1986/09/01 by L. L. Campbell · 7 citations
Physics and Astronomy · Computer Science · #Statistical Mechanics and Entropy #Topological and Geometric Data Analysis #Bayesian Modeling and Causal Inference
paper · pdf · doi:10.1090/s0002-9939-1986-0848890-5
Čencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In Čencov’s theorem, the underlying differentiable manifold is the probability simplex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 1 Superscript n Baseline x Subscript i Baseline equals 1 comma x Subscript i Baseline greater-than 0"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">Σ 1nxi = 1, xi > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For some purposes of using geometry to obtain insights about probability, it is more convenient to regard the simplex as a hypersurface in the positive cone. In the present paper Čencov’s result is extended to the positive cone. The proof uses standard techniques of differential geometry but does not use the language of category theory.