--- title: Generalized Extropy Divergence Ratio (GEDR) url: https://www.emergentmind.com/topics/generalized-extropy-divergence-ratio-gedr type: topic --- # Generalized Extropy Divergence Ratio (GEDR) Searching arXiv for papers on extropy, GEDR, and related divergence ratios. arXiv search query: "Generalized Extropy Divergence Ratio extropy divergence ratio" Generalized Extropy Divergence Ratio (GEDR) is a class of extropy-based relative information measures for comparing two probability distributions through normalized cross-extropy. In the formulation introduced in "Extropy-Based Generalized Divergence and Similarity Ratios" [2508.13696], GEDR quantifies directional discrepancy by dividing generalized extropy inaccuracy by generalized extropy itself, yielding a dimensionless ratio defined for density, distribution, and survival representations. Its construction belongs to the broader extropy program initiated by extropy as the complementary dual of entropy [1109.6440], but it differs from earlier extropy functionals in a crucial respect: it is explicitly comparative, directional, and normalized, whereas several antecedent measures in the literature were either single-distribution uncertainty indices or unnormalized divergence functionals. ## 1. Extropy foundations and the emergence of ratio-based comparison Extropy was introduced as the complementary dual of entropy. For a finite discrete distribution \(p_N=(p_1,\dots,p_N)\), entropy is \[ H(p_N)=-\sum_{i=1}^N p_i\log p_i, \] whereas extropy is \[ J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i). \] These coincide in the binary case, but for \(N\ge 3\) they bifurcate into distinct measures; extropy is permutation-invariant, maximized at the uniform distribution, and its maximum remains bounded as \(N\to\infty\) [1109.6440]. In the continuous setting, the density analogue of extropy is \[ j(f)=-\frac{1}{2}\int f^2(x)\,dx, \] which places extropy in an \(L_2\)-type geometry rather than the logarithmic geometry of entropy [1109.6440]. A central structural observation in the later GEDR framework is that many divergences can be written as the difference between an inaccuracy term and an average uncertainty term. Extropy naturally supplies such terms through inner products of probabilistic representations. Motivated by normalization for interpretability, in a manner analogous to correlation relative to covariance, GEDR was proposed as a ratio version of extropy-based comparison [2508.13696]. The distinction from earlier extropy generalizations is important. "On Tsallis extropy with an application to pattern recognition" introduces Tsallis extropy, \[ JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right), \] as a dual uncertainty measure to Tsallis entropy for a single discrete random variable, with limits to Shannon extropy and several order and boundedness properties, but it does not define a two-distribution divergence or any ratio-based construct named GEDR [2103.07168]. This suggests that GEDR should be understood not as a direct consequence of Tsallis extropy, but as a later normalization program built on extropy-based inaccuracy. ## 2. Formal definition and representational scope GEDR is defined on general probabilistic representations \(\phi_1(x)\) and \(\phi_2(x)\), where \(\phi\) may denote a PDF \(f\), a CDF \(F\), or a survival function \(\bar F\), for nonnegative, absolutely continuous random variables on \([0,\infty)\). The generalized extropy and generalized extropy inaccuracy are \[ U(\phi_1(X))=-\frac{1}{2}\int_0^\infty \phi_1^2(x)\,dx,\qquad U(\phi_1(X),\phi_2(Y))=-\frac{1}{2}\int_0^\infty \phi_1(x)\phi_2(x)\,dx. \] The Generalized Extropy Divergence Ratio of \(X\) with \(Y\) is then \[ I(\phi_1(X)\mid \phi_2(Y))=\frac{U(\phi_1(X),\phi_2(Y))}{U(\phi_1(X))}, \] with reverse ratio \[ I(\phi_2(Y)\mid \phi_1(X))=\frac{U(\phi_1(X),\phi_2(Y))}{U(\phi_2(Y))}. \] Its symmetric companion, the Generalized Extropy Similarity Ratio (GESR), is \[ S(\phi_1(X),\phi_2(Y)) =\frac{U(\phi_1(X),\phi_2(Y))^2}{U(\phi_1(X))\,U(\phi_2(Y))} =I(\phi_1(X)\mid \phi_2(Y))\,I(\phi_2(Y)\mid \phi_1(X)). \] These are the core definitions of the framework [2508.13696]. For discrete distributions with pmfs \(p=(p_i)\) and \(q=(q_i)\) on a common support, the same construction becomes \[ U(p)=-\frac{1}{2}\sum_i p_i^2,\qquad U(p,q)=-\frac{1}{2}\sum_i p_iq_i, \] and \[ I(p\mid q)=\frac{U(p,q)}{U(p)},\qquad I(q\mid p)=\frac{U(p,q)}{U(q)},\qquad S(p,q)=\frac{U(p,q)^2}{U(p)\,U(q)}. \] An explicit example given in the literature uses \(p=(0.6,0.4)\) and \(q=(0.5,0.5)\), yielding \(U(p)=-0.26\), \(U(q)=-0.25\), \(U(p,q)=-0.25\), hence \(I(p\mid q)\approx 0.961538\), \(I(q\mid p)=1\), and \(S(p,q)\approx 0.961538\) [2508.13696]. The three principal continuous specializations are as follows. | Representation | Self-term | GEDR notation | |---|---|---| | Density \(f\) | \(J(X)=-\frac12\int_0^\infty f^2(x)\,dx\) | \(I_E(X\mid Y)\) | | Survival \(\bar F\) | \(J_s(X)=-\frac12\int_0^\infty \bar F^2(x)\,dx\) | \(I_{SE}(X\mid Y)\) | | CDF \(F\) | \(\bar{\xi}J(X)=-\frac12\int_0^\infty F^2(x)\,dx\) | \(I_{CE}(X\mid Y)\) | In each case the cross-term is obtained by replacing the square with the product of the two representations, and the associated similarity ratio is the squared normalized cross-term [2508.13696]. The domain requires finite negative denominators \(U(\phi_1(X))\) and finite cross-terms. For proper PDFs, CDFs, and survival functions this is ensured under mild square-integrability assumptions; support mismatches contribute zero through the product \(\phi_1\phi_2\), and the ratio is undefined only when the denominator vanishes, which is excluded by assumption [2508.13696]. ## 3. Axioms, range, and geometric interpretation GEDR is asymmetric in general: \[ I(\phi_1\mid \phi_2)\neq I(\phi_2\mid \phi_1). \] It is strictly positive whenever the defining integrals are finite and the denominator is nonzero. It satisfies the identity condition \[ I(\phi_1\mid \phi_1)=1, \] and more generally \(I(\phi_1\mid \phi_2)=1\) when \(\phi_1\equiv \phi_2\). Its range is therefore \((0,\infty)\), not \([0,\infty)\) with zero at equality; equality of distributions corresponds to the neutral value \(1\), because GEDR is a normalized ratio rather than a difference-type divergence [2508.13696]. A useful order relation links the two directions: \[ I(\phi_1(X)\mid\phi_2(Y))>(<)\,1 \iff I(\phi_2(Y)\mid\phi_1(X))<(>)\,1. \] The ratio identity \[ \frac{I(\phi_1\mid\phi_2)}{I(\phi_2\mid\phi_1)} = \frac{U(\phi_2)}{U(\phi_1)} \] shows that directional imbalance is determined by the relative magnitudes of the two self-extropy terms [2508.13696]. GESR removes directionality and restores boundedness: \[ 00, \] and location invariance under common shift, \[ I(X+a\mid Y+a)=I(X\mid Y),\qquad S(X+a,Y+a)=S(X,Y), \] for nonnegative random variables under appropriate support behavior. Continuity under \(L_2\) perturbations is also stated [2508.13696]. ## 4. Structural bounds under stochastic models A major part of the GEDR theory concerns bounds under structured stochastic relationships. Under the proportional generalized extropy model, \[ U(\phi_2(Y))=c\,U(\phi_1(X)),\qquad c>0, \] the reverse directional ratio satisfies \[ I(\phi_2(Y)\mid \phi_1(X))=\frac{1}{c}\,I(\phi_1(X)\mid \phi_2(Y)). \] Consequently, \[ S(\phi_1,\phi_2)=\frac{1}{c}\,\big(I(\phi_1\mid\phi_2)\big)^2, \] together with \[ I(\phi_1\mid\phi_2)<\sqrt c,\qquad I(\phi_2\mid\phi_1)<\frac{1}{\sqrt c}. \] This gives a direct parametric link between asymmetry and similarity [2508.13696]. For the proportional hazards model (PHM), where \[ h_Y(x)=c\,h_X(x),\qquad c>0, \] equivalently \(\bar G(x)=\bar F(x)^c\), the density-based and survival-based specializations admit one-sided bounds. In the density case, if \(c>(<)1\), \[ I_E(X\mid Y)<(>)\,c^2\,I_E(Y\mid X), \qquad J(Y)>(<)\,c^2\,J(X), \] and for \(c>1\), \[ I_E(X\mid Y)1/c, \] with the inequalities reversed for \(c<1\). In the survival case, if \(c>(<)1\), \[ I_{SE}(X\mid Y)<(>)\,I_{SE}(Y\mid X), \qquad J_s(X)<(>)\,J_s(Y), \] and for \(c>1\), \[ I_{SE}(X\mid Y)<1,\qquad I_{SE}(Y\mid X)>1, \] again reversed for \(c<1\) [2508.13696]. For the proportional reversed hazards model (PRHM), where \[ \lambda_Y(x)=c\,\lambda_X(x),\qquad c>0, \] equivalently \(G(x)=F(x)^c\), the CDF-based specialization satisfies, for \(c>(<)1\), \[ I_{CE}(X\mid Y)>(<)\,I_{CE}(Y\mid X), \qquad \bar{\xi}J(X)<(>)\,\bar{\xi}J(Y), \] and for \(c>1\), \[ I_{CE}(X\mid Y)>1,\qquad I_{CE}(Y\mid X)<1, \] with reversal for \(c<1\). The corresponding similarity ratios remain bounded by \(1\), with equality only at \(c=1\) [2508.13696]. These results clarify how GEDR reacts to systematic stochastic ordering. A plausible implication is that the sign of deviation from \(1\) can encode model-relative dominance: in PHM the survival-based direction with the higher hazard tends to produce a ratio on one side of \(1\), while the reverse direction lies on the opposite side. ## 5. Estimation, simulation, and applied use For density-based comparison, GEDR is estimated via kernel density estimators. With independent samples \(X_1,\dots,X_n\sim f\) and \(Y_1,\dots,Y_m\sim g\), Parzen estimators \[ \hat f(x)=\frac{1}{nb_n}\sum_{j=1}^n k\!\left(\frac{x-X_j}{b_n}\right),\qquad \hat g(x)=\frac{1}{mb_m}\sum_{j=1}^m k\!\left(\frac{x-Y_j}{b_m}\right) \] lead to \[ \widehat{S}_E = \frac{\big(\int_0^\infty \hat f(x)\hat g(x)\,dx\big)^2} {\big(\int_0^\infty \hat f^2(x)\,dx\big)\big(\int_0^\infty \hat g^2(x)\,dx\big)}, \] and hence \[ \widehat{I}_E(X\mid Y) = \frac{\int_0^\infty \hat f(x)\hat g(x)\,dx} {\int_0^\infty \hat f^2(x)\,dx}. \] For survival and cumulative extropy versions, empirical survival functions and empirical CDFs are computed on a pooled grid \(z_k\), and the required integrals are approximated by Riemann sums. The paper recommends choosing the probabilistic representation according to the application, constructing a common grid, using bandwidths \(b\to 0\) with \(nb\to\infty\), applying boundary correction on \([0,\infty)\), and ensuring nonzero denominators and numerical stability [2508.13696]. Simulation studies are reported for the similarity ratios. For \(S_E\) comparing \(\mathrm{Beta}(3,2)\) and \(\mathrm{Beta}(2,3)\), with true value \(0.5625\), the bias decreases from \(0.0448\) at \(n=50\) to \(0.0049\) at \(n=200\), and the MSE decreases from \(0.00201\) to \(2.4\times 10^{-5}\). For \(S_{SE}\) comparing \(\mathrm{Exp}(1)\) and \(\mathrm{Exp}(2)\), with true value \(0.8889\), the bias drops from \(0.0125\) to \(0.0002\), and the MSE from \(0.000156\) to \(2\times 10^{-7}\). For \(S_{CE}\) comparing \(U(0,1)\) and \(\mathrm{Beta}(3,2)\), with true value \(0.94501\), the bias also decreases with sample size [2508.13696]. These results are reported as confirmation of consistency. The applications emphasize bounded similarity, but they are directly relevant to GEDR because \(S=I(\phi_1\mid\phi_2)I(\phi_2\mid\phi_1)\). In lifetime data analysis of the FD\&C Red No. 40 mouse experiment, self-similarity is approximately \(1\) on the diagonal, and similarity with the control group declines as dose increases; for example, \[ S_{SE}(\text{Control},\text{Low})=0.963,\quad S_{SE}(\text{Control},\text{Medium})=0.942,\quad S_{SE}(\text{Control},\text{High})=0.906. \] In image analysis, uniform intensity scaling by \(c\in(0,1]\) preserves similarity: \[ S_{SE}(A,B)=S_{SE}(cA,cB), \] and similarly for \(S_{CE}\) and \(S_E\). A classification strategy based on a black reference image \(Z\) uses \(S_{SE}(Z,A)\) and \(S_{SE}(Z,B)\) as exposure-invariant signatures [2508.13696]. Since GEDR is the directional factorization of the same similarity, these applications also illustrate the operational context in which directional discrepancy can be extracted when symmetry is not desired. ## 6. Relation to neighboring extropy measures and recurrent confusions GEDR belongs to a family of extropy-based comparison measures, but the surrounding terminology is heterogeneous. A persistent source of confusion is the word “divergence.” In the Tsallis extropy literature, \(JS_\alpha(X)\) is repeatedly called a “measure of discrimination,” yet it is a single-distribution uncertainty measure rather than a divergence between two distributions; the paper does not define a \(P\)-versus-\(Q\) construct and does not introduce any ratio named GEDR [2103.07168]. Likewise, "Further results on relative, divergence measures based on extropy and their applications" develops extropy, relative extropy, extropy inaccuracy, extropy divergence, and dynamic residual and past variants, but explicitly does not define or use the term GEDR [2503.07123]. A second distinction concerns unnormalized difference-type divergences. The foundational extropy work defines discrete relative extropy \[ D^c(p_N\|s_N)=\sum_{i=1}^N (1-p_i)\log\frac{1-p_i}{1-s_i}, \] and for densities identifies the extropic dual to KL divergence with half the \(L_2\) metric, \[ d^c(f\|g)=\frac{1}{2}\int (f-g)^2\,dx. \] These are directed or symmetric discrepancy functionals, but they are not normalized ratios [1109.6440]. Similarly, the survival-extropy divergence framework based on \[ D(F,G)=\int_0^\infty (\bar F(x)-\bar G(x))^2\,dx \] supports U-statistic, EDF, and kernel estimators, as well as jackknife empirical likelihood inference, yet the paper states that it does not use the term GEDR; its main object is a squared \(L_2\) survival-function divergence rather than a ratio [2507.15810]. A third distinction is with precursor ratio constructions. "Inaccuracy and divergence measures based on survival extropy" introduces the survival extropy inaccuracy ratio \[ I\xi(X,Y)=\frac{\xi J_s(X,Y)}{J_s(X)}, \] together with asymmetric and symmetric survival-extropy divergences and dynamic versions, but not the generalized name GEDR [2410.22747]. This suggests that the survival-based specialization of GEDR, \[ I_{SE}(X\mid Y)=\frac{\xi J_s(X,Y)}{J_s(X)}, \] can be viewed as a direct generalization and unification of such earlier ratio constructs, now placed within a common PDF/CDF/SF framework [2508.13696]. Within this landscape, GEDR is best characterized as a normalized, directional extropy comparison functional with three distinguishing features: it is ratio-based rather than difference-based; it accommodates density, survival, and cumulative representations in a single formalism; and it is paired with a symmetric cosine-square similarity ratio. Those features mark the point at which extropy-based comparison shifts from isolated divergences and inaccuracies to a consolidated theory of relative information ratios. Source: https://www.emergentmind.com/topics/generalized-extropy-divergence-ratio-gedr