Instatistics, theprecision matrix orconcentration matrix is thematrix inverse of thecovariance matrix or dispersion matrix,${\displaystyle P={\mathrm{\Sigma }}^{-1}}$.^[1][2][3]Forunivariate distributions, the precision matrix degenerates into ascalarprecision, defined as thereciprocal of thevariance,${\displaystyle p=\frac{1}{{\sigma }^2}}$.^[4]

Othersummary statistics ofstatistical dispersion also calledprecision (orimprecision^[5][6])include the reciprocal of thestandard deviation,${\displaystyle p=\frac{1}{\sigma }}$;^[3] the standard deviation itself and therelative standard deviation;^[7]as well as thestandard error^[8] and theconfidence interval (or its half-width, themargin of error).^[9]

One particular use of the precision matrix is in the context ofBayesian analysis of themultivariate 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.^[10] For instance, if both theprior and thelikelihood haveGaussian 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 theposterior will simply be the sum of the precision matrices of the prior and the likelihood.

As the inverse of aHermitian matrix, the precision matrix of real-valued random variables, if it exists, ispositive definite and symmetrical.

Another reason the precision matrix may be useful is that if two dimensions${\displaystyle i}$ and${\displaystyle j}$ of a multivariate normal areconditionally independent, then the${\displaystyle ij}$ and${\displaystyle ji}$ elements of the precision matrix are${\displaystyle 0}$. 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. It also means that precision matrices are closely related to the idea ofpartial correlation.

The precision matrix plays a central role ingeneralized least squares, compared toordinary least squares, where${\displaystyle P}$ is theidentity matrix, and toweighted least squares, where${\displaystyle P}$ is diagonal (theweight matrix).

The termprecision in this sense ("mensura praecisionis observationum") first appeared in the works ofGauss (1809) "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (page 212). Gauss's definition differs from the modern one by a factor of${\displaystyle \sqrt{2}}$. He writes, for the density function of anormal distribution with precision${\displaystyle h}$ (reciprocal of standard deviation),

${\displaystyle \phi \mathrm{\Delta }=\frac{h}{\sqrt{\pi }}{e}^{-hh\mathrm{\Delta }\mathrm{\Delta }}.}$

where${\displaystyle hh=h^2}$ (seemodern exponential notation).Later Whittaker & Robinson (1924) "Calculus of observations" called this quantitythe modulus (of precision), but this term has dropped out of use.^[11]

• Statistical deviation and dispersion
• Bayesian statistics

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