Why Are There No Triple Affine Hecke Algebras?

In differential geometry, what are "dual affine connections"?

• What makes one affine connection "dual" to another?  In what sense is dual being used here? About the E&M algorithm: "In information geometry, the E step and the M step are interpreted as projections under dual affine connections" What does this mean?

• Answer:

  Let M be a manifold.  [math] \nabla [/math] and [math] \nabla^\ast [/math] are dual with respect to the metric [math] g = \langle , \rangle [/math] if [math] Z \langle X,Y \rangle = \langle \nabla_Z X, Y \rangle + \langle X, \nabla_Z^\ast Y \rangle [/math] for all vector fields X, Y, Z in the tangent space to M.  In Information Geometry, g is the Fisher information metric.  It is a Riemannian metric on a smooth statistical manifold (where each point denotes a probability distribution).  Chentsov's theorem also lends weight to the idea that it's a very natural metric to consider (if you want some invariance with respect to sufficient statistics).