Precision (statistics)

precisionprecision matrixconcentrationprecision parameter
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.wikipedia
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Covariance matrix

variance-covariance matrixcovariance matricescovariance
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.
The inverse of this matrix, if it exists, is the inverse covariance matrix, also known as the concentration matrix or precision matrix.

Multivariate normal distribution

multivariate normalbivariate normal distributionjointly normally distributed
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.
:such that The inverse of the covariance matrix is called the precision matrix, denoted by.

Likelihood function

likelihoodlikelihood ratiolog-likelihood
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.
The second derivative evaluated at, known as Fisher information, determines the curvature of the likelihood surface, and thus indicates the precision of the estimate.

Partial correlation

It also means that precision matrices are closely related to the idea of partial correlation.
If we define the precision matrix P = (p ij ) = Ω −1, we have:

Statistics

statisticalstatistical analysisstatistician
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.

Multiplicative inverse

reciprocalinversereciprocals
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.

Variance

sample variancepopulation variancevariability
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.

Invertible matrix

invertibleinversenonsingular
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.

Random variable

random variablesrandom variationrandom
Thus, if we are considering a single random variable in isolation, its precision is the inverse of its variance: p=1/σ².

Bayesian inference

BayesianBayesian analysisBayesian method
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.

Prior probability

prior distributionpriorprior probabilities
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.

Gaussian function

Gaussianbell curveGaussian kernel
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.

Posterior probability

posterior distributionposteriorposterior probability distribution
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.

Hermitian matrix

HermitianHermitian matricesHermitian conjugate matrix
As the inverse of a Hermitian matrix, the precision matrix of real-valued random variables, if it exists, is positive definite and symmetrical.

Definiteness of a matrix

positive definitepositive semidefinitepositive-definite
As the inverse of a Hermitian matrix, the precision matrix of real-valued random variables, if it exists, is positive definite and symmetrical.

Conditional independence

conditionally independentconditional dependenciesconditional independencies
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.

Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
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).

Normal distribution

normally distributedGaussian distributionnormal
Some authors advocate using the precision \tau as the parameter defining the width of the distribution, instead of the deviation \sigma or the variance \sigma^2.

Wald test

WaldWald estimatorWald statistic
In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.

Pooled variance

Pooled standard deviationpooledpooling
Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances.

Normal-gamma distribution

gamma-normal distributionGaussian-gamma distributionNormal-gamma
It is the conjugate prior of a normal distribution with unknown mean and precision.

Graphical lasso

In statistics, the graphical lasso is a sparse penalized maximum likelihood estimator for the concentration or precision matrix (inverse of covariance matrix) of a multivariate elliptical distribution.

Inverse-gamma distribution

inverse gamma distributionInverse gammaInv-gamma
However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior.

Normal-Wishart distribution

Gaussian-Wishart distributionnormal-Wishart
It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).