Editing Generalized linear model (section)

==Overview==

In a generalized linear model (GLM), each outcome '''Y''' of the [[dependent variable]]s is assumed to be generated from a particular [[probability distribution|distribution]] in an [[exponential family]], a large class of [[probability distributions]] that includes the [[normal distribution|normal]], [[binomial distribution|binomial]], [[poisson distribution|Poisson]] and [[gamma distribution|gamma]] distributions, among others. The conditional mean '''''μ''''' of the distribution depends on the independent variables '''X''' through:

: <math>\operatorname{E}(\mathbf{Y}\mid\mathbf{X}) = \boldsymbol{\mu} = g^{-1}(\mathbf{X}\boldsymbol{\beta}), </math>

where E('''Y'''&nbsp;|&nbsp;'''X''') is the [[expected value]] of '''Y''' [[conditional expectation|conditional]] on '''X'''; '''X''&beta;''''' is the ''linear predictor'', a linear combination of unknown parameters '''''&beta;'''''; ''g'' is the link function.

In this framework, the variance is typically a function, '''V''', of the mean:

:<math> \operatorname{Var}(\mathbf{Y}\mid\mathbf{X})  = \operatorname{V}(g^{-1}(\mathbf{X}\boldsymbol{\beta})). </math>

It is convenient if '''V''' follows from an exponential family of distributions, but it may simply be that the variance is a function of the predicted value.

The unknown parameters, '''''β''''', are typically estimated with [[maximum likelihood]], maximum [[quasi-likelihood]], or [[Bayesian probability|Bayesian]] techniques.