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Precision matrix

The precision matrix of a random vector is the inverse of its covariance matrix.

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Synonyms

The precision matrix is sometimes called concentration matrix.

Definition

The following is a precise definition.

Definition Let X be a Kx1 random vector. Let V be its covariance matrix:[eq1]If V is invertible, then the precision matrix of X is the $K	imes K$ matrix H defined as[eq2]

When X is a random variable ($K=1$), then the precision matrix becomes a scalar and it is equal to the reciprocal of the variance of $X,$. In this case, it is often denoted by the lowercase letter $h$:[eq3]and it is simply called the precision of X.

Thus, in the univariate case precision is inversely proportional to variance: when variance tends to infinity, we have zero precision; on the contrary, when variance tends to zero, we have infinite precision.

Precision matrix and normal densities

The joint probability density function of a multivariate normal random vector is often written in terms of its precision matrix.

If X has a multivariate normal distribution with mean mu and covariance matrix V, then its joint probability density function is[eq4]

By using the precision matrix, this can be written as[eq5]because, by elementary properties of the determinant, we have that[eq6]

In the univariate case, when X is a normal random variable with mean mu and variance sigma^2, the density[eq7]becomes[eq8]

Parametrizing a normal density in terms of its precision matrix often has significant advantages. For example, it can simplify the algebra of calculations involving normal densities. Or, when the values of a multivariate normal density need to be computed several times by numerical methods, employing the precision matrix can spare the computationally burdensome task of performing several matrix inversions to calculate $V^{-1}$.

More details

You can read more details about covariance matrices in the lecture entitled Covariance matrix.

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