In the signal detection paradigm, the non-parametric index of sensitivity A′, as first introduced by Pollack and Norman (1964), is a popular alternative to the more traditional d′ measure of sensitivity. Smith (1995) clarified a confusion about the interpretation of A′ in relation to the area beneath proper receiver operating characteristic (ROC) curves, and provided a formula (which he called A′′) for this commonly held interpretation. However, he made an error in his calculations. Here, we rectify this error by providing the correct formula (which we call A) and compare the discrepancy that would have resulted. The corresponding measure for bias b is also provided. Since all such calculations apply to “proper” ROC curves with non-decreasing slopes, we also prove, as a separate result, the slope-monotonicity of ROC curves generated by likelihood-ratio criterion.

From a smooth, strictly convex function Φ: Rⁿ → R, a parametric family of divergence function D_Φ^(α) may be introduced: [ ] for x, y, ε int dom(Φ) and for α ε R, with D_Φ^(±1 defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, D(±1)_Φ reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate Φ * and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari & Nagaoka, 2000). This formulation based on convex analysis naturally extends the information-geometric interpretation of divergence functions (Eguchi, 1983) to allow the distinction between two different kinds of duality: referential duality (α -α) and representational duality (Φ  Φ *). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that ± α-connections defined on probability densities embody both referential and representational duality and are hence themselves bidual. When restricted to a finite-dimensional affine submanifold, the natural parameters of a certain representation of densities and the expectation parameters under its conjugate representation form biorthogonal coordinates. The alpha representation (indexed by β now, β ε [−1, 1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals D^{(α, β)}, (α, β) ε [−1, 1] × [-1, 1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when α =±1 or when β = 1, but to the family of Jensen difference (Rao, 1987) when β = 1.

Divergence functions are the non-symmetric “distance” on the manifold, Μ_θ, of parametric probability density functions over a measure space, (Χ,μ). Classical information geometry prescribes, on Μ_θ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μ_θ x Μ_θ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μ_θ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μ_θ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).