Derive and Implement Analytic Hessian for Nested Logit

[timothyb0912]

I plan to get around to this at some point in the coming months. Pull requests, in the meantime, are always welcome though.

At the moment, the sum of the outer products of the gradient are used as an approximation to the actual hessian.

[Eh2406]

Do we know that the Hessian has a closed form? If so do we know what it is? ("Derive" suggests we may not.) All answers are okay. Just trying to assess how much work this will be.

[timothyb0912]

Hey, sorry for the confusion.

I am pretty sure the hessian exists in closed-form, although its form is a bit complicated. I say this based on the fact that the gradient exists in closed form and nothing indicates that the derivative of the gradient would be undefined or lose its closed-form nature.

I say derive simply because I haven't seen explicit formulas for the hessian written down anywhere but it doesn't seem impossible (just tedious).