# Statistical inference

inferential statisticsinferenceinferencesinferentialstatisticalformal statistical inferencehypothesis testinductiveinductive statisticsinfer
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.wikipedia
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### Model selection

statistical model selectionselectingchoose a model
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.
state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling".

### Data analysis

data analyticsanalysisdata analyst
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.
Inferential statistics includes techniques to measure relationships between particular variables.

### Statistical hypothesis testing

hypothesis testingstatistical teststatistical tests
A statistical hypothesis test is a method of statistical inference.

### Confidence interval

confidence intervalsconfidence levelconfidence
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.
The principle behind confidence intervals was formulated to provide an answer to the question raised in statistical inference of how to deal with the uncertainty inherent in results derived from data that are themselves only a randomly selected subset of a population.

### Nonparametric statistics

non-parametricnon-parametric statisticsnonparametric
Nonparametric statistics includes both descriptive statistics and statistical inference.

### Statistical population

populationsubpopulationsubpopulations
Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates.
In statistical inference, a subset of the population (a statistical sample) is chosen to represent the population in a statistical analysis.

### Bayesian inference

BayesianBayesian analysisBayesian method
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. The classical (or frequentist) paradigm, the Bayesian paradigm, the likelihoodist paradigm, and the AIC-based paradigm are summarized below.
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.

### Statistical classification

classificationclassifierclassifiers

### Normal distribution

normally distributedGaussian distributionnormal
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.
Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data.

### Likelihood function

likelihoodlikelihood ratiolog-likelihood
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).
Later, Barnard and Birnbaum led a school of thought that advocated the likelihood principle, postulating that all relevant information for inference is contained in the likelihood function.

### Frequentist inference

frequentistfrequentist statisticsclassical
Frequentist inference is a type of statistical inference that draws conclusions from sample data by emphasizing the frequency or proportion of the data.

### Biostatistics

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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.
Because of that, the sampling process is very important for statistical inference.

### Akaike information criterion

AICAIC-basedAICc
It now forms the basis of a paradigm for the foundations of statistics; as well, it is widely used for statistical inference.

### Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
Others, however, propose inference based on the likelihood function, of which the best-known is maximum likelihood estimation.
The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

### Randomized experiment

randomized trialrandomizationrandomized
)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.
Randomization also produces ignorable designs, which are valuable in model-based statistical inference, especially Bayesian or likelihood-based.

### Information theory

information-theoreticinformation theoristinformation
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. The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.
The theory has also found applications in other areas, including statistical inference, natural language processing, cryptography, neurobiology, human vision, the evolution and function of molecular codes (bioinformatics), model selection in statistics, thermal physics, quantum computing, linguistics, plagiarism detection, pattern recognition, and anomaly detection.

### P-value

p''-valuepp''-values
The p-value is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing.

### Data mining

data-miningdataminingknowledge discovery in databases
The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.
Aside from the raw analysis step, it also involves database and data management aspects, data pre-processing, model and inference considerations, interestingness metrics, complexity considerations, post-processing of discovered structures, visualization, and online updating.

### Fiducial inference

fiducialfiducial distributionfaith
Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution".
Fiducial inference is one of a number of different types of statistical inference.

### Likelihoodist statistics

likelihoodistlikelihoodismschool of thought
Beyond this, there are significant differences within likelihood approaches: "orthodox" likelihoodists consider data only as evidence, and do not use it as the basis of statistical inference, while others make inferences based on likelihood, but without using Bayesian inference or frequentist inference.

### Confidence distribution

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.
In statistical inference, the concept of a confidence distribution (CD) has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest.

### Statistical model

modelprobabilistic modelstatistical modeling
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. 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).
More generally, statistical models are part of the foundation of statistical inference.

### Linear regression

regression coefficientmultiple linear regressionregression
The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining.

### Algorithmic inference

sampling mechanism
Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst.

### Descriptive statistics

descriptivedescriptive statisticstatistics
Inferential statistics can be contrasted with descriptive statistics.
Descriptive statistics is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent.