Uncertainty Measures

evidencelib quantifies the uncertainty and information content of a mass function with a family of entropy-style measures.

m.deng_entropy()
m.tfb_entropy(order=2)
m.fractal_belief_entropy()
m.information_volume()
m.nonspecificity()
m.strife()

Measure	Meaning	Reference
`deng_entropy()`	Total uncertainty; Shannon entropy for Bayesian masses	Deng, Chaos, Solitons & Fractals 91 (2016)
`tfb_entropy(order=k)`	k-order time fractal-based entropy; `order=1` is Deng entropy	Zhou & Deng, Information Sciences 586 (2022)
`fractal_belief_entropy()`	Shannon entropy of the fractal spread of masses over sub-propositions	Zhou & Deng, arXiv:2012.00235
`information_volume()`	Limit of Deng entropy under iterative maximum-entropy splitting	Deng, IJCCC 15(6) (2020)
`nonspecificity()`	Generalized Hartley measure of imprecision	Klir & Wierman (1999)
`strife()`	Conflict-based part of total uncertainty	Klir & Wierman (1999)

All measures require m(empty) = 0; normalize a TBM-style result first.

Example

from evidencelib import Frame

frame = Frame.dst(["a", "b", "c"])
a, b, c = frame.symbols()
m = frame.mass({a: 0.5, b: 0.2, a | b | c: 0.3})

m.deng_entropy()          # 2.328...
m.nonspecificity()        # 0.475...
m.information_volume()    # 3.425... (>= Deng entropy)

DSm cardinality on DSmT frames

On free and hybrid DSm frames the measures replace the set cardinality |A| with the DSm cardinality: the number of Venn regions the proposition covers. On DST frames both cardinalities coincide, so the classical formulas are recovered.

free = Frame.dsmt(["p", "q"])
p, q = free.symbols()

free.mass({p & q: 1.0}).deng_entropy()   # 0.0  (single Venn region)
free.mass({p: 1.0}).deng_entropy()       # 1.585 (p covers two regions)

The k-order maximum of tfb_entropy on a DST frame with n hypotheses is the higher order information volume of a mass function (HOIVMF), log2((k+2)**n - (k+1)**n).

Notes

information_volume(epsilon=1e-3, max_iterations=1000) matches the convergence threshold used in the defining paper.
fractal_belief_entropy() enumerates the 2**c - 1 sub-propositions of each focal element; keep focal cardinalities moderate.