Statistical inference is the process of using

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

*Charles Sanders Peirce, 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, ) * * * * * *

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Statistical Inference

NPTEL Statistical Inference

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Statistical induction and prediction

{{Authority control Statistical inference, Inductive reasoning Deductive reasoning Logic and statistics Philosophy of science Psychometrics de:Mathematische Statistik

data analysis
Data analysis is a process of inspecting, cleansing, transforming, and modelling
In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century Eng ...

to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population
Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...

, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled
Sample or samples may refer to:
Base meaning
* Sample (statistics)
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sci ...

from a larger population.
Inferential statistics can be contrasted with descriptive statistics
A descriptive statistic (in the count noun
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The trad ...

. 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. In machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); see also predictive inference Predictive inference is an approach to statistical inference that emphasizes the prediction of future observations based on past observations.
Initially, predictive inference was based on ''observable'' parameters and it was the main purpose of stu ...

.
Introduction

Statistical inference makes propositions about a population, using data drawn from the population with some form ofsampling
Sampling may refer to:
*Sampling (signal processing), converting a continuous signal into a discrete signal
*Sample (graphics), Sampling (graphics), converting continuous colors into discrete color components
*Sampling (music), the reuse of a sound ...

. Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model
A statistical model is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system ...

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 translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".
The conclusion of a statistical inference is a statistical proposition
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...

. Some common forms of statistical proposition are the following:
* a point estimate In statistics, point estimation involves the use of statistical sample, sample data to calculate a single value (known as a point estimate since it identifies a Point (geometry), point in some parameter space) which is to serve as a "best guess" or ...

, i.e. a particular value that best approximates some parameter of interest;
* an interval estimateIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

, e.g. a confidence interval
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

(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
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

at the stated confidence level
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

;
* a credible interval
In Bayesian statistics, a credible interval is an interval within which an unobserved parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''para'': "beside", "subsidiary"; and wikt:μέ ...

, i.e. a set of values containing, for example, 95% of posterior belief;
* rejection of a hypothesis
A hypothesis (plural hypotheses) is a proposed explanation
An explanation is a set of statements usually constructed to describe a set of facts which clarifies the causes, context
Context may refer to:
* Context (language use), the rel ...

;
* clustering or classification Classification is a process related to categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience
Experience refers to conscious
, an English Paracels ...

of data points into groups.
Models and 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.Cox (2006) page 2 Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.Degree of models/assumptions

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-parametricNonparametric statistics is the branch of statistics that is not based solely on Statistical parameter, parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on ...

: 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
In statistics, a vector of random variables is heteroscedastic (or heteroskedastic; from Ancient Greek "different" and "dispersion") if the variability of the Errors and residuals, random disturbance is different across elements of the vector. H ...

). 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 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 stimatorsbased on very large samples, where the central limit theorem ensures that these stimatorswill 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.Approximate distributions

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 thenormal distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

approximates (to two digits of accuracy) the distribution of the sample mean
The sample mean (or "empirical mean") and the sample covariance are statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes incl ...

for many population distributions, by the Berry–Esseen theorem.
Jörgen Hoffman-Jörgensen's ''Probability With a View Towards Statistics'', Volume I. Page 399
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
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler function (mathematics), functions, and with Quantitative property, quantitatively characterization (ma ...

and functional analysis
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...

to quantify the error of approximation. In this approach, the metric geometry
In mathematics, a metric space is a non empty Set (mathematics), set together with a Metric (mathematics)#Definition, metric on the set. The metric is a function (mathematics), function that defines a concept of ''distance'' between any two Elemen ...

of probability distribution
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

s is studied; this approach quantifies approximation error with, for example, the Kullback–Leibler divergence
In mathematical statistics, the Kullback–Leibler divergence, D_\text(P \parallel Q) (also called relative entropy), is a statistical distance: a measure of how one probability distribution ''Q'' is different from a second, reference probability ...

, Bregman divergence, and the Hellinger distanceHellinger is a surname. Notable people with the surname include:
*Bert Hellinger (1925–2019), German psychotherapist
*Ernst Hellinger (1883–1950), German mathematician
**Hellinger distance, used to quantify the similarity between two probabilit ...

.
With indefinitely large samples, limiting results like the central limit theorem
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in ...

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 momentsIn econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dic ...

and the use of generalized estimating equation
In statistics, a generalized estimating equation (GEE) is used to Estimator, estimate the parameters of a generalized linear model with a possible unknown Correlation and dependence, correlation between outcomes.
Parameter estimates from the GEE ar ...

s, which are popular in econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Econ ...

and biostatistics
Biostatistics (also known as biometry) are the development and application of statistical
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data are units of in ...

. 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 likelihood
In statistics, the likelihood function (often simply called the likelihood) measures the goodness of fit of a statistical model to a Sample (statistics), sample of data for given values of the unknown Statistical parameter, parameters. It is formed ...

s (such as with one-parameter exponential families
In theory of probability, probability and statistics, an exponential family is a parametric model, parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, based on ...

).
Randomization-based models

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, the 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.Hinkelmann and Kempthorne(2008) Statistical inference from randomized studies is also more straightforward than many other situations. InBayesian inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and e ...

, randomization is also of importance: in survey samplingIn statistics, survey sampling describes the process of selecting a sample of elements from a target statistical population, population to conduct a survey.
The term "Statistical survey, survey" may refer to many different types or techniques of obs ...

, use of sampling without replacement ensures the exchangeabilityIn statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when ...

of the sample with the population; in randomized experiments, randomization warrants a missing at random assumption for covariate
Dependent and independent variables are variables in mathematical modeling
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to f ...

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 experiment
In science
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is something that is truth, true. The usual test for a ...

s 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.
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.
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, 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 inference

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 $(X\_1,Y\_1),\; (X\_2,Y\_2),\; \backslash cdots\; ,\; (X\_n,Y\_n)$ are independent and identically distributed (iid), or * a ''deterministic design'', where the variables $X\_1,\; X\_2,\; \backslash cdots,\; X\_n$ are deterministic, but the corresponding response variables $Y\_1,Y\_2,\; \backslash cdots,\; Y\_n$ are random and independent with a common conditional distribution, i.e., $P\backslash left\; (Y\_j\; \backslash leq\; y\; ,\; X\_j\; =x\backslash right\; )\; =\; D\_x(y)$, which is independent of the index $j$. In either case, the model-free randomization inference for features of the common conditional distribution $D\_x(.)$ relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature ''conditional mean'', $\backslash mu(x)=E(Y\; ,\; X\; =\; x)$, can be consistently estimated via local averaging or local polynomial fitting, under the assumption that $\backslash mu(x)$ is smooth. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the ''conditional mean'', $\backslash mu(x)$.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 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 (orfrequentist
Frequentist inference is a type of statistical inference
Statistical inference is the process of using data analysis
Data analysis is a process of inspecting, Data cleansing, cleansing, Data transformation, transforming, and Data modeling, ...

) paradigm, the Bayesian
Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister.
Bayesian () refers to a range of concepts and approaches that are ultimately based on a degree-of-belief interpretation of probability, ...

paradigm, the likelihoodist paradigm, and the AIC-based paradigm are summarized below.
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 *Confidence interval
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

* Null hypothesis significance testing
Frequentist inference, objectivity, and decision theory

One interpretation offrequentist inferenceFrequentist inference is a type of statistical inference that draws conclusions from sample data by emphasizing the frequency or proportion of the data. An alternative name is frequentist statistics. This is the inference framework in which the wel ...

(or classical inference) is that it is applicable only in terms of frequency probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of a sequence, limit of its relative frequency in many trials. Probabilities can be found (in principle) by a repeatable o ...

; 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 function
As a topic of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...

s. However, some elements of frequentist statistics, such as statistical decision theory, do incorporate utility function
As a topic of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...

s. In particular, frequentist developments of optimal inference (such as minimum-variance unbiased estimators, or uniformly most powerful testing) make use of loss function In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimizat ...

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. However, loss-functions are often useful for stating optimality properties: for example, median-unbiased estimators are optimal under absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

loss functions, in that they minimize expected loss, and least squares
The method of least squares is a standard approach in regression analysis
In ing, regression analysis is a set of statistical processes for the relationships between a (often called the 'outcome' or 'response' variable) and one or more s ...

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
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...

/test statistic
A test statistic is a statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population
Population t ...

to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.
Bayesian inference

The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate into 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 inference

*Credible interval
In Bayesian statistics, a credible interval is an interval within which an unobserved parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''para'': "beside", "subsidiary"; and wikt:μέ ...

for interval estimationIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

* Bayes factor
In statistics, the use of Bayes factors is a Bayesian probability, Bayesian alternative to classical Hypothesis Testing, hypothesis testing. Bayesian model comparison is a method of model selection based on Bayes factors. The models under cons ...

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'sutility function
As a topic of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...

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 decision
An optimal decision is a decision that leads to at least as good a known or expected outcome as all other available decision options. It is an important concept in decision theory
Decision theory (or the theory of choice not to be confused with cho ...

s 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
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and e ...

assert that inference ''must'' take place in this decision-theoretic framework, and that Bayesian inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and e ...

should not conclude with the evaluation and summarization of posterior beliefs.
Likelihood-based inference

Likelihoodism approaches statistics by using thelikelihood function
The likelihood function (often simply called the likelihood) describes the joint probability
Given random variables X,Y,\ldots, that are defined on a probability space, the joint probability distribution for X,Y,\ldots is a probability distribut ...

. 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
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. T ...

.
AIC-based inference

The ''Akaike information criterion
The Akaike information criterion (AIC) is an estimator
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...

'' (AIC) is an estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguishe ...

of the relative quality of statistical model
A statistical model is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system ...

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
Model selection is the task of selecting a statistical model
A statistical model is a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act ...

.
AIC is founded on information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...

: 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
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures ...

of the model and the simplicity of the model.)
Other paradigms for inference

Minimum description length

The minimum description length (MDL) principle has been developed from ideas ininformation theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...

Soofi (2000) and the theory of Kolmogorov complexity
In algorithmic information theory
Algorithmic information theory (AIT) is a branch of theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science
Computer science deals with the theoretical ...

.Hansen & Yu (2001) 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
In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy.
Named after Claude Shannon, the source coding theorem ...

it provides the MDL description of the data, on average and asymptotically.Hansen and Yu (2001), page 747. In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. T ...

and maximum a posteriori estimation
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution
In Bayesian statistics, the posterior probability of a random event or an u ...

(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.Rissanen (1989), page 84
The MDL principle has been applied in communication-coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

in information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...

, in linear regression
In statistics, linear regression is a Linearity, linear approach for modelling the relationship between a Scalar (mathematics), scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of ...

, and in data mining
Data mining is a process of extracting and discovering patterns in large data set
A data set (or dataset) is a collection of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

.
The evaluation of MDL-based inferential procedures often uses techniques or criteria from computational complexity theory
Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by ...

.
Fiducial inference

Fiducial inference
Fiducial inference is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with f ...

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
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

and, since this has not invalidated the application of confidence interval
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

s, 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 probabilitiesUpper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the even ...

.
Structural inference

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. Donald A. S. Fraser developed a general theory for structural inference based ongroup theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and applied this to linear models. The theory formulated by Fraser has close links to decision theory and Bayesian statistics and can provide optimal frequentist decision rules if they exist.
Inference topics

The topics below are usually included in the area of statistical inference. #Statistical assumptionsStatistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a st ...

# Statistical decision theory
#Estimation theory
Estimation theory is a branch of statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of inf ...

#Statistical hypothesis testing
A statistical hypothesis test is a method of statistical inference
Statistical inference is the process of using data analysis
Data analysis is a process of inspecting, Data cleansing, cleansing, Data transformation, transforming, and Data ...

# 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 * StylometryNotes

References

Citations

Sources

* . * * David R. Cox, Cox, D. R. (2006). ''Principles of Statistical Inference'', Cambridge University Press. . * 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) * * David A. Freedman, Freedman, D. A. (2010). ''Statistical Models and Causal Inferences: A Dialogue with the Social Sciences'' (Edited by David Collier (political scientist), David Collier, Jasjeet S. Sekhon, Jasjeet Sekhon, and Philip B. Stark), Cambridge University Press. * * * * Reprinted as * Konishi S., Kitagawa G. (2008), ''Information Criteria and Statistical Modeling'', Springer. * * Lucien Le Cam, Le Cam, Lucian. (1986) ''Asymptotic Methods of Statistical Decision Theory'', Springer. * David S. Moore, Moore, D. S.; McCabe, G. P.; Craig, B. A. (2015), ''Introduction to the Practice of Statistics'', Eighth Edition, Macmillan. * (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, pp604

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

*Charles Sanders Peirce, 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, ) * * * * * *

Further reading

* George Casella, Casella, G., Roger Lee Berger, Berger, R. L. (2002). ''Statistical Inference''. Duxbury Press. * * Held L., Bové D.S. (2014). ''Applied Statistical Inference—Likelihood and Bayes'' (Springer). * * *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 * * Young, G.A., Smith, R.L. (2005). ''Essentials of Statistical Inference'', CUP.External links

* MIOpenCourseWare

Statistical Inference

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Statistical induction and prediction

{{Authority control Statistical inference, Inductive reasoning Deductive reasoning Logic and statistics Philosophy of science Psychometrics de:Mathematische Statistik