How do I compare multivariate normal distributions and get a p-value?

Question

I have sample-data for two multivariate normal distributions. From this sample-data, I can calculate each distribution’s parameters (means and standard deviations).

1. How do I quantify the distance (or divergence) between these two multivariate normal distributions? I don’t mind whether I use the sample-data or the distribution parameters to make this quantification.
2. Given the chosen test in (1), how do I calculate an accompanying p-value which can be used to reject the null hypothesis for the given pair of multivariate normal distributions?

While I can implement solutions in Python, I’m no stats expert, and would appreciate explanations formatted as such; for complex explanations, corresponding simplifications would be very useful.

Answer 1

I'm not sure whether I understand your question correctly, but from your description I think you might want to do a multivariate analysis of variance (MANOVA). It is used to compare sample means of multivariate distributions and returns p values as well.

Again, if I understand your question correctly, for an implementation you would need to assign a variable to each sample, which indicates whether the values belong to sample one or sample two. You could then use this variable as your independent variable and each of your sampled variables as a dependent variable.