AltDistributions
AltDistributions.AltMvNormal
— TypeAltMvNormal(_, μ, L)
Inner constructor used internally, for specifying L
directly when the first argument is Val{:L}
.
You don't want to use this unless you obtain L
directly. Use a Cholesky
factorization instead.
AltMvNormal(μ, Σ)
Multivariate normal distribution with mean μ
and covariance matrix Σ
, which can be an abstract matrix (eg a factorization) or I
. If Σ
is not symetric because of numerical error, wrap in LinearAlgebra.Symmetric
.
Use the AltMvNormal(Val(:L), μ, L)
constructor for using LL'=Σ
directly.
Also, see StdCorrFactor
for formulating L
from standard deviations and a Cholesky factor of a correlation matrix:
AltMvNormal(μ, StdCorrFactor(σ, S))
AltDistributions.StdCorrFactor
— TypeStdCorrFactor(σ, F)
A factor L
of a covariance matrix Σ = LL'
given as L = Diagonal(σ) * F
. Can be used in place of L
, without performing the multiplication.
AltDistributions.LKJL
— TypeLKJL(η)
The LKJ distribution (Lewandowski et al 2009) for the Cholesky factor L of correlation matrices.
A correlation matrix $Ω=LL'$ has the density $|Ω|^{η-1}$. However, it is usually not necessary to construct $Ω$, so this distribution is formulated for the Cholesky decomposition L*L'
, and takes L
directly.
Note that the methods does not check if L
yields a valid correlation matrix.
Valid values are $η > 0$. When $η > 1$, the distribution is unimodal at Ω=I
, while $0 < η < 1$ has a trough. $η = 2$ is recommended as a vague prior.
When $η = 1$, the density is uniform in Ω
, but not in L
, because of the Jacobian correction of the transformation.