Return to Answer

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature stand point. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028

Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.

http://www.amazon.com/Information-Translations-Mathematical-Monographs-Tanslations/dp/0821843028