zabell (1992)
- Cox (2006) page 66
- Hampel 2003.
- Davison, page 12.[full citation needed]
- Barnard, G.A. (1995) "Pivotal Models and the Fiducial Argument", International Statistical Review, 63 (3), 309–323. JSTOR 1403482
- Bandyopadhyay, P. S.; Forster, M. R., eds. (2011), Philosophy of Statistics, Elsevier.
- Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical statistics: Basic and selected topics. 1 (Second (updated printing 2007) ed.). Prentice Hall. ISBN 978-0-13-850363-5. MR 0443141.
- Cox, D. R. (2006). Principles of Statistical Inference, Cambridge University Press. ISBN 0-521-68567-2.
- 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)
- Freedman, D. A. (2009). Statistical Models: Theory and practice (revised ed.). Cambridge University Press. pp. xiv+442 pp. ISBN 978-0-521-74385-3. MR 2489600.
- 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.
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- Moore, D. S.; McCabe, G. P.; Craig, B. A. (2015), Introduction to the Practice of Statistics, Eighth Edition, Macmillan.
- Neyman, Jerzy (1956). "Note on an article by Sir Ronald Fisher". Journal of the Royal Statistical Society, Series B. 18 (2): 288–294. JSTOR 2983716. (reply to Fisher 1955)
- 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. 604–615. Internet Archive Eprint.
- (1878 April), "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705–718. Internet Archive Eprint.
- (1878 June), "The Order of Nature", Popular Science Monthly, v. 13, pp. 203–217.Internet Archive Eprint.
- (1878 August), "Deduction, Induction, and Hypothesis", Popular Science Monthly, v. 13, pp. 470–482. Internet Archive Eprint.
- Peirce, C. S. (1883), "A Theory of probable inference", Studies in Logic, pp. 126-181, Little, Brown, and Company. (Reprinted 1983, John Benjamins Publishing Company, ISBN 90-272-3271-7)
- Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Berlin: Walter de Gruyter. ISBN 978-3-11-013863-4. MR 1291393.
- Rissanen, Jorma (1989). Stochastic Complexity in Statistical Inquiry. Series in Computer Science. 15. Singapore: World Scientific. ISBN 978-9971-5-0859-3. MR 1082556.
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