[Paper Review] On Mean Divergence Measures

Inder J. Taneja|ArXiv.org|Jan 19, 2005

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TL;DR

This paper introduces new divergence measures based on differences between classical means—arithmetic, geometric, harmonic, and square-root means—using Csiszár's f-divergence framework. It establishes a refined inequality chain among these measures, improving upon known bounds and linking them to established divergences like Hellinger and triangular discrimination.

ABSTRACT

Arithmetic, geometric and harmonic means are the three classical means famous in the literature. Another mean such as square-root mean is also known. In this paper, we have constructed divergence measures based on nonnegative differences among these means, and established an interesting inequality by use of properties of Csiszar's f-divergence. Connections of new mean divergences measures with classical divergence measures such as Jeffreys-Kullback-Leibler J-divergence, Sibson-Burbea-Rao Jensen difference divergence measure and Taneja's arithmetic - geometric mean divergence are also established.

Motivation & Objective

• To develop new divergence measures based on nonnegative differences between classical means (arithmetic, geometric, harmonic, square-root).
• To establish a comprehensive inequality chain relating these new mean divergence measures.
• To improve existing inequalities involving classical divergences such as J-divergence, Jensen-Shannon, and arithmetic-geometric mean divergence.
• To demonstrate that even non-convex mean divergence measures (e.g., M_GH) can be meaningfully bounded within the inequality framework.
• To unify and refine existing bounds using Csiszár’s f-divergence formalism and convexity properties.

Proposed method

• Define six new divergence measures as sums of pairwise mean differences over probability distributions: M_SA, M_SG, M_SH, M_AG, M_AH, M_GH.
• Express these divergences as special cases of Csiszár’s f-divergence by constructing appropriate convex functions f_i(x) for each measure.
• Use the convexity and normalization properties of f-divergence to ensure nonnegativity and convexity of the derived measures.
• Derive and prove a refined inequality chain: D_f8 ≤ (1/3)D_f1 ≤ (1/4)D_f3 ≤ (1/3)D_f2 ≤ D_f6, using algebraic manipulation of mean expressions.
• Simplify and re-express the inequality in terms of known divergences (e.g., h(P||Q) = Hellinger, Δ(P||Q) = triangular discrimination).
• Leverage the nonnegativity of the function f5(x) = (S + H - A - G)/6 to validate each step in the inequality chain.

Experimental results

Research questions

• RQ1 How can classical means (arithmetic, geometric, harmonic, square-root) be used to define new divergence measures between probability distributions?
• RQ2 What is the relationship between these new mean divergence measures and established divergences like Hellinger and triangular discrimination?
• RQ3 Can a tighter inequality chain be established among these mean-based divergences than previously known?
• RQ4 To what extent can non-convex mean divergence measures (e.g., M_GH) be bounded within a convex inequality framework?
• RQ5 How do the derived inequalities improve upon existing results in the literature, such as those by Dragomir et al. and Taneja (2004)?

Key findings

• A new inequality chain is established: D_f8(P||Q) ≤ (1/3)D_f1(P||Q) ≤ (1/4)D_f3(P||Q) ≤ (1/3)D_f2(P||Q) ≤ D_f6(P||Q), where each D_f corresponds to a mean divergence via Csiszár’s f-divergence.
• The inequality improves upon earlier bounds, such as h(P||Q) ≤ (1/2)Δ(P||Q), by introducing tighter intermediate terms.
• The measure M_GH(P||Q) is not convex, yet it is bounded below by M_SA(P||Q) and above by (1/3)M_SH(P||Q), showing robustness of the framework.
• The chain is re-expressed in terms of known divergences: M_GH ≤ M_SA ≤ (1/3)M_SH ≤ (1/4)Δ ≤ (3Δ + 2M_SG)/16 ≤ (h + 3M_SA)/4 ≤ (h + M_SH)/4 ≤ (6M_SG + Δ)/4 ≤ (1/2)M_SG ≤ h ≤ (1/2)Δ.
• The proof relies on the nonnegativity of f5(x) = (S + H - A - G)/6, which validates each step in the inequality chain.
• The results improve upon prior work, including Taneja (2004) and Dragomir et al. (2004), by refining bounds on J-divergence and related measures.