Describe the workflow you want to enable

The sklearn.mixture.GaussianMixture has built in functions to get the Akaike Information Criterion (AIK) and Bayesian Information Criterion (BIC) scores. However, the sklearn.mixture.BayesianGaussianMixture class does not.

Describe your proposed solution

With some modifications to the functions for Gaussian Mixture, it would be possible to implement BIC and AIC scores for the Bayesian Gaussian Mixture class as well.

Describe alternatives you've considered, if relevant

Additional context

I encountered this when working on a ML project where I wanted to use the BIC and AIC scores to evaluate my model when fed different combinations of data. I wanted to use the fact that with the Bayesian Gaussian Mixture class, the effective number of components can be inferred from the data, to speed up my computations.

I'm wondering if this is something that should be incorporated? If there is interest, I made a pull request to solve this.

Thanks in advance for your time!