The precision matrix of a random vector is the inverse of its covariance matrix.
The precision matrix is sometimes called concentration matrix.
The following is a precise definition.
Definition
Let
be a
random vector. Let
be its covariance
matrix:![[eq1]](/images/precision_matrix__4.png){kind=small}
If
is invertible, then the precision matrix of
is the
matrix
defined
as![[eq2]](/images/precision_matrix__9.png){kind=small}
When
is a random variable
(
),
then the precision matrix becomes a scalar and it is equal to the reciprocal
of the variance of
.
In this case, it is often denoted by the lowercase letter
:![[eq3]](/images/precision_matrix__14.png)
and
it is simply called the precision of
.
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.
The joint probability density function of a multivariate normal random vector is often written in terms of its precision matrix.
If
has a multivariate normal distribution with mean
and covariance matrix
,
then its joint probability density function
is![[eq4]](/images/precision_matrix__19.png){kind=small}
By using the precision matrix, this can be written
as![[eq5]](/images/precision_matrix__20.png){kind=small}
because,
by elementary properties of the determinant, we have
that![[eq6]](/images/precision_matrix__21.png){kind=small}
In the univariate case, when
is a normal random variable with mean
and variance
,
the
density![[eq7]](/images/precision_matrix__25.png)
becomes![[eq8]](/images/precision_matrix__26.png)
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
.
You can read more details about covariance matrices in the lecture entitled Covariance matrix.
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Please cite as:
Taboga, Marco (2021). "Precision matrix", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/precision-matrix.
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