Fisher information metric

In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements. In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements. The metric is interesting in several respects. By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. It can also be understood to be the infinitesimal form of the relative entropy (i.e., the Kullback–Leibler divergence); specifically, it is the Hessian of the divergence. Alternately, it can be understood as the metric induced by the flat space Euclidean metric, after appropriate changes of variable. When extended to complex projective Hilbert space, it becomes the Fubini–Study metric; when written in terms of mixed states, it is the quantum Bures metric. Considered purely as a matrix, it is known as the Fisher information matrix. Considered as a measurement technique, where it is used to estimate hidden parameters in terms of observed random variables, it is known as the observed information. Given a statistical manifold with coordinates θ = ( θ 1 , θ 2 , … , θ n ) {displaystyle heta =( heta _{1}, heta _{2},ldots , heta _{n})} , one writes p ( x , θ ) {displaystyle p(x, heta )} for the probability distribution as a function of θ {displaystyle heta } . Here x {displaystyle x} is drawn from the value space R for a (discrete or continuous) random variable X. The probability is normalized by ∫ X p ( x , θ ) d x = 1 {displaystyle int _{X}p(x, heta ),dx=1}