ADVI: Option to save family parameters to JSON

[WardBrian]

Summary:

Currently the mean is saved in the CSV output above the samples, but there is no way to get the estimated (co-)variance out, short of approximating it through the samples.

We could save it to JSON, similar to issues #3181 #3260

Description:

Option to save the non-mean parameters of the family for ADVI. For meanfield this is

stan/src/stan/variational/families/normal_meanfield.hpp

Lines 28 to 30 in 9de06d4

	* Log standard deviation (log scale) vector.
	*/
	Eigen::VectorXd omega_;

For fullrank this is

stan/src/stan/variational/families/normal_fullrank.hpp

Lines 28 to 31 in 9de06d4

	* Cholesky factor of covariance:
	* Sigma = L_chol * L_chol.transpose()
	*/
	Eigen::MatrixXd L_chol_;

These could be used to e.g. draw further samples from the approximation from outside the algorithmic run.

Additional Information:

Originally requested on the forums here: https://discourse.mc-stan.org/t/cmdstanpy-variational-how-to-get-density-parameters/37280/

Current Version:

v2.35.0