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.[1][2][3] Thus, if we are considering a single random variable in isolation, its precision is the inverse of its variance: p=1/σ². Some particular statistical models define the term precision differently.

For broader coverage of this topic, see Accuracy and precision.

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.[4] 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 matrix, the precision matrix of real-valued random variables, if it exists, is positive definite and symmetrical.

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. It also means that precision matrices are closely related to the idea of partial correlation.

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 . He writes, for the density function of a normal random variable with precision h,

Later Whittaker & Robinson (1924) “Calculus of observations” called this quantity the modulus, but this term has dropped out of use.[5]

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References
  1. DeGroot, Morris H. (1969). Optimal Statistical Decisions. New York: McGraw-Hill. p. 56.
  2. Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. p. 144. ISBN 0-19-506011-3.
  3. Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.
  4. Bernardo, J. M. & Smith, A.F.M. (2000) Bayesian Theory, Wiley ISBN 0-471-49464-X
  5. "Earliest known uses of some of the words in mathematics".
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