Editing Sufficient statistic (section)

==Background==
Roughly, given a set <math> \mathbf{X}</math> of [[independent identically distributed]] data conditioned on an unknown parameter <math>\theta</math>, a sufficient statistic is a function <math>T(\mathbf{X})</math> whose value contains all the information needed to compute any estimate of the parameter (e.g. a [[maximum likelihood]] estimate).  Due to the factorization theorem ([[#Fisher–Neyman factorization theorem|see below]]), for a sufficient statistic <math>T(\mathbf{X})</math>, the probability density can be written as <math>f_{\mathbf{X}}(x;\theta) = h(x) \, g(\theta, T(x))</math>.  From this factorization, it can easily be seen that the maximum likelihood estimate of <math>\theta</math> will interact with <math>\mathbf{X}</math> only through <math>T(\mathbf{X})</math>.  Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points.

More generally, the "unknown parameter" may represent a [[Euclidean vector|vector]] of unknown quantities or may represent everything about the model that is unknown or not fully specified.  In such a case, the sufficient statistic may be a set of functions, called a ''jointly sufficient statistic''.  Typically, there are as many functions as there are parameters.  For example, for a [[Gaussian distribution]] with unknown [[mean]] and [[variance]], the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the [[sample mean]] and [[sample variance]]).

In other words, '''the [[joint probability distribution]] of the data is conditionally independent of the parameter given the value of the sufficient statistic for the parameter'''. Both the statistic and the underlying parameter can be vectors.