A **statistical model** is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.^{ [1] }

- Introduction
- Formal definition
- An example
- General remarks
- Dimension of a model
- Nested models
- Comparing models
- See also
- Notes
- References
- Further reading

A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).^{ [2] }

All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference.

Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sided dice. We will study two different statistical assumptions about the dice.

The first statistical assumption is this: for each of the dice, the probability of each face (1, 2, 3, 4, 5, and 6) coming up is 1/6. From that assumption, we can calculate the probability of both dice coming up 5: 1/6 × 1/6 = 1/36. More generally, we can calculate the probability of any event: e.g. (1 and 2) or (3 and 3) or (5 and 6).

The alternative statistical assumption is this: for each of the dice, the probability of the face 5 coming up is 1/8 (because the dice are weighted). From that assumption, we can calculate the probability of both dice coming up 5: 1/8 × 1/8 = 1/64. We cannot, however, calculate the probability of any other nontrivial event, as the probabilities of the other faces are unknown.

The first statistical assumption constitutes a statistical model: because with the assumption alone, we can calculate the probability of any event. The alternative statistical assumption does *not* constitute a statistical model: because with the assumption alone, we cannot calculate the probability of every event.

In the example above, with the first assumption, calculating the probability of an event is easy. With some other examples, though, the calculation can be difficult, or even impractical (e.g. it might require millions of years of computation). For an assumption to constitute a statistical model, such difficulty is acceptable: doing the calculation does not need to be practicable, just theoretically possible.

In mathematical terms, a statistical model is usually thought of as a pair (), where is the set of possible observations, i.e. the sample space, and is a set of probability distributions on .^{ [3] }

The intuition behind this definition is as follows. It is assumed that there is a "true" probability distribution induced by the process that generates the observed data. We choose to represent a set (of distributions) which contains a distribution that adequately approximates the true distribution.

Note that we do not require that contains the true distribution, and in practice that is rarely the case. Indeed, as Burnham & Anderson state, "A model is a simplification or approximation of reality and hence will not reflect all of reality"^{ [4] }—whence the saying "all models are wrong".

The set is almost always parameterized: . The set defines the parameters of the model. A parameterization is generally required to have distinct parameter values give rise to distinct distributions, i.e. must hold (in other words, it must be injective). A parameterization that meets the requirement is said to be * identifiable *.^{ [3] }

Suppose that we have a population of children, with the ages of the children distributed uniformly, in the population. The height of a child will be stochastically related to the age: e.g. when we know that a child is of age 7, this influences the chance of the child being 1.5 meters tall. We could formalize that relationship in a linear regression model, like this: height_{i} = *b*_{0} + *b*_{1}age_{i} + ε_{i}, where *b*_{0} is the intercept, *b*_{1} is a parameter that age is multiplied by to obtain a prediction of height, ε_{i} is the error term, and *i* identifies the child. This implies that height is predicted by age, with some error.

An admissible model must be consistent with all the data points. Thus, a straight line (height_{i} = *b*_{0} + *b*_{1}age_{i}) cannot be the equation for a model of the data—unless it exactly fits all the data points, i.e. all the data points lie perfectly on the line. The error term, ε_{i}, must be included in the equation, so that the model is consistent with all the data points.

To do statistical inference, we would first need to assume some probability distributions for the ε_{i}. For instance, we might assume that the ε_{i} distributions are i.i.d. Gaussian, with zero mean. In this instance, the model would have 3 parameters: *b*_{0}, *b*_{1}, and the variance of the Gaussian distribution.

We can formally specify the model in the form () as follows. The sample space, , of our model comprises the set of all possible pairs (age, height). Each possible value of = (*b*_{0}, *b*_{1}, *σ*^{2}) determines a distribution on ; denote that distribution by . If is the set of all possible values of , then . (The parameterization is identifiable, and this is easy to check.)

In this example, the model is determined by (1) specifying and (2) making some assumptions relevant to . There are two assumptions: that height can be approximated by a linear function of age; that errors in the approximation are distributed as i.i.d. Gaussian. The assumptions are sufficient to specify —as they are required to do.

A statistical model is a special class of mathematical model. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e. some of the variables are stochastic. In the above example with children's heights, ε is a stochastic variable; without that stochastic variable, the model would be deterministic.

Statistical models are often used even when the data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process).

Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses. Relatedly, the 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".^{ [5] }

There are three purposes for a statistical model, according to Konishi & Kitagawa.^{ [6] }

- Predictions
- Extraction of information
- Description of stochastic structures

Those three purposes are essentially the same as the three purposes indicated by Friendly & Meyer: prediction, estimation, description.^{ [7] } The three purposes correspond with the three kinds of logical reasoning: deductive reasoning, inductive reasoning, abductive reasoning.

Suppose that we have a statistical model () with . The model is said to be * parametric * if has a finite dimension. In notation, we write that where k is a positive integer ( denotes the real numbers; other sets can be used, in principle). Here, k is called the **dimension** of the model.

As an example, if we assume that data arise from a univariate Gaussian distribution, then we are assuming that

- .

In this example, the dimension, k, equals 2.

As another example, suppose that the data consists of points (x, y) that we assume are distributed according to a straight line with i.i.d. Gaussian residuals (with zero mean): this leads to the same statistical model as was used in the example with children's heights. The dimension of the statistical model is 3: the intercept of the line, the slope of the line, and the variance of the distribution of the residuals. (Note that in geometry, a straight line has dimension 1.)

Although formally is a single parameter that has dimension k, it is sometimes regarded as comprising k separate parameters. For example, with the univariate Gaussian distribution, is formally a single parameter with dimension 2, but it is sometimes regarded as comprising 2 separate parameters—the mean and the standard deviation.

A statistical model is *nonparametric* if the parameter set is infinite dimensional. A statistical model is *semiparametric* if it has both finite-dimensional and infinite-dimensional parameters. Formally, if k is the dimension of and n is the number of samples, both semiparametric and nonparametric models have as . If as , then the model is semiparametric; otherwise, the model is nonparametric.

Parametric models are by far the most commonly used statistical models. Regarding semiparametric and nonparametric models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".^{ [8] }

Two statistical models are **nested** if the first model can be transformed into the second model by imposing constraints on the parameters of the first model. As an example, the set of all Gaussian distributions has, nested within it, the set of zero-mean Gaussian distributions: we constrain the mean in the set of all Gaussian distributions to get the zero-mean distributions. As a second example, the quadratic model

*y*=*b*_{0}+*b*_{1}*x*+*b*_{2}*x*^{2}+ ε, ε ~ 𝒩(0,*σ*^{2})

has, nested within it, the linear model

*y*=*b*_{0}+*b*_{1}*x*+ ε, ε ~ 𝒩(0,*σ*^{2})

—we constrain the parameter *b*_{2} to equal 0.

In both those examples, the first model has a higher dimension than the second model (for the first example, the zero-mean model has dimension 1). Such is often, but not always, the case. As a different example, the set of positive-mean Gaussian distributions, which has dimension 2, is nested within the set of all Gaussian distributions.

Comparing statistical models is fundamental for much of statistical inference. Indeed, Konishi & Kitagawa (2008 , p. 75) state this: "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling. They are typically formulated as comparisons of several statistical models."

Common criteria for comparing models include the following: *R*^{2}, Bayes factor, Akaike information criterion, and the likelihood-ratio test together with its generalization, the relative likelihood.

- ↑ Cox 2006 , p. 178
- ↑ Adèr 2008 , p. 280
- 1 2 McCullagh 2002
- ↑ Burnham & Anderson 2002 , §1.2.5
- ↑ Cox 2006 , p. 197
- ↑ Konishi & Kitagawa 2008 , §1.1
- ↑ Friendly & Meyer 2016 , §11.6
- ↑ Cox 2006 , p. 2

This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations .(September 2010) |

In statistics, the **likelihood function** measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.

In statistics, the **likelihood-ratio test** assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

**Pattern recognition** is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. Pattern recognition has its origins in statistics and engineering; some modern approaches to pattern recognition include the use of machine learning, due to the increased availability of big data and a new abundance of processing power. However, these activities can be viewed as two facets of the same field of application, and together they have undergone substantial development over the past few decades. A modern definition of pattern recognition is:

The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.

In statistics, **maximum likelihood estimation** (**MLE**) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In probability theory and statistics, a **Gaussian process** is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

In mathematical statistics, the **Fisher information** is a way of measuring the amount of information that an observable random variable *X* carries about an unknown parameter *θ* of a distribution that models *X*. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher. The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics.

The **Akaike information criterion** (**AIC**) is an estimator of prediction error and thereby 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.

In statistics, a **mixture model** is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information.

In Bayesian probability, the **Jeffreys prior**, named after Sir Harold Jeffreys, is a non-informative (objective) prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher information matrix:

In statistics, a **parametric model** or **parametric family** or **finite-dimensional model** is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.

**Robust statistics** is statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard-deviations; under this model, non-robust methods like a t-test work poorly.

In statistics, a **semiparametric model** is a statistical model that has parametric and nonparametric components.

**Uncertainty quantification** (**UQ**) is the science of quantitative characterization and reduction of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense.

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

In probability theory, **Dirichlet processes** are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.

**Algorithmic inference** gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computing, bioinformatics, and, long ago, structural probability . The main focus is on the algorithms which compute statistics rooting the study of a random phenomenon, along with the amount of data they must feed on to produce reliable results. This shifts the interest of mathematicians from the study of the distribution laws to the functional properties of the statistics, and the interest of computer scientists from the algorithms for processing data to the information they process.

In statistics, **identifiability** is a property which a model must satisfy for precise inference to be possible. A model is **identifiable** if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameters must generate different probability distributions of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the **identification conditions**.

In statistics: **asymptotic theory**, or **large sample theory**, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size *n* may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of *n* → ∞. In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too.

In Bayesian inference, the **Bernstein-von Mises theorem** provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a posterior distribution converges in the limit of infinite data to a multivariate normal distribution centered at the maximum likelihood estimator with covariance matrix given by , where is the true population parameter and is the Fisher information matrix at the true population parameter value.

In statistics, suppose that we have been given some data, and we are constructing a statistical model of that data. The **relative likelihood** compares the relative plausibilities of different candidate models or of different values of a parameter of a single model.

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