Constructor for PDMat with Cholesky pre- and post- multiplied by diagonal matrix

[sethaxen]

Suppose a user has a Cholesky decomposition of a matrix R and a vector of positive entries σ and wishes to create Σ = PDMat(cholesky(σ .* Matrix(R) .* σ')). This is common in Bayesian inference. e.g. a Cholesky factor of a correlation matrix is drawn from a LKJCholesky distribution, while a vector of standard deviations is drawn from some other distribution, and one wants to construct the Cholesky factorization of the covariance matrix for use in MvNormal.

This can be done much more efficiently with Σ = PDMat(Cholesky(R.uplo == 'U' ? R.U * Diagonal(σ) : Diagonal(σ) * R.L, R.uplo, 0)). But this is still quite long. It would be ideal to have either a constructor for PDMat or some other utility function like the following:

function PDMat(fac::Cholesky, scale::AbstractVector)
    factors_scaled = fac.uplo == 'U' ? fac.factors .* scale' : scale .* fac.factors
    return PDMat(Cholesky(factors_scaled, fac.uplo, 0))
end
function PDMat(fac::Cholesky, scale::UniformScaling)
    return PDMat(Cholesky(fac.factors * scale, fac.uplo, 0))
end
PDMat(fac::Cholesky, scale::Number) = PDMat(fac, scale * I)

See also discussions in JuliaStats/Distributions.jl#1336 and TuringLang/Turing.jl#1629 (comment).