Your notation is clear to me! It can be shown that: Df(p⋆,q)=∫dxq(x)f⋄(f⋄#(⟨λ,T(x)⟩))
Even though f⋄ and f⋄# are both convex, their composition is not necessarily convex. Sorry, I don’t have any clear counterexamples at the ready. (The Hessian determinants all vanish in the graphene/bit-string model.) It’s likely that I’ve configured my numerical solvers badly for this example.
As far as binding to reality:
Power laws are ubiquitous in nature. Exponential family doesn’t admit power laws; f-div does. (It admits many other important non-exponential distributions as well—see the B-E example above.)
Symmetries in your data that are weaker than—or just different from—independence are ubiquitous in nature. Exponential family is only congruent with independence; not so, f-div.
KL / Gibbs entropy is inadequate for many observable systems. This typically appears under the heading nonextensive entropy (Gell-Mann & Tsallis w/ contributions by Touchette and others) or Tsallis entropy. This paper is far enough outside my field that I can only loosely validate it, but it’s an attempt to explain dark energy by way of applying observational data to determine Bayes factors of different nonextensive entropy models of our cosmological horizon.
I consider Jensen’s, DPI, and coarse-graining monotonicity to be pretty important desiderata (see Nielsen). f-div is the biggest family that supports these, uniquely determined.
[I could write several more posts worth of material]
Your notation is clear to me! It can be shown that:
Df(p⋆,q)=∫dxq(x)f⋄(f⋄#(⟨λ,T(x)⟩))
Even though f⋄ and f⋄# are both convex, their composition is not necessarily convex. Sorry, I don’t have any clear counterexamples at the ready. (The Hessian determinants all vanish in the graphene/bit-string model.) It’s likely that I’ve configured my numerical solvers badly for this example.
As far as binding to reality:
Power laws are ubiquitous in nature. Exponential family doesn’t admit power laws; f-div does. (It admits many other important non-exponential distributions as well—see the B-E example above.)
Symmetries in your data that are weaker than—or just different from—independence are ubiquitous in nature. Exponential family is only congruent with independence; not so, f-div.
KL / Gibbs entropy is inadequate for many observable systems. This typically appears under the heading nonextensive entropy (Gell-Mann & Tsallis w/ contributions by Touchette and others) or Tsallis entropy. This paper is far enough outside my field that I can only loosely validate it, but it’s an attempt to explain dark energy by way of applying observational data to determine Bayes factors of different nonextensive entropy models of our cosmological horizon.
I consider Jensen’s, DPI, and coarse-graining monotonicity to be pretty important desiderata (see Nielsen). f-div is the biggest family that supports these, uniquely determined.
[I could write several more posts worth of material]