Shun'ichi Amari (1985) Differential-geometrical methods in statistics, Lecture notes in statistics, Springer-Verlag, Berlin.
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Again, the variable is understood as a coordinate on the Riemann manifold.
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In particular, one can introduce Hessian metrics like the Fisher information metric, the Weinhold metric, the Ruppeiner metric and others, whose components are calculated as the Hessian of a particular thermodynamic potential.