Parameter estimation and the space of random variables

The procedure of parameter estimation for nonlinear models, e.g. in biomedicine, is typically carried out by the minimization of a cost functional (e.g. Ordinary Least Squares, OLS) of the differences between predicted and experimentally observed values. As such, this process, of an exquisitely computational nature, is carried out in n-dimensional Euclidean space (case space). Theoretically, however, what is desired and what is indeed the philosophical essence of the modelling effort, is to approximate some hypothetical random variable expressing the possible outcomes of the process of interest with some other random variable, computable as a (very possibly nonlinear) function of more readily observable random variables. The aim of the present work is to describe the logical relationship between the Hilbert space of random variables and n-dimensional case space. Index Terms--statistical inference, nonlinear modeling, functional analysis, differential geometry, measure theory