Replace PPE/MKD with new noun (Homotopy?) + Serialization Design

[Affie]

Criteria

• Select new noun HomotopyDensity
• Use .reprkind to recognize PPE replacements,
  • note there may be other research topologies, but statelabel gives freedom to explore, since PPE friendly export to new statelabel always possible.
• Replace PPE as part of DFG v1 work downstream
• JSON-serde with HomotopyDensityDFG, also see Live type

From the original needs

• Ploting - function calls can convert to required type
• Searching or indexing only applicable to certain types.

Description

Related names are Homotopy*, MKD, GMM, Mixture, ...?

Storied History

See discussion below

[Affie]

• In old world KDE was not on manifold, but had a balanced tree and could approximate non-parametric products.
• Rewrite was Manifold Kernel Density (MKD), now using ManellicTree for on manifold unbalanced tree, using Mahony parallel transport for approximating products. Cleaner rewrite of serialization.
• Parametric Point Estimates PPE was window dressing for non-parametric beliefs, but never really supported multiple modes properly.
• Working towards hybrid parametric non-parametric belief representations.

As an example, the current PPE mean/max functionality can be achieved with getters such as getBelief[Max/Mean]

[Affie]

Maybe also something with a tree such as BeliefTree, ModeTree, or MixtureTree

[Affie]

The plan is to deprecate PPEs in the meantime, that way we can always add this after DFGv1 as well.