Invariant Fisher information (Q1330589)

Let \(t = t(p,\omega): M \times \Omega \to \mathbb{R}^ k\) be a family of \(\mathbb{R}^ k\)-valued random variables parametrized by an \(n\)-dimensional manifold \(M\), where \(x = (x^ 1,\dots,x^ n)\) is a chart around \(p \in M\), \(\Omega\) a probability space and \(\omega \in \Omega\). Assuming \(t\) possesses a density \(\rho(p,t)\), the Fisher information associated with \(t\) is defined to be \(g(p) = \mathbb{E}_ p[(\partial_ i l)(\partial_ j l)]dx^ i \otimes dx^ j\) where \(\mathbb{E}_ p\) is the expectation with respect to \(\rho(p,t),l(p,t) = \log \rho(p,t)\) and \(\partial_ i = {\partial\over \partial x^ i} \cdot g(p)\) is, by definition, invariant under a change of parameters \(x \mapsto x'\) 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 \(g_{\text{inv}}(p)\) which is invariant under a general change of both parameters and random variables. We can, in the end, express the difference \(g_{\text{inv}}(p) - g(p)\) in terms of two types of connections which are purely geometrical objects. If we further impose a certain ``linearly'' on our construction, we can express \(g_{\text{inv}}(p) - g(p)\) 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 \(g_{\text{inv}}(p) = g(p)\) holds.