[abhgh]

I'm sorry I didn't have a chance to respond before -you've been very patient with your responses! Thanks!

Yes, this is quite neat!

I will look up the rule/theorem you mention. It seems "kind of" reasonable in the sense the same number of independent parameters should show up in any representation, but (and this is probably me just being dumb) I need to think about why wouldn't the form of the representations not affect the number of parameters.

Although, if m parameters in a representation can be mapped to n (m>n) parameters in another, the m parameters aren't really independent.

OK, now I am just talking to myself :-)

Also, in this context, the results around discriminative classifiers learning better than their generative counterparts (asymptotically) is something I need to think about. [1]

Well, there goes my next few evenings.

[1] https://ai.stanford.edu/~ang/papers/nips01-discriminativegen...