statistical-divergence

A statistical divergence is defined as a functional D[P||Q] mapping two probability distributions to a non-negative real number; remove probability distributions and the divergence has no inputs and ceases to operate — operational dependency per Law 8 removal test.

The Rényi divergence is a named family of statistical divergences — the umbrella's own accepted definition names it explicitly: 'the Kullback–Leibler, Jensen–Shannon, chi-squared, and Rényi families are all special cases or variants of the broader concept of statistical divergence.' Law 11e nearest kind: it is NOT an f-divergence (its (1/(α−1))·log Σ P^α Q^(1−α) form does not reduce to the f-divergence form f(t) = t·f′(1/(1−t)) family), so statistical-divergence — the accepted umbrella that names it — is the nearest existing kind. A statistician says 'Rényi divergence is a statistical divergence'. Specific → general.

f-divergence is a specific family of statistical divergences parameterized by a convex function f; every f-divergence is a statistical divergence, and a competent speaker would classify it as 'a type of statistical divergence'.

Tsallis divergence is a specific family of statistical divergences parameterized by order q; the accepted definition of statistical-divergence explicitly names it ('Tsallis divergence via q') as a distinct generating functional alongside f-divergence, Rényi divergence, and chi-squared divergence. A statistician would classify it as 'a type of statistical divergence.' Law 11e nearest kind: statistical-divergence is the closest existing kind. Specific → general.