 # Sufficient Statistic (Graphical)

Sufficient Statistic:

Suppose X is a random vector with probability distribution (or density) P(X | V), where V is a vector of parameters, and Xo is a realization of X. A statistic T(X) is called a sufficient statistic if the conditional probability (density) does not depend upon V for any possible Xo.

In other words, the observed value T(Xo) of a sufficient statistic T bears all the information (about the vector V of parameters) the data Xo contain.

A trivial example of a sufficient statistic is T(X) = X, but such a statistic is useless. Practically interesting cases of sufficient statistics are those when T has (much) smaller dimension than the vector X itself. Such statistics allow you to reduce all the observed data values to a smaller set of values preserving all the information about the parameter V.

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