Iterative Approximation of Statistical Distributions and Relation to Information Geometry

The optimal control of stochastic processes through sensor estimation of probability density functions is given a geometric setting via information theory and the information metric. Information theory identifies the exponential distribution as the maximum entropy distribution if only the mean is known and the Γ distribution if also the mean logarithm is known. The surface representing Γ models has a natural Riemannian information metric. The exponential distributions form a one-dimensional subspace of the two-dimensional space of all Γ distributions, so we have an isometric embedding of the random model as a subspace of the Γ models. This geometry provides an appropriate structure on which to represent the dynamics of a process and algorithms to control it. This short paper presents a comparative study on the parameter estimation performance between the geodesic equation and the B-spline function approximations when they are used to optimize the parameters of the Γ family distributions. In this case, the B-spline functions are first used to approximate the Γ probability density function on a fixed length interval; then the coefficients of the approximation are related, through mean and variance calculations, to the two parameters (i.e. μ and β) in Γ distributions. A gradient based parameter tuning method has been used to produce the trajectories for (μ, β) when B-spline functions are used, and desired results have been obtained which are comparable to the trajectories obtained from the geodesic equation. This revised version was published online in August 2006 with corrections to the Cover Date.