Precision (statistics)

In statistics, precision is the reciprocal of the variance, and the precision matrix (also known as concentration matrix) is the matrix inverse of the covariance matrix. Some particular statistical models define the term precision differently.

One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution: for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise. For instance, if both the prior and the likelihood have Gaussian form, and the precision matrix of both of these exist (because their covariance matrix is full rank and thus invertible), then the precision matrix of the posterior will simply be the sum of the precision matrices of the prior and the likelihood.

As the inverse of a Hermitian positive semidefinite matrix, the precision matrix, if it exists, is positive definite. Another reason the precision matrix may be useful is that if two dimensions i and j of a multivariate normal are conditionally independent, then the ij and ji elements of the precision matrix are 0. This means that precision matrices tend to be sparse when many of the dimensions are conditionally independent, which can lead to computational efficiencies when working with them.

History

The term precision in this sense (“mensura praecisionis observationum”) first appeared in the works of Gauss (1809) “Theoria motus corporum coelestium in sectionibus conicis solem ambientium” (page 212). Gauss’s definition differs from the modern one by a factor of $\scriptstyle {\sqrt {2}}$. He writes, for the density function of a normal random variable with precision h,

$\varphi \Delta ={\frac {h}{\sqrt {\pi }}}\,e^{-hh\Delta \Delta }.$

Later Whittaker & Robinson (1924) “Calculus of observations” called this quantity the modulus, but this term has dropped out of use.