Maximum likelihood estimationIn 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.
Likelihood functionFunction related to statistics and probability theoryA likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the models being compared are parameterized by a parameter, with the parameter often written as θ, or they are parameterized by multiple parameters given as the components of a vector.
Marginal likelihoodA marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample for all possible values of the parameters; it can be understood as the probability of the model itself and is therefore often referred to as model evidence or simply evidence. Due to the integration over the parameter space, the marginal likelihood does not directly depend upon the parameters.
Akaike information criterionThe 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. AIC is founded on information theory.Bayes factorThe Bayes factor (BF) is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis and an alternative, but this is not necessary; for instance, it could also be a non-linear model compared to its linear approximation.
Laplace's methodMethod for approximate evaluation of integralsIn mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form ∫abeMf(x)dx,{\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,}where f{\displaystyle f} is a twice-differentiablefunction, M{\displaystyle M} is a large number, and the endpoints a{\displaystyle a} and b{\displaystyle b} may be infinite. This technique was originally presented in the book by Laplace (1774).ParameterA parameter (from Ancient Greek παρά (pará) 'beside, subsidiary' and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.).
Multiple linear regressionMultiple linear regression, also known as multivariable linear regression, is a generalization of simple linear regression to the case of more than one independent variable. The basic model for multiple linear regression is Y i = β 0 + β 1 X i 1 + β 2 X i 2 + … + β p X i p + ϵ i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+\beta _{2}X_{i2}+\ldots +\beta _{p}X_{ip}+\epsilon _{i}} for each observation i = 1 , … , n {\textstyle i=1,\ldots ,n} .
Model selectionTask of selecting a statistical model from a set of candidate models Model selection is the task of selecting a model from among various candidates on the basis of performance criterion to choose the best one. In the context of machine learning and more generally statistical analysis, this may be the selection of a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered.
OverfittingIn mathematical modeling, overfitting is the production of an analysis that corresponds too closely or exactly to a particular set of data and may therefore fail to fit to additional data or predict future observations reliably. An overfitted model is a mathematical model that contains more parameters than can be justified by the data. In the special case of a model that consists of a polynomial function, these parameters represent the degree of a polynomial.
StatisticsStudy of collection and analysis of dataThe normal distribution, a very common probability density, is used extensively in inferential statistics.Scatter plots and line charts are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.Statistics (from German: Statistik, orig.ObservationObservation in the natural sciences refers to the active acquisition of information from a primary source. It involves the act of noticing or perceiving phenomena and gathering data based on direct engagement with the subject of study. In living organisms, observation typically occurs through the senses. In science, it often extends beyond unaided perception, involving the use of scientific instruments to detect, measure, and record data.
Likelihood functionFunction related to statistics and probability theoryA likelihood function (often simply called the likelihood) gives the relative merit of various statistical models for describing a data set. Often the models being compared are parameterized by a parameter, with the parameter often written as θ, or they are parameterized by multiple parameters given as the components of a vector.Marginal likelihoodA marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample for all possible values of the parameters; it can be understood as the probability of the model itself and is therefore often referred to as model evidence or simply evidence. Due to the integration over the parameter space, the marginal likelihood does not directly depend upon the parameters.Akaike information criterionThe 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. AIC is founded on information theory.Bayes factorThe Bayes factor (BF) is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis and an alternative, but this is not necessary; for instance, it could also be a non-linear model compared to its linear approximation.Laplace's methodMethod for approximate evaluation of integralsIn mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form ∫abeMf(x)dx,{\displaystyle \int _{a}^{b}e^{Mf(x)}\,dx,}where f{\displaystyle f} is a twice-differentiablefunction, M{\displaystyle M} is a large number, and the endpoints a{\displaystyle a} and b{\displaystyle b} may be infinite. This technique was originally presented in the book by Laplace (1774).ParameterA parameter (from Ancient Greek παρά (pará) 'beside, subsidiary' and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.).Multiple linear regressionMultiple linear regression, also known as multivariable linear regression, is a generalization of simple linear regression to the case of more than one independent variable. The basic model for multiple linear regression is Y i = β 0 + β 1 X i 1 + β 2 X i 2 + … + β p X i p + ϵ i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+\beta _{2}X_{i2}+\ldots +\beta _{p}X_{ip}+\epsilon _{i}} for each observation i = 1 , … , n {\textstyle i=1,\ldots ,n} .Model selectionTask of selecting a statistical model from a set of candidate models Model selection is the task of selecting a model from among various candidates on the basis of performance criterion to choose the best one. In the context of machine learning and more generally statistical analysis, this may be the selection of a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered.OverfittingIn mathematical modeling, overfitting is the production of an analysis that corresponds too closely or exactly to a particular set of data and may therefore fail to fit to additional data or predict future observations reliably. An overfitted model is a mathematical model that contains more parameters than can be justified by the data. In the special case of a model that consists of a polynomial function, these parameters represent the degree of a polynomial.StatisticsStudy of collection and analysis of dataThe normal distribution, a very common probability density, is used extensively in inferential statistics.Scatter plots and line charts are used in descriptive statistics to show the observed relationships between different variables, here using the Iris flower data set.Statistics (from German: Statistik, orig.ObservationObservation in the natural sciences refers to the active acquisition of information from a primary source. It involves the act of noticing or perceiving phenomena and gathering data based on direct engagement with the subject of study. In living organisms, observation typically occurs through the senses. In science, it often extends beyond unaided perception, involving the use of scientific instruments to detect, measure, and record data.*As an Amazon Associate I earn from qualifying purchases.