By Shun-ichi Amari

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60) which shows that F is a mixture family generated by the delta distributions δ(s − x), s ∈ R. Here, p(s) are mixing coefficients. 62) and ψ is a functional of θ(s) formally given by ψ[θ(s)] = log exp {θ(s)} ds . 63) Hence, F is an exponential family where θ(s) = log p(s) + ψ is the θ affine coordinates and η(s) = p(s) is the dual affine coordinates η. The dual convex function is ϕ [η(s)] = η(s) log η(s)ds. 66) and we have where ∇ is the Fréchet-derivative with respect to function θ(s). 58). 68) in the m-coordinates.

40 This gives a mapping from F to Sn , which approximates p(x) by p ∈ Sn . When the discretization is done in such a way that pi in each interval converges to 0 as n tends to infinity, the approximation looks fine. Then, the geometry of F would be defined by the limit of Sn consisting of discretized p. However, we have difficulty in this approach. The limit n → ∞ of the geometry of Sn might not be unique, depending on the method of discretization. Moreover, an admissible discretization would be different for different p(x).

84) θ ∗ (t) = at + b. This is the dual affine structure derived from the convex function ψ ∗ θ ∗ . Since the coordinate transformation between the two affine coordinate systems θ and θ ∗ is not linear in general, a geodesic is not a dual geodesic and vice versa. This implies that we have introduced two different criteria of straightness or flatness in M, namely primal and dual flatness. M is dually flat and the two flat coordinates are connected by the Legendre transformation. 2 Tangent Space, Basis Vectors and Riemannian Metric When dθ is an (infinitesimally) small line element, the square of its length ds is given by gi j dθi dθ j .