# AltDistributions

`AltDistributions.AltMvNormal`

— Type```
AltMvNormal(_, 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`

— Type```
StdCorrFactor(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`

— Type```
LKJL(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.