Let t = t(p, omega) : M x OMEGA --> R(k) be a family of R(k)-valued random variables parametrized by a n-dimensional manifold M, where x = (x1,...,x(n)) is a chart around p is-an-element-of M, OMEGA a probability space and omega is-an-element-of OMEGA. Assuming t possesses a density p(p, t), the Fisher information associated with t is defined to be g(p) = = epsilon(p)[partial derivative(i)l)(partial derivative(j)l)]dx(i) X dx(j) where epsilon(p) is the expectation with respect to rho(p, t), l(p, t) = log rho(p, t) and partial derivative(i) = partial derivative/partial derivative x(i). g(p) is, by definition, invariant under a change of parameters x --> x' and also a change of random variables of the form t = t(t'). However, it may not be invariant under a general change of random variables t = t(p,t'). The aim of this paper is to construct information g(inv)(p) which is invariant under a general change of both parameters and random variables. We can, in the end, express the difference g(inv)(P) - g(P) in terms of two types of connections which are purely geometrical objects. If we further impose a certain ''linearity'' on our construction, we can express g(inv)(p) - g(p) in terms of a single linear connection on a vector bundle so that the vanishing of the curvature would insure the existence of a ''special'' t in which g(inv)(p) = g(p) holds.
