Reparametrization-covariant theory for on-line learning of probability distributions

: We discuss the on-line learning of probability distributions in a reparametrization covariant formulation. Reparametrization covariance plays an essential role not only to respect an intrinsic property of "information" but also for pattern recognition problems. We can obtain an optimal on-line learning algorithm with reparametrization invariance, where the conformal gauge connects a covariant formulation with a noncovariant one in a natural way.