Hamlitonian Sampling for Balmer Lines

[dgahle]

Update BaySAR to be compatible with inference.mcmc.HamiltonianChain to resolve the effective sample size in demo/balmer_series.py` flatlining at about 7.

Useful links:

• SymPy - https://python.plainenglish.io/how-is-symbolic-differentiation-done-in-python-using-sympy-6484554f25b0
• My maths notes - https://www.overleaf.com/project/64fc839f4d91b9553e1f581b

Task list

• [ ] Write out the mathematics
• [ ] Plan out the objects that require a gradient class method
• [ ] Update this list with the above and cross out as going along
• [ ] Add unit tests to verify functionality
• [ ] Update demo/balmer_series.py with the HamiltonianChain

[dgahle]

It's likely that each function that needs to be differentiated will need to be a class that can store all the dependent/nested function gradient methods can be accessed for.

Or cache the gradients of some of the lowest level functions like emission, temperature, and density profiles.